Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆

e obtain a graded character formula for certain graded modules for the current algebra over a simple Lie algebra of type E₆. For certain values of their highest weight, these modules were conjectured to be isomorphic to the classical limit of the corresponding minimal affinizations of the associated...

Повний опис

Збережено в:

Бібліографічні деталі
Дата:	2011
Автори:	Moura, A., Pereira, F.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2011
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/154775
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆ / A. Moura, F. Pereira // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 69–115. — Бібліогр.: 24 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-154775
record_format	dspace
spelling	irk-123456789-1547752019-09-01T11:38:37Z Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆ Moura, A. Pereira, F. e obtain a graded character formula for certain graded modules for the current algebra over a simple Lie algebra of type E₆. For certain values of their highest weight, these modules were conjectured to be isomorphic to the classical limit of the corresponding minimal affinizations of the associated quantum group. We prove that this is the case under further restrictions on the highest weight. Under another set of conditions on the highest weight, Chari and Greenstein have recently proved that they are projective objects of a full subcategory of the category of graded modules for the current algebra. Our formula applies to all of these projective modules. 2011 Article Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆ / A. Moura, F. Pereira // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 69–115. — Бібліогр.: 24 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:17B10, 17B70, 20G42. http://dspace.nbuv.gov.ua/handle/123456789/154775 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	e obtain a graded character formula for certain graded modules for the current algebra over a simple Lie algebra of type E₆. For certain values of their highest weight, these modules were conjectured to be isomorphic to the classical limit of the corresponding minimal affinizations of the associated quantum group. We prove that this is the case under further restrictions on the highest weight. Under another set of conditions on the highest weight, Chari and Greenstein have recently proved that they are projective objects of a full subcategory of the category of graded modules for the current algebra. Our formula applies to all of these projective modules.
format	Article
author	Moura, A. Pereira, F.
spellingShingle	Moura, A. Pereira, F. Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆ Algebra and Discrete Mathematics
author_facet	Moura, A. Pereira, F.
author_sort	Moura, A.
title	Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆
title_short	Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆
title_full	Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆
title_fullStr	Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆
title_full_unstemmed	Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆
title_sort	graded limits of minimal affinizations and beyond: the multiplicity free case for type e₆
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/154775
citation_txt	Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆ / A. Moura, F. Pereira // Algebra and Discrete Mathematics. — 2011. — Vol. 12, № 1. — С. 69–115. — Бібліогр.: 24 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT mouraa gradedlimitsofminimalaffinizationsandbeyondthemultiplicityfreecasefortypee6 AT pereiraf gradedlimitsofminimalaffinizationsandbeyondthemultiplicityfreecasefortypee6
first_indexed	2025-07-14T06:53:04Z
last_indexed	2025-07-14T06:53:04Z
_version_	1837604258410659840
fulltext	Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 12 (2011). Number 1. pp. 69 – 115 c© Journal “Algebra and Discrete Mathematics” Graded limits of minimal affinizations and beyond: the multiplicity free case for type E6 Adriano Moura and Fernanda Pereira Communicated by V. M. Futorny Abstract. We obtain a graded character formula for certain graded modules for the current algebra over a simple Lie algebra of type E6. For certain values of their highest weight, these modules were conjectured to be isomorphic to the classical limit of the corresponding minimal affinizations of the associated quantum group. We prove that this is the case under further restrictions on the highest weight. Under another set of conditions on the highest weight, Chari and Greenstein have recently proved that they are projective objects of a full subcategory of the category of graded modules for the current algebra. Our formula applies to all of these projective modules. Introduction The problem of determining the structure of the minimal affinizations of quantum groups is one of the most studied problems in the finite- dimensional representation theory of quantum affine algebras in recent years (see [6] for a recent survey with a comprehensive list of references). In particular, determining the character of such representations when regarded as modules for the quantum group Uq(g) over the underlying semisimple Lie algebra g is of special interest. Determining the character is theoretically equivalent to determining the multiplicity of the irreducible constituents of these representations when regarded as Uq(g)-modules. 2000 Mathematics Subject Classification: 17B10, 17B70, 20G42. Key words and phrases: minimal affinizations of quantum groups, character formulae, affine Kac-Moody algebras. 70 Graded limits of minimal affinizations and beyond In practice, computing the multiplicities out of a given character is a laborious task which can be performed algorithmically. One of the methods which have been used to approach this problem is that of considering the classical limit of the given module and regard it as a representation for the current algebra g[t] = g⊗C[t]. This approach was first considered in [2, 7] and it was then further developed in [9, 10, 22]. In this paper, we apply this method for g of type E6 and obtain a formula for the multiplicities of the irreducible constituents of the graded pieces of these modules assuming certain conditions on the highest weight. Our formula actually holds for a larger class of g[t]-modules. Namely, given a dominant integral weight λ of g, the first author defined in [22] a g[t]- module denoted by M(λ). The definition is by generator and relations which naturally generalize the relations of the classical limits of Kirillov- Reshetikhin modules obtained in [2]. It was conjectured in [22] that M(λ) is isomorphic to the classical limit of the minimal affinizations of the irreducible Uq(g)-module of highest-weight λ provided that there exists a unique equivalence class of minimal affinizations associated to λ. Our main results are a formula for the multiplicities of the irreducible constituents of the graded pieces of the modules M(λ) and the proof of the conjecture of [22] assuming certain conditions on λ. To explain these conditions, let us label the nodes of the Dynkin diagram of g as follows. ❡ 1 ❡ 2 ❡ 3 ❡ 4 ❡ 5 ❡ 6 Let I = {1, 2, . . . , 6} and identify it with the set of nodes of the Dynkin diagram of g following the above labeling. For an integral weight µ, the support of µ is the subset of I consisting of labels such that the value of µ on the corresponding co-root is nonzero. The connected closure of the support is the minimal connected subdiagram of the Dynkin diagram of g containing the nodes in the support of µ. We mostly focus our study on the modules M(λ) with λ not supported in the trivalent node and prove that the character formula (3.12) below holds for all λ with support con- tained in one of the following subsets of I: {1, 2, 5, 6}, {1, 4, 5, 6}, {2, 4, 6}. Following the conjecture of [22], we conjecture that (3.12) holds for all λ not supported in the trivalent node and prove in such generality that (3.12) gives an upper bound for the multiplicities of the g-irreducible constituents of the graded pieces of M(λ) (see (3.10)). In particular, it follows from (3.9) that all irreducible constituents are multiplicity free (even if the grading is not taken into account). As a byproduct of the proof of (3.12), we obtain a realization of M(λ) as a submodule of the tensor product A. Moura, F. Pereira 71 of the classical limits of certain Kirillov-Reshetikhin modules (Theorem 3.14(a)), thus establishing part of the conjecture of [22] for such λ. Keeping the above conditions on λ and further assuming that the connected closure of the support of λ is of type A, we prove that M(λ) is isomorphic to the classical limit of the corresponding minimal affinizations when regarded as g[t]-modules (Theorem 3.14(b)). This establishes the other part of the conjecture of [22] for these values of λ. In particular, (3.12) gives the multiplicities of the irreducible constituents of the minimal affinizations when the support of λ is contained in one of the following subsets of I: {1, 2, 5}, {1, 4, 5}, {1, 2, 6}, {4, 5, 6}, {2, 4}. Moreover, we also prove that, if (3.12) indeed holds for any λ not supported in the trivalent node as conjectured, then we can include {1, 2, 4, 5} in this list. Dropping all the assumptions on λ except that the connected closure of its support is of type A, we prove that the classical limit of the corresponding minimal affinizations are quotients of M(λ) (Proposition 3.15). This is a further step towards the proof of the conjecture of [22] in general. However, the graded character formula for the Kirillov-Reshetikhin modules associated to the trivalent node given in [16] implies that, if λ is supported on that node, then these modules are not multiplicity free. We remark that, in [23], Nakajima developed an algorithm for computing the t-analogue of the q-character of any finite-dimensional irreducible representation of the quantum affine algebra associated to any simply laced simple Lie algebra g. In particular, without any assumption on λ, the graded character of the classical limits of the minimal affinizations associated to λ can be computed using this algorithm. Theoretically, one can then compute the multiplicities from the character as mentioned in the first paragraph of this introduction. On the other hand, with the above assumptions on λ, formula (3.12) gives these multiplicities directly. Let us explain the reasons behind the several aforementioned restric- tions on λ. First we recall that, for simply laced g, there exists a unique equivalence class of minimal affinizations associated to λ if and only if the connected closure of its support is of type A. Let θ be the highest root of g and, given i ∈ I, let ǫi(θ) be its coordinate in the basis of simple roots. Given a positive integer r, let g[t : r] be the quotient of g[t] by the ideal g ⊗ trC[t]. It turns out that M(λ) factors to a module for g[t : r] where r is the maximum of ǫi(θ) for i running on the support of λ. In particular, if g is of type E6, M(λ) can be regarded as a module for g[t : 3]. Moreover, if λ is not supported on the trivalent node, then M(λ) factors to a module for g[t : 2]. The category G2 of graded g[t : 2]-modules with finite-dimensional graded pieces has been recently studied in [4, 5] by exploring its interplay with the theory of Koszul algebras and quiver representations. The literature on the representation theory of g[t : r] 72 Graded limits of minimal affinizations and beyond for r > 2 is more limited and results such as the ones from [4, 5] are yet to be established. Thus, we focus on the case that M(λ) factors to a g[t : 2]-module which, for type E6, is equivalent to assuming that λ is not supported on the trivalent node (as mentioned above, this is also the necessary and sufficient condition for the modules M(λ) to be multiplicity free). It follows from [5, Theorem 1] that, if λ satisfies certain conditions, then M(λ) is a projective object of a full subcategory of G2 naturally attached to λ. Moreover, [5, Theorem 2] gives a graded character formula for M(λ) provided λ satisfies the conditions of [5, Theorem 1]. We remark that [5, Theorem 2] expresses the graded character of M(λ) in terms of an alternating sum of the graded characters of M(µ) with µ strictly smaller than λ with respect to the usual partial order on the weight lattice of g. Hence, the formula of [5, Theorem 2] is of recursive nature. For g of type E6, we prove that the conditions on λ required on [5, Theorem 1] is equivalent to requiring that the support of λ be contained in one of the following subsets of I: {1, 2, 5, 6}, {1, 4, 5, 6}. Therefore, (3.12) holds beyond the cases covered by [5, Theorem 2]. This latter list of subsets of I also hints that it should be expected that when the support of λ contains {2, 4} the situation should be more complicated than otherwise. Indeed, the proof of (3.12) for this case is significantly more technically involved than for the others. The paper is organized as follows. In Section 1, we review the basic notation on simply laced simple Lie algebras and the associated loop algebras, current algebras, quantum groups, and quantum affine algebras. In Section 2, we review the relevant facts on the finite-dimensional rep- resentation theory of these algebras. After reviewing the classification of minimal affinizations in Subsection 3.1, the main results (Theorem 3.14, Proposition 3.15, the multiplicity free property (3.10), and the character formula (3.12)) are stated in Subsection 3.2. The relation of our results with those of [5] is explained in Subsection 3.3. The proofs are given in Section 4. Acknowledgements: The work of the first author was partially supported by CNPq. The M.Sc. studies of the second author, during which part of this work was done, were supported by FAPESP. 1. Quantum and classical loop algebras Throughout the paper, let C,R,Z,Z≥m denote the sets of complex num- bers, reals, integers, and integers bigger or equal m, respectively. Given a ring A, the underlying multiplicative group of units is denoted by A×. A. Moura, F. Pereira 73 The dual of a vector space V is denoted by V ∗. The symbol ∼= means “isomorphic to”. The cardinality of a set S will be denoted by \|S\|. 1.1. Classical algebras Let I = {1, . . . , n} be the set of vertices of a finite-type simply laced Dynkin diagram and let g be the associated semisimple Lie algebra over C with a fixed Cartan subalgebra h. Fix a set of positive roots R+ and let n± = ⊕ α ∈ R+ g±α where g±α = {x ∈ g : [h, x] = ±α(h)x, ∀ h ∈ h}. The simple roots will be denoted by αi and the fundamental weights by ωi, i ∈ I. Q,P,Q+, P+ will denote the root and weight lattices with corresponding positive cones, respectively. Let also hi ∈ h, be the co-root associated to αi, i ∈ I. We equip h∗ with the partial order λ ≤ µ iff µ−λ ∈ Q+. Let C = (cij)i,j∈I be the Cartan matrix of g, i.e., cij = αj(hi). The Weyl group is denoted by W . The subalgebras g±α, α ∈ R+, are one-dimensional and [g±α, g±β] = g±α±β for every α, β ∈ R+. We denote by x±α any generator of g±α and, in case α = αi for some i ∈ I, we may also use the notation x±i in place of x±αi . In particular, if α+β ∈ R+, [x±α , x ± β ] is a nonzero generator of g±α±β and we simply write [x±α , x ± β ] = x±α+β. For each subset J of I let gJ be the Lie subalgebra of g generated by x±αj , j ∈ J , and define n±J , hJ in the obvious way. Let also QJ be the subgroup of Q generated by αj , j ∈ J , and R+ J = R+ ∩QJ . Given λ ∈ P , let λJ be the restriction of λ to h∗J and λJ ∈ P be such that λJ(hj) = λ(hj) if j ∈ J and λJ(hj) = 0 otherwise. By abuse of language, we will refer to any subset J of I as a subdiagram of the Dynkin diagram of g. The support of µ ∈ P is defined to be the subdiagram supp(µ) ⊆ I given by supp(µ) = {i ∈ I : µ(hi) 6= 0}. Let also supp(µ) be the minimal connected subdiagram of I containing supp(µ). If a is a Lie algebra overC, define its loop algebra to be ã = a⊗CC[t, t−1] with bracket given by [x ⊗ tr, y ⊗ ts] = [x, y] ⊗ tr+s. Clearly a ⊗ 1 is a subalgebra of ã isomorphic to a and, by abuse of notation, we will continue denoting its elements by x instead of x⊗ 1. We also consider the current algebra a[t] which is the subalgebra of ã given by a[t] = a ⊗ C[t]. Then g̃ = ñ− ⊕ h̃⊕ ñ+ and h̃ is an abelian subalgebra and similarly for g[t]. The elements x±α ⊗ tr, x±i ⊗ tr, and hi ⊗ tr will be denoted by x±α,r, x ± i,r, and hi,r, respectively. Also, Diagram subalgebras g̃J are defined in the obvious way. Let U(a) denote the universal enveloping algebra of a Lie algebra a. Then U(a) is a subalgebra of U(ã). Given a ∈ C, let τa be the Lie algebra automorphism of a[t] defined by τa(x⊗ f(t)) = x⊗ f(t− a) for 74 Graded limits of minimal affinizations and beyond every x ∈ a and every f(t) ∈ C[t]. If a 6= 0, let eva : ã → a be the evaluation map x ⊗ f(t) 7→ f(a)x. We also denote by τa and eva the induced maps U(a[t]) → U(a[t]) and U(ã) → U(a), respectively. Given a nonzero x ∈ a we shall denote by U(x) the universal enveloping algebra of the one-dimensional subalgebra generated by x regarded as a subalgebra of U(a). For each i ∈ I and r ∈ Z, define elements Λi,r ∈ U(h̃) by the following equality of formal power series in the variable u: ∞∑ r=0 Λi,±ru r = exp ( − ∞∑ s=1 hαi,±s s us ) . (1.1) 1.2. Quantum algebras Let C(q) be the ring of rational functions on an indeterminate q and A = C[q, q−1]. Set [m] = qm − q−m q − q−1 , [m]! = [m][m− 1] . . . [2][1], [mr ] = [m]! [r]![m− r]! , for r,m ∈ Z≥0, m ≥ r. Notice that [m], [mr ] ∈ A. The quantum loop algebra Uq(g̃) of g is the associative C(q)-algebra with generators x±i,r (i ∈ I, r ∈ Z), k±1 i (i ∈ I), hi,r (i ∈ I, r ∈ Z\{0}) and the following defining relations: kik −1 i = k−1 i ki = 1, kikj = kjki, kihj,r = hj,rki, kix ± j,rk −1 i = q±cijx±j,r, [hi,r, hj,s] = 0, [hi,r, x ± j,s] = ± 1 r [rcij ]x ± j,r+s, x±i,r+1x ± j,s − q±cijx±j,sx ± i,r+1 = q±cijx±i,rx ± j,s+1 − x±j,s+1x ± i,r, [x+i,r, x − j,s] = δi,j ψ+ i,r+s − ψ− i,r+s q − q−1 , ∑ σ∈Sm m∑ k=0 (−1)k[mk ]x±i,rσ(1) . . . x±i,rσ(k) x±j,sx ± i,rσ(k+1) . . . x±i,rσ(m) = 0, if i 6= j, for all sequences of integers r1, . . . , rm, where m = 1 − cij , Sm is the symmetric group on m letters, and the ψ± i,r are determined by equating powers of u in the formal power series Ψ± i (u) = ∞∑ r=0 ψ± i,±ru r = k±1 i exp ( ±(q − q−1) ∞∑ s=1 hi,±su s ) . A. Moura, F. Pereira 75 Denote by Uq(ñ ±), Uq(h̃) the subalgebras of Uq(g̃) generated by {x±i,r}, {k±1 i , hi,s}, respectively. Let Uq(g) be the subalgebra generated by x±i := x±i,0, k ±1 i , i ∈ I, and define Uq(n ±), Uq(h) in the obvious way. Uq(g) is a subalgebra of Uq(g̃) and multiplication establishes isomorphisms of C(q)-vectors spaces: Uq(g) ∼= Uq(n −)⊗Uq(h)⊗Uq(n +) and Uq(g̃) ∼= Uq(ñ −)⊗Uq(h̃)⊗Uq(ñ +). Let J ⊆ I and consider the subalgebra Uq(g̃J ) generated by k±1 j , hj,r, x ± j,s for all j ∈ J, r, s ∈ Z, r 6= 0. If J = {j}, the algebra Uq(g̃j) := Uq(g̃J) is isomorphic to Uq(s̃l2). Similarly we define the subalgebra Uq(gJ), etc. For i ∈ I, r ∈ Z, k ∈ Z≥0, define (x±i,r) (k) = (x± i,r) k [k]! . Define also elements Λi,r, i ∈ I, r ∈ Z by ∞∑ r=0 Λi,±ru r = exp ( − ∞∑ s=1 hi,±s [s] us ) . (1.2) Although we are denoting the elements x±i,r, hi,r, and Λi,r above by the same symbol as their classical counterparts, this will not create confusion as it will be clear from the context. Let UA(g̃) be the A-subalgebra of Uq(g̃) generated by the elements (x±i,r) (k), k±1 i for i ∈ I, r ∈ Z, and k ∈ Z≥0. Define UA(g) similarly and notice that UA(g) = UA(g̃)∩Uq(g). Henceforth a will denote a Lie algebra of the following set: g, n±, h, g̃, ñ±, h̃. For the proof of the next proposition see [2, Lemma 2.1] and the locally cited references. Proposition 1.1. The canonical map C(q) ⊗A UA(a) → Uq(a) is an isomorphism. Regard C as an A-module by letting q act as 1 and set Uq(a) = C⊗A UA(a). (1.3) Denote by x̄ the image of x ∈ UA(g̃) in Uq(g̃). For a proof of the next proposition see [11, Proposition 9.2.3] and the locally cited references. Proposition 1.2. U(g̃) is isomorphic to the quotient of Uq(g̃) by the ideal generated by ki − 1. In particular, the category of Uq(g̃)-modules on which ki act as the identity operator for all i ∈ I is equivalent to the category of all g̃-modules. The algebra Uq(g̃) is a Hopf algebra and induces a Hopf algebra structure (over A) on UA(g̃). Moreover, the induced Hopf algebra structure on U(g̃) coincides with the usual one (see [11, 21]). On Uq(g) we have ∆(x+i ) = x+i ⊗1+ki⊗x + i , ∆(x−i ) = x−i ⊗k−1 i +1⊗x−i , ∆(ki) = ki⊗ki (1.4) for all i ∈ I. The next lemma is easily established (cf. [22, Lemma 1.5]). 76 Graded limits of minimal affinizations and beyond Lemma 1.3. Suppose x = [x−i1 , [x − i2 , · · · [x−il−1 , x−il ] · · · ]]. Then x ∈ UA(n −) and ∆(x) ∈ x⊗ ( l∏ j=1 k−1 ij ) + 1⊗ x+ f(q)y for some y ∈ UA(g)⊗ UA(g) and some f(q) ∈ A such that f(1) = 0. An expression for the comultiplication ∆ of Uq(g̃) in terms of the generators x±i,r, hi,r, k ±1 i is not known. The following partial information will suffice for our purposes (see [22, Lemma 1.6] and the locally cited references). Lemma 1.4. ∆(x−i,1) = x−i,1⊗ki+1⊗x−i,1+x for some x ∈ UA(g)⊗UA(g) such that x̄ = 0. 1.3. The ℓ-weight lattice Given a field F consider the multiplicative group PF of n-tuples of rational functions µ = (µ1(u), · · · ,µn(u)) with values in F such that µi(0) = 1 for all i ∈ I. We shall often think of µi(u) as a formal power series in u with coefficients in F. Given a ∈ F× and i ∈ I, let ωi,a be defined by (ωi,a)j(u) = 1− δi,jau. Clearly, if F is algebraically closed,PF is the free abelian group generated by these elements which are called fundamental ℓ-weights. It is also convenient to introduce elements ωλ,a, λ ∈ P, a ∈ F, defined by ωλ,a = ∏ i∈I (ωi,a) λ(hi). (1.5) If F is algebraically closed, introduce the group homomorphism (weight map) wt : PF → P by setting wt(ωi,a) = ωi. Otherwise, let K be an algebraically closed extension of F so that PF can be regarded as a subgroup of PK and define the weight map on PF by restricting the one on PK. Define the ℓ-weight lattice of Uq(g̃) to be Pq := PC(q). The submonoid P+ q of Pq consisting of n-tuples of polynomials is called the set of dominant ℓ-weights of Uq(g̃). Given λ ∈ P+ q with λi(u) = ∏ j(1− ai,ju), where ai,j belongs to some algebraic closure of C(q), let λ− ∈ P+ q be defined by λ− i (u) = ∏ j(1− a−1 i,j u). We will also use the notation λ+ = λ. Given ν ∈ Pq, say ν = λµ−1 with λ,µ ∈ P+ q , define a C(q)-algebra homomorphism Ψν : Uq(h̃) → C(q) by setting Ψν(k ±1 i ) = q ±wt(ν)(hi) i and ∑ r≥0 Ψν(Λi,±r)u r = (λ±)i(u) (µ±)i(u) . (1.6) A. Moura, F. Pereira 77 One easily checks that the map Ψ : Pq → (Uq(h̃)) ∗ given by ν 7→ Ψν is injective. Define the ℓ-weight lattice P of g̃ to be the subgroup of Pq generated by ωi,a for all i ∈ I and all a ∈ C× or, equivalently, P = PC. Set also P+ = P ∩ P+ q . From now on we will identify Pq with its image in (Uq(h̃)) ∗ under Ψ. Similarly, P will be identified with a subset of U(h̃)∗ via the homomorphism Ψν : U(h̃) → C determined by (1.6) and Ψν(hi) = wt(ν)(hi). It will be convenient to introduce the following notation. Given i ∈ I, a ∈ C(q)×, r ∈ Z≥0, define ωi,a,r = r−1∏ j=0 ωi,aqr−1−2j . (1.7) If J ⊆ I and λ ∈ Pq, let λJ be the associated J-tuple of rational functions. Notice that λJ can be regarded as an element of the ℓ-weight lattice of Uq(g̃J). Let also λJ ∈ Pq be such that (λJ)j(u) = λj(u) for every j ∈ J and (λJ)j(u) = 1 otherwise. Given i ∈ I and a ∈ C(q)×, define the simple ℓ-root αi,a by αi,a = ωi,aq,2 ∏ j 6=i ω−1 j,aq,−cj,i . (1.8) The subgroup of Pq generated by the simple ℓ-roots is called the ℓ-root lattice of Uq(g̃) and will be denoted by Qq. Let also Q+ q be the submonoid generated by the simple ℓ-roots. Quite clearly wt(αi,a) = αi. Define a partial order on Pq by µ ≤ λ if λµ−1 ∈ Q+ q . Remark. The elements αi,a were first defined in [15] where they were denoted by Ai,aq. The term simple ℓ-root was introduced in [8] where an alternate definition in terms of an action of the braid group of g on Pq was given. For more details on the ℓ-weight lattice see [20, Section 3] and the references therein. 2. Finite-dimensional representations 2.1. Simple Lie algebras For the sake of fixing notation, we now review some basic facts about the representation theory of g and Uq(g). For the details see [19] and [11] for instance. 78 Graded limits of minimal affinizations and beyond Given a Uq(g)-module V and µ ∈ P , let Vµ = {v ∈ V : kiv = qµ(hi)v for all i ∈ I}. A nonzero vector v ∈ Vµ is called a weight vector of weight µ. If v is a weight vector such that x+i v = 0 for all i ∈ I, then v is called a highest- weight vector. If V is generated by a highest-weight vector of weight λ, then V is said to be a highest-weight module of highest weight λ. A Uq(g)-module V is said to be a weight module if V = ⊕ µ ∈ P Vµ. Denote by Cq be the category of all finite-dimensional weight modules of Uq(g). Analogous concepts for g-modules are defined similarly after setting Vµ = {v ∈ V : hv = µ(h)v for all h ∈ h}. Denote by C the category of finite-dimensional g-modules. Let Z[P ] be the integral group ring over P and denote by e : P → Z[P ], λ 7→ eλ, the inclusion of P in Z[P ] so that eλeµ = eλ+µ. The character of an object V from Cq or C is defined by char(V ) = ∑ µ∈P dim(Vµ)e µ. (2.1) The following theorem summarizes the basic facts about the categories Cq and C. Theorem 2.1. Let V be an object either of Cq or of C. Then: (a) dimVµ = dimVwµ for all w ∈ W. (b) V is completely reducible. (c) For each λ ∈ P+, the g-module V (λ) generated by a vector v satisfying x+i v = 0, hiv = λ(hi)v, (x−i ) λ(hi)+1v = 0, ∀ i ∈ I, is irreducible and finite-dimensional. If V ∈ C is irreducible, then V is isomorphic to V (λ) for some λ ∈ P+. (d) For each λ ∈ P+ the Uq(g)-module Vq(λ) generated by a vector v satisfying x+i v = 0, kiv = qλ(hi)v, (x−i ) λ(hi)+1v = 0, ∀ i ∈ I, is irreducible and finite-dimensional. If V ∈ Cq is irreducible, then V is isomorphic to Vq(λ) for some λ ∈ P+. A. Moura, F. Pereira 79 (e) For all λ ∈ P+, char(Vq(λ)) = char(V (λ)). If J ⊆ I we shall denote by Vq(λJ) the simple Uq(gJ)-module of highest weight λJ . Similarly V (λJ ) denotes the corresponding irreducible gJ -module. Proposition 2.2. Let λ ∈ P+, J ⊆ I, and suppose v ∈ Vq(λ)λ (re- spectively v ∈ V (λ)λ) is nonzero. Then Uq(gJ)v ∼= Vq(λJ) (respectively U(gJ)v ∼= V (λJ)). Assume g = g1 ⊕ g2 where gj are semisimple Lie algebras. Then P = P1 × P2 where Pj is the weight lattice of gj for j = 1, 2, and so on. Given λ ∈ P+ j , denote by Vj(λ) the irreducible gj-module of highest-weight λ. If V1 is a g1-module and V2 is a g2-module, then V1 ⊗ V2 is naturally a g-module. Proposition 2.3. Let λ = (λ1, λ2) ∈ P+ and µ = (µ1, µ2) ∈ P . Then: (a) V (λ) ∼= V1(λ1)⊗ V2(λ2) as g-modules. (b) V (λ)µ ∼= (V1(λ1)µ1)⊗ (V2(λ2)µ2) as h-modules. We will need the following elementary lemma (a proof can be found in [22, Lemma 2.3]). Lemma 2.4. Let V be a finite-dimensional g-module and suppose l ∈ Z≥1, νk ∈ P, vk ∈ Vνk , for k = 1, . . . , l, are such that V = ∑l k=1 U(n −)vk. Fix a decomposition V = m⊕ j = 1 Vj where m ∈ Z≥1, Vj ∼= V (µj) for some µj ∈ P+, and let πj : V → Vj be the associated projection for j = 1, . . . ,m. Then, there exist distinct k1, . . . , km ∈ {1, . . . , l} such that νkj = µj and πj(vkj ) 6= 0. 2.2. Loop algebras Let V be a Uq(g̃)-module. We say that a nonzero vector v ∈ V is an ℓ-weight vector if there exists λ ∈ Pq and k ∈ Z>0 such that (η −Ψλ(η)) kv = 0 for all η ∈ Uq(h̃). In that case, λ is said to be the ℓ-weight of v. V is said to be an ℓ-weight module if every vector of V is a linear combination of ℓ-weight vectors. In that case, let Vλ denote the subspace spanned by all ℓ-weight vectors of ℓ-weight λ. An ℓ-weight vector v is said to be a highest-ℓ-weight vector if ηv = Ψλ(η)v for every η ∈ Uq(h̃) and x+i,rv = 0 for all i ∈ I and all r ∈ Z. V is said to be a highest-ℓ-weight module if it is generated by a highest-ℓ-weight vector. Denote by C̃q the category 80 Graded limits of minimal affinizations and beyond of all finite-dimensional ℓ-weight modules of Uq(g̃). Quite clearly C̃q is an abelian category. Observe that if V ∈ C̃q, then V ∈ Cq and Vλ = ⊕ λ : wt(λ) = λ Vλ. (2.2) Moreover, if V is a highest-ℓ-weight module of highest ℓ-weight λ, then dim(Vwt(λ)) = 1 and Vµ 6= 0 ⇒ µ ≤ wt(λ). (2.3) Define the concepts of ℓ-weight vector, etc., for g̃ in a similar way and denote by C̃ the category of all finite-dimensional g̃-modules. The next proposition is easily established using (2.3). Proposition 2.5. If V is a highest-ℓ-weight module, then it has a unique proper submodule and, hence, a unique irreducible quotient. Definition 2.6. Let λ ∈ P+ q and λ = wt(λ). The Weyl module Wq(λ) of highest ℓ-weight λ is the Uq(g̃)-module defined by the quotient of Uq(g̃) by the left ideal generated by the elements x+i,r, (x − i,r) λ(hi)+1, and η −Ψλ(η) for every i ∈ I, r ∈ Z, and η ∈ Uq(h̃). Denote by Vq(λ) the irreducible quotient of Wq(λ). The Weyl module W (λ),λ ∈ P+, of g̃ is defined in a similar way. Its irreducible quotient will be denoted by V (λ). The next theorem was proved in [13]. Theorem 2.7. For every λ ∈ P+ q (resp. P+) the module Wq(λ) (resp. W (λ)) is the universal finite-dimensional Uq(g̃)-module (resp. g̃-module) with highest ℓ-weight λ. Every simple object of C̃q (resp. C̃) is highest-ℓ- weight. We shall need the following lemma which is a consequence of the proof of Theorem 2.7. Lemma 2.8. If V is a highest-ℓ-weight module of g̃ and v be a highest-ℓ- weight vector. Then V = U(g[t])v. If J ⊆ I we shall denote by Vq(λJ) the Uq(g̃J)-irreducible module of highest ℓ-weight λJ . Similarly V (λJ ) denotes the corresponding irreducible g̃J -module. Similar notations for the Weyl modules are defined in the obvious way. The next theorem was conjectured in [15] and proved in [14]. Theorem 2.9. Let V be a quotient of Wq(λ) for some λ ∈ P+ q . If Vµ 6= 0, then µ ≤ λ. A. Moura, F. Pereira 81 Given V in C̃q, let wtℓ(V ) = {µ ∈ Pq : Vµ 6= 0}. We will need the following proposition proved in [22, Section 4.8]. Proposition 2.10. Suppose g is of type A, λ ∈ P+, λ = ∏ i∈I ωi,ai,λ(hi), µ ∈ wtℓ(Vq(λ)), and λµ−1 = αj,bjαj+1,bj+1 · · ·αk,bk for some j ≤ k and some ai, bl ∈ C(q)×, i ∈ I, l = j, . . . , k. (a) If ai+1 ai = qλ(hi)+λ(hi+1)+1 for all i < n, then bk = akq λ(hk)−1. (b) If ai+1 ai = q−(λ(hi)+λ(hi+1)+1) for all i < n, then bj = ajq λ(hj)−1. 2.3. Classical limits Denote by P+ A the subset of Pq consisting of n-tuples of polynomials with coefficients in A. Let also P× A be the subset of P+ A consisting of n-tuples of polynomials whose leading terms are in CqZ\{0} = A×. Given λ ∈ P+ A , let λ be the element of P+ obtained from λ by evaluating q at 1. Recall that an A-lattice (or form) of a C(q)-vector space V is a free A-submodule L of V such that C(q)⊗A L = V . If V is a Uq(g̃)-module, a UA(g̃)-admissible lattice of V is an A-lattice of V which is also a UA(g̃)- submodule of V . Given a UA(g̃)-admissible lattice of a Uq(g̃)-module V , define L̄ = C⊗A L, (2.4) where C is regarded as an A-module by letting q act as 1. Then L̄ is a g̃-module by Proposition 1.2 and dim(L̄) = dim(V ). The next theorem is essentially a corollary of the proof of Theorem 2.7. Theorem 2.11. Let V be a nontrivial quotient ofWq(λ) for some λ ∈ P× A , v a highest-ℓ-weight vector of V , and L = UA(g̃)v. Then, L is a UA(g̃)- admissible lattice of V and char(L̄) = char(V ). In particular, L̄ is a quotient of W (λ). Definition 2.12. Let λ ∈ P× A , v be a highest-ℓ-weight vector of Vq(λ) and L = UA(g̃)v. We denote by Vq(λ) the g̃-module L̄. 3. Minimal affinizations and Beyond 3.1. Classification of minimal affinizations We now review the notion of minimal affinizations of an irreducible Uq(g)- module introduced in [1]. Given λ ∈ P+, a Uq(g̃)-module V is said to be an affinization of Vq(λ) if, as a Uq(g)-module, V ∼= Vq(λ)⊕ ⊕ µ < λ Vq(µ) ⊕mµ(V ) (3.1) 82 Graded limits of minimal affinizations and beyond for some mµ(V ) ∈ Z≥0. Two affinizations of Vq(λ) are said to be equivalent if they are isomorphic as Uq(g)-modules. If λ ∈ P+ q is such that wt(λ) = λ, then Vq(λ) is quite clearly an affinization of Vq(λ). The partial order on P+ induces a natural partial order on the set of (equivalence classes of) affinizations of Vq(λ). Namely, if V and W are affinizations of Vq(λ), say that V ≤W if one of the following conditions hold: (a) mµ(V ) ≤ mµ(W ) for all µ ∈ P+; (b) for all µ ∈ P+ such that mµ(V ) > mµ(W ) there exists ν > µ such that mν(V ) < mν(W ). A minimal element of this partial order is said to be a minimal affinization. Theorem 3.1 ([12]). Let λ ∈ P+ q , λ = wt(λ), and V = Vq(λ). Suppose g is of type A. Then V is a minimal affinization of Vq(λ) iff there exist a ∈ C(q)× and ǫ ∈ {1,−1} such that λ = n∏ i=1 ωi,ai,λ(hi) with a1 = a and ai+1 ai = qǫ(λ(hi)+λ(hi+1)+1) for all i ∈ I, i < n. If g is of type D or E, suppose the support of λ is contained in a connected subdiagram J ⊆ I of type A. Then, V is a minimal affinization of Vq(λ) iff Vq(λJ ) is a minimal affinization of Vq(λJ ). The next corollaries are easily established (recall from §1.1 that supp(λ) is the minimal connected subdiagram of I containing supp(λ)). Corollary 3.2. Suppose λ ∈ P+ is such that supp(λ) is of type A. Then, Vq(λ) has a unique equivalence class of minimal affinizations. Corollary 3.3. Given i ∈ I and m ∈ Z≥0, the modules Vq(ωi,a,m), a ∈ C(q)×, are the only minimal affinizations of Vq(mωi). The modules Vq(ωi,a,m) are known as Kirillov-Reshetikhin modules. We now state a few results which were used in the proof of Theorem 3.1 and will be useful for us as well. The proofs can be found in [12]. Lemma 3.4. Suppose ∅ 6= J ⊆ I is a connected subdiagram of the Dynkin diagram of g. Let V = Vq(λ), v a highest-ℓ-weight vector of V , and VJ = Uq(g̃J)v. Then, VJ ∼= Vq(λJ). Definition 3.5. Suppose g is of type A. A connected subdiagram J ⊆ I is said to be an admissible subdiagram. If g is of type D or E, let i0 ∈ I be the trivalent node. A connected subdiagram J ⊆ I is said to be admissible if J is of type A and J\{i0} is connected. A. Moura, F. Pereira 83 Proposition 3.6. Suppose J ⊆ I is admissible and that λ ∈ P+ q is such that Vq(λ) is a minimal affinization of Vq(λ) where λ = wt(λ). Then Vq(λJ) is a minimal affinization of Vq(λJ). The next proposition was proved in [22, Proposition 3.7]. Proposition 3.7. Let λ ∈ P+ q and λ = wt(λ). If Vq(λ) is a minimal affinization of Vq(λ), then there exist ai ∈ C(q)×, i ∈ I, such that λ =∏ i∈I ωi,ai,λ(hi) and ai aj ∈ qZ for all i, j ∈ I. Corollary 3.8. For every λ ∈ P+ there exist λ ∈ P× A such that Vq(λ) is a minimal affinization of Vq(λ). Moreover, λ = ωλ,a for some a ∈ C×. 3.2. Graded characters Recall the definition of the maps τa : g[t] → g[t] from subsection 1.1. Definition 3.9. Let λ ∈ P× A , λ = wt(λ), and a ∈ C× be such that λ = ωλ,a. The g[t]-module L(λ) is defined to be the pullback of Vq(λ) by τa. It is immediate from Theorem 2.11 that char(L(λ)) = char(Vq(λ)). (3.2) Let V be a Z≥0-graded vector space and denote its r-th graded piece by V [r]. A g[t]-module V is said to be Z≥0-graded if V is a Z≥0-graded vector space and x⊗ tsv ∈ V [r + s] for every v ∈ V [r], x ∈ g, r, s ∈ Z≥0. Observe that if V is a Z≥0-graded g[t]-module, then each graded peace is a g-module. Given s ∈ Z≥0, denote by V (s) the quotient of V by its g[t]-submodule ⊕ r > s V [r]. We shall refer to V (s) as the truncation of V at degree s. If V is a finite-dimensional Z≥0-graded g[t]-module, define the graded character of V by chart(V ) = ∑ r≥0 char(V [r]) tr ∈ Z[P ][t]. Let also mµ,r(V ) be the multiplicity of V (µ) as an irreducible constituent of V [r]. Definition 3.10. Let m ∈ Z≥0 and i ∈ I. The g[t]-module M(mωi) is the quotient of U(g[t]) by the left ideal generated by n+[t], h⊗ tC[t], hj , hi −m, x−αj , (x−αi )m+1, x−αi,1 for all j 6= i. (3.3) Define T (mωi) to be the g[t]-submodule of M(ωi) ⊗m generated by the top weight space. 84 Graded limits of minimal affinizations and beyond Quite clearly M(mωi) is a Z≥0-graded g[t]-module. Given λ ∈ P+ one can consider the modules A(λ) defined in [22]. These are graded g[t]- modules which were proved to be finite-dimensional in [22, Proposition 3.15]. One can proceed similarly to prove that the modules M(mωi) are finite-dimensional. Moreover, it was proved in [24, Proposition 5.2.5] that A(mωi) ∼=M(mωi) (for a general simple Lie algebra g). We shall not need the modules A(λ) in this paper. Given i ∈ I,m, r ∈ Z≥0, let vi,m be the image of 1 in M(mωi) and set R(i,m, r) = {α ∈ R+ : x−α,rvi,m = 0}. (3.4) Since (h⊗ tC[t])vi,m = 0, it follows that R(i,m, r) ⊆ R(i,m, s) for all s ≥ r. (3.5) In particular, it follows that M(0) is the trivial representation and R+(i, 0, s) = R+ for all i ∈ I, s ∈ Z≥0. Now, given λ ∈ P+ and r ∈ Z≥0, set R(λ, r) = ⋂ i∈I R(i, λ(hi), r). (3.6) Notice R(mωi, r) = R(i,m, r) for all i ∈ I and m, r ∈ Z≥0. Definition 3.11. Let λ ∈ P+. The g[t]-module M(λ) is the quotient of U(g[t]) by the left ideal generated by n+[t], h⊗ tC[t], hi − λ(hi), (x−αi )λ(hi)+1, x−α,r (3.7) for all i ∈ I, r ∈ Z≥0, and α ∈ R(λ, r). Define T (λ) to be the g[t]- submodule of ⊗ i ∈ I M(λ(hi)ωi) generated by the top weight space. Definitions 3.10 and 3.11 of M(mωi) coincide since R(mωi, r) = R(i,m, r) for all i ∈ I,m, r ∈ Z≥0. The modules M(λ) are clearly Z≥0- graded. It follows from [22, Proposition 3.13] that M(λ) is a quotient of the module A(λ) of [22] and, hence, finite-dimensional. Moreover, one easily sees that T (λ) is a graded quotient of M(λ) for all λ ∈ P+ (the details can be found in [24, Proposition 5.2.10]). Proposition 3.12 ([22, Proposition 3.21]). Let λ ∈ P× A be such that Vq(λ) is a minimal affinization of Vq(λ) where λ = wt(λ). Then, T (λ) is a quotient of L(λ). The following is the main conjecture of [22]. Conjecture 3.13. Let λ ∈ P+. Then,M(λ) ∼= T (λ). Moreover, if supp(λ) is of type A and λ ∈ P× A is such that Vq(λ) is a minimal affinization of Vq(λ), then, M(λ) ∼= L(λ). A. Moura, F. Pereira 85 For the rest of the subsection assume that g is of type E6 and that the nodes of the Dynkin diagram are labeled as in the introduction. We now state our main results. Theorem 3.14. Let λ ∈ P+ be such that λ(h3) = 0. Suppose that either {2, 4} * supp(λ) or supp(λ) ⊆ {2, 4, 6}. Then: (a) The first isomorphism in Conjecture 3.13 holds. (b) The second isomorphism in Conjecture 3.13 holds provided that supp(λ) is of type A. Notice that part (a) of Theorem 3.14 and Proposition 3.12 together with the following proposition which will be proved in Subsection 4.4 imply part (b) of Theorem 3.14. Proposition 3.15. Let λ ∈ P+ be such that is of type A. Then, L(λ) is a quotient of M(λ). As a byproduct of the proof of Theorem 3.14 we are able to compute chart(M(λ)) for λ as in the theorem. In particular, we compute char(Vq(λ)) for all λ ∈ P+ q such that wt(λ) satisfies the hypothesis of part (b) of the theorem. Let us now present these formulas and, along the way, explain the strategy of the proof of Theorem 3.14(a). Fix λ ∈ P+ and, given µ ∈ P and r ∈ Z≥0, set mµ,r = mµ,r(M(λ)) and tµ,r = mµ,r(T (λ)). We have already seen that tµ,r ≤ mµ,r. Therefore, in order to prove the first isomorphism of Conjecture 3.13, it suffices to show that mµ,r ≤ tµ,r for all µ ∈ P+, r ∈ Z≥0. (3.8) For r ∈ Z6, set wt(r) = λ− r1(ω2 − ω5)− r2(ω4 − ω1)− r3(ω2 − ω4 + ω5)− − r4(ω1 − ω2 + ω4)− r5(ω2 − ω3 + ω4)− r6ω6 and gr(r) = r1 + r2 + r3 + r4 + r5 + r6. Let also A = {r ∈ Z6 ≥0 : r6 ≤ m6, r3 ≤ m5, r4 ≤ m1, r1+r3+r5 ≤ m2, r2+r4+r5 ≤ m4}, Aµ = {r ∈ A : wt(r) = µ}, Ar = {r ∈ A : gr(r) = r}, and Aµ,r = Aµ∩Ar. 86 Graded limits of minimal affinizations and beyond The omission of the dependence of wt and A on λ in the notation will not create confusion. One easily checks that the function wt : Z6 → P is injective and, if r ∈ A, then wt(r) ∈ P+. In particular, \|Aµ\| ≤ 1 for all µ ∈ P+. (3.9) The basic idea for proving (3.8) is the same one used in [9, 10, 22]. Namely, in Subsection 4.5, we will use the defining relations of M(λ) to show that, if λ(h3) = 0, then mµ,r ≤ \|Aµ,r\|. (3.10) Moreover, for λ as in Theorem 3.14, by performing some explicit compu- tations in T (λ), we show in Subsection 4.7 that tµ,r ≥ \|Aµ,r\|. (3.11) Clearly (3.10) and (3.11) together imply (3.8). Moreover, chart(M(λ)) = ∑ r∈A char(V (wt(r)))tgr(r) (3.12) for all λ as in Theorem 3.14. In particular, for λ as in Theorem 3.14(b) and λ ∈ P+ q such that Vq(λ) is a minimal affinization of Vq(λ), we have char(Vq(λ)) = ∑ r∈A char(V (wt(r))). (3.13) Remark. Similar results in the case that g is of classical type or G2 were obtained in [9, 10, 22] (however, the definition of the modules T (mωi) requires some extra care in the non simply laced case). Equation (3.12) (and similar ones for general g) was predicted in [16] in the case that λ = mωi for some i ∈ I,m ∈ Z≥0. However, the meaning of the gradation in [16] is related to the quantum context, whereas here it appears by computing the classical limit. It is not clear to us why these two gradations coincide. The formulas in [16] were obtained by assuming the Kirillov- Reshetikhin conjecture whose proof was later completed in [17, 18]. Our results give an alternate proof of these formulas for g of type E6 and i 6= 3. As mentioned in the introduction, M(mω3) is not multiplicity free in general. Using the methods of this paper, we are able to prove that the isotypical components of M(mω3)[r] are exactly as given by [16]. However, so far we could only obtain an upper bound for mµ,r which is most often larger than the actual value of mµ,r. A. Moura, F. Pereira 87 We end this subsection by reviewing a construction used in [9, §2.6] which will be useful for us as well. Let Vr, 0 ≤ r ≤ k, be g-modules such that Homg(g⊗ Vr, Vr+1) 6= 0, Homg(∧ 2(g)⊗ Vr, Vr+2) = 0, 0 ≤ r ≤ k − 1, (3.14) where we assume that Vk+1 = 0. Fix non-zero elements pr ∈ Homg(g ⊗ Vr, Vr+1), 0 ≤ r ≤ k − 1, and set pk = 0. It is easily checked that the following formulas extend the canonical g-module structure to a graded g[t]-module structure on V = ⊕k r=1Vr: (x⊗ t)w = pr(x⊗ w), (x⊗ ts)w = 0, (3.15) for all x ∈ g, w ∈ Vr, 1 ≤ r ≤ k, s ≥ 2. Moreover, V [r] ∼= Vr for all 0 ≤ r ≤ k. Also, if V0 = U(g)w0 and the maps pr for r < k are all surjective, then V = U(n−[t])w0. 3.3. Projectivity If supp(λ) is not of type A, then Proposition 3.15 is probably false. In fact, most likely, M(λ) is then a proper quotient of L(λ). We now explain the motivation for studying the modules M(λ) beyond the cases associated to minimal affinizations from the perspective of [5]. We begin with following straightforward lemma which has been implicitly used in [5]. Lemma 3.16. Let r ∈ Z>0 and V be a g[t]-module generated by a vector v satisfying (g⊗ trC[t])v = 0. Then, (g⊗ trC[t])V = 0. Proof. Let x ∈ g, s ≥ r, and w = (x1 ⊗ tr1) · · · (x2 ⊗ trm)v for some m, rj ∈ Z≥0, xj ∈ g, j = 1, . . . ,m. We proceed by induction on m. If m = 0, we have (x⊗ts)w = 0 by hypothesis. Assume m > 0, let w′ = (x2⊗ tr2) · · · (xm⊗trm)v and assume, by induction hypothesis, that (y⊗ts)w′ = 0 for all y ∈ g, s ≥ r. Then, given x ∈ g and s ≥ r, we have (x⊗ ts)w = (x1 ⊗ tr1)(x⊗ ts)w′ + ([x, x1]⊗ ts+s1)w′. Both summands are zero by the induction hypothesis on m. The next proposition follows immediately from the above lemma and the definition of M(λ). Proposition 3.17. Let λ ∈ P+ and r > 0 be such that R(λ, r) = R+. Then, (g⊗ trC[t])M(λ) = 0. 88 Graded limits of minimal affinizations and beyond If V is a g[t]-module as in Lemma 3.16, then the canonical projection g[t] → g[t : r] := g[t]/g⊗ trC[t] induces a g[t : r]-module structure on V . Chari and Greenstein in [4, 5] initiated the study of the category G2 of graded g[t : 2]-modules with finite-dimensional graded pieces (they do not assume g is simply laced). Given a subset Γ of P+ × Z≥0, they consider the full subcategories G2(Γ) of G2 consisting of modules V such that V (µ) is an irreducible constituent of V [r] only if (µ, r) ∈ Γ. In particular, they consider subsets Γ of the following form. Given Ψ ⊆ R+ and λ ∈ P , set Γ(λ,Ψ) = {(µ, r) ∈ P+ ×Z≥0 : λ−µ = ∑ β∈Ψ nββ, nβ ∈ Z≥0, ∑ β∈Ψ nβ = r}. Notice that (λ, 0) ∈ Γ(λ,Ψ) for any choice of Ψ and that Γ(λ, ∅) = {(λ, 0)}. If we regard V (λ) as a module for g[t : 2] by pulling back the canonical projection g[t : 2] → g[t : 1] = g, then V (λ) is an object of G2(Γ(λ,Ψ)). The full strength of the results of [5] is realized when Ψ is either empty or of the form Ψν for some ν ∈ P where Ψν = {α ∈ R+ : (α, ν) = max{(β, ν) : β ∈ R+}} and (·, ·) is the bilinear form on P ×P induced from the Killing form of g. For λ ∈ P+ such that R(λ, 2) = R+, set Ψλ = R+\R(λ, 1). The following theorem is a particular case of [5, Theorem 1]. Theorem 3.18. Let λ ∈ P+ be such that R(λ, 2) = R+ and suppose that either Ψλ = ∅ or Ψλ = Ψν for some ν ∈ P . Then, M(λ) is the projective cover of V (λ) in the category G2(Γ(λ,Ψ λ)). For λ as in Theorem 3.18, [5, Theorem 2] gives a formula for computing the graded character of M(λ) by induction on the cardinality of the set Γ(λ,Ψλ). Let us return to the case that g is of type E6. It follows from the proof of Theorem 3.14 (see Lemma 4.11 below) that M(λ) is a module as in Lemma 3.16 with r = 3. Moreover, if λ(h3) = 0, then we can take r = 2. Lemma 3.19. Let λ ∈ P+ be such that λ(h3) = 0 and {2, 4} * supp(λ). Then, either Ψλ = ∅ or there exists ν ∈ P such that Ψλ = Ψν . Proof. Recalling that (αi, ν) = 1 2(αi, αi)ν(hi) and using the characteriza- tions of R(λ, 1) given by (4.7), one easily checks by inspection of Table 1 below that (a) supp(λ) ⊆ {1, 5} ⇒ Ψλ = ∅. (b) 6 ∈ supp(λ) ⊆ {1, 5, 6} ⇒ Ψλ = Ψω6 . A. Moura, F. Pereira 89 (c) 2 ∈ supp(λ) ⊆ {1, 2, 5, 6} ⇒ Ψλ = Ψω2 . (d) 4 ∈ supp(λ) ⊆ {1, 4, 5, 6} ⇒ Ψλ = Ψω4 . Clearly λ satisfies the hypothesis of the lemma iff it satisfies one of the conditions (a)-(d) above. This immediately implies the following corollary of Theorem 3.18. Corollary 3.20. Let λ be as in Lemma 3.19. Then,M(λ) is the projective cover of V (λ) in the category G2(Γ(λ,Ψ λ)). Similarly to the proof of Lemma 3.19, one easily checks that if {2, 4} ⊆ supp(λ), then Ψλ 6= ∅ and Ψλ 6= Ψν for all ν ∈ P . Therefore, λ satisfies the hypothesis of Theorem 3.18 iff it satisfies the hypothesis of Lemma 3.19. It follows that every λ as in Theorem 3.18 satisfies the hypothesis of Theorem 3.14. On the other hand, if λ satisfies the hypothesis of Theorem 3.14 but not the one of Theorem 3.18, then {2, 4} ⊆ supp(λ) ⊆ {2, 4, 6}. In this case, we cannot conclude that M(λ) is a projective object in some subcategory of G2 nor can we use [5, Theorem 2] to compute its graded character. Remark. It is worth remarking that we will perform most of the proof of (3.11) using only the hypothesis λ(h3) = 0. This provides some evidence that Conjecture 3.13 holds in complete generality. In particular, we con- jecture that (3.12) is the graded character of M(λ) for all λ ∈ P+ such that λ(h3) = 0. 4. Proofs 4.1. On characters for type A2 We now record some lemmas about the characters of certain finite- dimensional sl3-modules which will be needed in the proof of (3.11). To simplify some formulas, we introduce the notation of divided powers. If A is an associative algebra, x ∈ A, and r ∈ Z≥0, set x(r) = 1 r!x r. We will make use of the following result on representations of the 3-dimensional Heisenberg algebra which will also be used in the proof of (3.10). Thus, consider the three-dimensional Heisenberg Lie algebra H spanned by elements x, y, z where z is central and [x, y] = z. Part (a) of the following lemma is standard while a proof of part (b) can be found in [10, Lemma 1.5]. Lemma 4.1. Let r, s ∈ Z≥0, V a representation of H, and suppose 0 6= v ∈ V is such that xrv = 0. 90 Graded limits of minimal affinizations and beyond (a) The following identity holds in U(H): x(r)y(s) = min{r,s}∑ k=0 z(k)y(s−k)x(r−k). (b) For all k ∈ Z≥0, the element yszkv is in the span of elements of the form xaybzcv with 0 ≤ c < r, a+ c = k, and b+ c = k+ s. Moreover, if xv = 0, then yszv = 1 s+1 xy s+1v. Recall that U(n−) is Q+-graded and denoted by U(n−)η the piece of degree η. For the remainder of this subsection we assume g = sl3 and I = {1, 2}. Observe that the map n− → H given by x−i 7→ x and x−j 7→ y, where i, j ∈ I are distinct, is an isomorphism. Lemma 4.2. Let i, j ∈ I, i 6= j, and η = kiαi + kjαj ∈ Q+. Then {(x−i ) (r)(x−j ) (kj)(x−i ) (ki−r) : 0 ≤ r ≤ min{ki, kj}} is a basis of U(n−)η. Proof. Since dim(U(n−)η) = p(η) = min{ki, kj} + 1, it suffices to show that this set is linearly independent. Let us write x = x−i , y = x−j , and z = [x, y]. Then, by part (a) of Lemma 4.1 we have x(r)y(kj)x(ki−r) = min{r,kj}∑ k=0 ( ki − k r − k ) z(k)y(kj−k)x(ki−k). One now easily uses the PBW theorem to prove that these vectors, with 0 ≤ r ≤ min{ki, kj}, are linearly independent. Lemma 4.3. Let λ = m1ω1 +m2ω2 ∈ P+, 0 ≤ k1 ≤ m1, 0 ≤ k2 ≤ m2, and µ = λ− k1α1 − k2α2. Then, dim(V (λ)µ) = min{k1, k2}+ 1. Proof. Straightforward using Kostant’s multiplicity formula (cf. [24, Propo- sition 5.3.10]). Lemma 4.4. Let V be a finite-dimensional g-module, l ∈ Z≥1, and µ1, . . . , µl ∈ P+. Assume µl < µs for all s < l, write ηs = µs − µl = ks,1α1 + ks,2α2, and suppose ks,i ≤ µs(hi), i ∈ I. Suppose also that there exists vs ∈ Vµs such that V = ∑l s=1 U(n −)vs. Let i, j ∈ I be distinct. Then, V ∼= l⊕ s = 1 V (µs) iff the vectors (x−i ) (r)(x−j ) (ks,j)(x−i ) (ks,i−r)vs for s = 1, . . . , l and 0 ≤ r ≤ min{ks,1, ks,2} are linearly independent. A. Moura, F. Pereira 91 Proof. By Lemma 4.3 we have dim(V (µs))µl = min{ks,1, ks,2}+ 1 and by Lemma 2.4 there exists m ≤ l and s1, . . . , sm such that V ∼= m⊕ r = 1 V (µsr). Hence, dim(Vµl ) = m∑ r=1 dim(V (µsr)µl ) = m∑ r=1 (min{ksr,1, ksr,2}+ 1). The if part follows since the cardinality of the set {(s, r) : s = 1, . . . , l, 0 ≤ r ≤ min{ks,1, ks,2}} is ∑l s=1(min{ks,1, ks,2}+ 1). Conversely, assume that V ∼= l⊕ s = 1 V (µs) and let Vs, s = 1, . . . , l, be a submodule of V isomorphic to V (µs) and such that V = l⊕ s = 1 Vs. Let also πs : V → Vs be the associated projection. By Lemma 2.4 we can assume πs(vs) is a highest weight vector of Vs. Observe that the set (x−i ) (r)(x−j ) (ks,j)(x−i ) (ks,i−r)πs(vs) with 0 ≤ r ≤ min{ks,1, ks,2} is a basis of (Vs)µl . Indeed, the set (x−i ) (r)(x−j ) (ks,j)(x−i ) (ks,i−r) is a basis for U(n−)ηs by Lemma 4.2. In particular, the vectors (x−i ) (r)(x−j ) (ks,j)(x−i ) (ks,i−r)πs(vs) with 0 ≤ r ≤ min{ks,1, ks,2} span (Vs)µl . Since we already know that dim((Vs)µl ) = min{ks,1, ks,2}+ 1, the claim follows. Let ar,s ∈ C be such that l∑ s=1 min{ks,1,ks,2}∑ r=0 ar,s(x − i ) (r)(x−j ) (ks,j)(x−i ) (ks,i−r)vs = 0. Given 1 ≤ t ≤ l, we get πt( l∑ s=1 min{ks,1,ks,2}∑ r=0 ar,s(x − i ) (r)(x−j ) (ks,j)(x−i ) (ks,i−r)vs) = min{kt,1,kt,2}∑ r=0 ar,t(x − i ) (r)(x−j ) (kt,j)(x−i ) (kt,i−r)πt(vt) = 0. It follows that ar,t = 0 for all t = 1, . . . , l and 0 ≤ r ≤ min{kt,1, kt,2}. Lemma 4.5. Let a, b, c,m ∈ Z≥0, i, j ∈ I, j 6= i, λ = mωi, and v ∈ V (λ)λ\{0}. Then, (x−i ) a(x−j ) b(x−i ) cv 6= 0 ⇔ b ≤ c and a+ c ≤ m. Moreover, (x−i ) a(x−j ) b(x−i ) cv = ( a∏ s=1 c+ s− b c+ s ) (x−j ) b(x−i ) a+cv. 92 Graded limits of minimal affinizations and beyond Proof. From the sl2 representation theory we have (x−i ) cv 6= 0 iff c ≤ m. Since x+j (x − i ) cv = 0 and hj(x − i ) cv = c(x−i ) cv, it follows from the sl2 representation theory once more that (x−j ) b(x−i ) cv 6= 0 iff b ≤ c (and c ≤ m). Notice that this together with the second statement implies the first statement. We prove the second statement by induction on a ≥ 0. The case a = 0 is obvious. The induction step will however depend on the knowledge of the case a = 1. For convenience set x = x−j , y = x−i , and z = [x, y]. Using the well-known commutation relation in U(n−) yxb = xby − bxb−1z we get yxbycv = xbyc+1v − bxb−1yczv = = xbyc+1v − b c+ 1 xbyc+1v = c+ 1− b c+ 1 xbyc+1v where, in the second equality, we used that xv = 0 and the last statement of Lemma 4.1. The case a = 1 follows. Then, for a > 1, using the induction hypothesis we get yaxbycv = y(ya−1xbycv) = ( a−1∏ s=1 c+ s− b c+ s ) yxbyc+a−1v. Since, by the case a = 1, we have yxbyc+a−1v = ( c+ a− b c+ a ) xbya+cv the second statement follows. Remark. Notice that if b ≤ c the number ∏a s=1 c+s−b c+s is a positive rational number. 4.2. Root data Henceforth we assume g is of type E6, set λ = ∑ i∈I miωi ∈ P+, and assume λ ∈ P× A is such that Vq(λ) is a minimal affinization of Vq(λ). We will need the expression of every positive root in terms of the simple roots and of some of them in terms of the fundamental weights. These expressions are given by Tables 1 and 2 below, respectively. A. Moura, F. Pereira 93 Table 1 β1 = α1 + α2 β2 = α4 + α5 β3 = α2 + α3 β4 = α3 + α4 β5 = α3 + α6 β6 = α1 + α2 + α3 β7 = α3 + α4 + α5 β8 = α2 + α3 + α6 β9 = α3 + α4 + α6 β10 = α2 + α3 + α4 β11 = α1 + α2 + α3 + α6 β12 = α3 + α4 + α5 + α6 β13 = α2 + α3 + α4 + α6 β14 = α1 + α2 + α3 + α4 β15 = α2 + α3 + α4 + α5 β16 = α1 + α2 + α3 + α4 + α5 β17 = α1 + α2 + α3 + α4 + α6 β18 = α2 + α3 + α4 + α5 + α6 β19 = α1 + α2 + α3 + α4 + α5 + α6 β20 = α2 + 2α3 + α4 + α6 β21 = α1 + α2 + 2α3 + α4 + α6 β22 = α2 + 2α3 + α4 + α5 + α6 β23 = α1 + α2 + 2α3 + α4 + α5 + α6 β24 = α1 + 2α2 + 2α3 + α4 + α6 β25 = α2 + 2α3 + 2α4 + α5 + α6 β26 = α1+2α2+2α3+α4+α5+α6 β27 = α1+α2+2α3+2α4+α5+α6 β28 = α1+2α2+2α3+2α4+α5+α6 β29 = α1+2α2+3α3+2α4+α5+α6 β30 = α1+2α2+3α3+2α4+α5+2α6 Table 2 α1 = 2ω1 − ω2 α2 = 2ω2 − ω1 − ω3 α3 = 2ω3 − ω2 − ω4 − ω6 α4 = 2ω4 − ω3 − ω5 α5 = 2ω5 − ω4 α6 = 2ω6 − ω3 β23 = ω1 − ω2 + ω3 − ω4 + ω5 β24 = ω2 − ω5 β25 = ω4 − ω1 β26 = ω2 − ω4 + ω5 β27 = ω1 − ω2 + ω4 β28 = ω2 − ω3 + ω4 β29 = ω3 − ω6 β30 = ω6 94 Graded limits of minimal affinizations and beyond 4.3. A smaller set of relations for M(λ) In order to prove Proposition 3.15, we need a version of [22, Proposition 4.6]. Proposition 4.6. Suppose that either m3 6= 0 or supp(λ) is of type A. Then, M(λ) is isomorphic to the g[t]-module N(λ) generated by a vector v satisfying hiv = miv and n+[t]v = h⊗tC[t]v = (x−αi )mi+1v = x−α,1v = 0 for all α ∈ R+ 1 := {α ∈ R+ : α = ∑ i∈I niαi with ni ≤ 1 for all i ∈ I} = R+\{βj : j ≥ 20}. Proof. It follows from Lemma 4.11 that R+ 1 ⊆ R(λ, 1) and, hence, M(λ) is a quotient of N(λ). Let us now show that, under the hypothesis assumed on λ, we have an epimorphism in the opposite direction. Thus, we need to show that x−α,rv = 0 for all α ∈ R(λ, r). In fact, after (3.5), given α ∈ R+, it suffices to show that x−α,rαv = 0 where rα = min{r : α ∈ R(λ, r)}. (4.1) If rα = 0 this follows immediately from the defining relations of N(λ) since they clearly imply that U(g)v ∼= V (λ). If α ∈ R+ 1 equation (4.1) is again immediate from the defining relations of N(λ). Therefore, we need to prove (4.1) for α ∈ R+\R+ 1 only. Notice also that Lemma 4.11 implies that rα ≤ 3 for all α ∈ R+. Assume first that m3 6= 0. It then follows from (4.7) that R+ 1 = R(λ, 1) and (4.1) is immediate for all α such that rα = 1. Equation (4.7) also implies that R(λ, 2) = {βj : 20 ≤ j ≤ 28} and R(λ, 3) = {β29, β30}. Therefore, we are left to show that x−βj ,2 v = 0 for all 20 ≤ j ≤ 28 and x−βj ,3 v = 0 for all 29 ≤ j ≤ 30. This follows from the following commutation relations together with (4.1) for α such that rα ≤ 1: x−β20,2 = [x−α3,1 , x−β13,1 ], x−β21,2 = [x−α3,1 , x−β17,1 ], x−β22,2 = [x−α3,1 , x−β18,1 ], x−β23,2 = [x−α3,1 , x−β19,1 ], x−β24,2 = [x−β3,1 , x−β17,1 ], x−β25,2 = [x−β4,1 , x−β18,1 ], x−β26,2 = [x−β3,1 , x−β19,1 ], x−β27,2 = [x−β4,1 , x−β19,1 ], x−β28,2 = [x−β10,1 , x−β19,1 ], x−β29,3 = [x−α3,1 , x−β28,2 ], x−β30,3 = [x−β18,1 , x−β21,2 ]. Now, assume m3 = 0. In this case, rα ≤ 2 for all α ∈ R+. We consider separately the cases supp(λ) ⊆ {1, 2, 4, 5} and supp(λ) ⊆ {1, 2, 6} (the case supp(λ) ⊆ {4, 5, 6} follows from the latter by the symmetry of the A. Moura, F. Pereira 95 Dynkin diagram). Thus, assume supp(λ) ⊆ {1, 2, 6} and consider the following relations x−β20,1 = [x−3 , x − β13,1 ], x−β21,1 = [x−3 , x − β17,1 ], x−β22,1 = [x−3 , x − β18,1 ], x−β23,1 = [x−3 , x − β19,1 ], x−β25,1 = [x−β4 , x−β18,1 ], x−β27,1 = [x−β4 , x−β19,1 ]. Since α3, β4 ∈ R(λ, 0) in this case, it follows that x−βj ,1 v = 0 for all 20 ≤ j ≤ 27, j 6= 24, 26. If m2 = 0, we need to show that x−βj ,1 v = 0 for j ∈ {24, 26, 28, 29} and x−β30,r v = 0 where r = 1 if m6 = 0 and r = 2 otherwise. Since, in this case, β3, β10 ∈ R(λ, 0), the former follows from the following relations x−β24,1 = [x−β3 , x−β17,1 ], x−β26,1 = [x−β3 , x−β19,1 ], x−β28,1 = [x−β10 , x−β19,1 ], x−β29,1 = [x−3 , x − β28,1 ]. The latter follows from the relations x−β30,1 = [x−β18 , x−β21,1 ] and x−β30,2 = [x−β18,1 , x−β28,1 ] using that β18 ∈ R(λ, 0) if m6 = 0. Assume supp(λ) ⊆ {1, 2, 4, 5}. As in the previous case, one sees that x−βj ,1 v = 0 for all 20 ≤ j ≤ 23. If both m2 and m4 are nonzero, we are left to show that x−βj ,2 v = 0 for all 24 ≤ j ≤ 30. For 24 ≤ j ≤ 28, this is done as in the case m3 6= 0 while for j = 29, 30 this then follows from the relations x−β29,2 = [x−3 , x − β28,2 ] and x−β30,2 = [x−β18,1 , x−β21,1 ]. If m2 = 0 and m4 6= 0, we need to show that x−β24,1 v = x−β26,1 v = 0. This is done as in the case supp(λ) ⊆ {1, 2, 6}. The case m2 6= 0 and m4 = 0 is treated similarly. In particular, if m4 = 0 we have x−β25,1 v = x−β27,1 v = 0. Finally, if m2 = m4 = 0, we need to prove in addition that x−βj ,1 v = 0 for j = 28, 29, 30. This is done as in the case supp(λ) ⊆ {1, 6}. 4.4. Quantized relations The goal of this subsection is to prove Proposition 3.15. We proceed as in the proof of [22, Proposition 3.22] where a similar statement for orthogonal Lie algebras was proved. First we record several previously proved results which will be used in the proof. Lemma 4.7 ([22, Lemma 4.18]). Suppose w is a highest-ℓ-weight vector of Vq(ωi,a,m) for some i ∈ I, a ∈ C(q)×, and m ∈ Z≥0. Then, x−i,1w = aqmx−i w. 96 Graded limits of minimal affinizations and beyond The following proposition follows from the results of [3, Section 6]. Proposition 4.8. Let l ∈ Z≥1, ij ∈ I,mj ∈ Z≥1, aj ∈ C(q)× for j = 1, . . . , l. If aj ak /∈ qZ>0 for j > k, then Vq(ωi1,a1,m1)⊗ · · · ⊗ Vq(ωil,al,ml ) is a highest-ℓ-weight module. Corollary 4.9 ([22, Corollary 4.4]). Let λ ∈ P+, ai ∈ C(q)×, i ∈ I, and λ = ∏ i∈I ωi,ai,λ(hi). Then, there exists an ordering i1, . . . , in of I such that Vq(λ) is isomorphic to the Uq(g̃)-submodule of Vq(ωi1,ai1 ,λ(hi1 )) ⊗ · · · ⊗ Vq(ωin,ain ,λ(hin ) ) generated by the top weight space. Proposition 4.10 ([22, Proposition 3.13]). Suppose λ ∈ P× A is such that Vq(λ) is a minimal affinization and that J ⊆ I is an admissible subdiagram. Let v be a highest-ℓ-weight vector of V = Vq(λ), λ = wt(λ), and a ∈ C× be such that λ = ωλ,a. Then x−α,rv = arx−α v for every α ∈ R+ J . If α ∈ R+ J for some admissible diagram J , we shall refer to α as an admissible root. Proof of Proposition 3.15. Let a ∈ C be such that λ̄ = ωλ,a. We fix a highest-ℓ-weight vector v of V = Vq(λ) and ai ∈ A×, i ∈ I, such that λ = ∏ i∈I ωi,ai,mi . Let also v̄ be the image of v in V and v′ be the image of v̄ in L(λ). By Proposition 4.6, we need to show that x−α,1v ′ = 0 for all α ∈ R+ 1 . This is equivalent to showing that x−α,1v̄ = av̄ for all α ∈ R+ 1 . (4.2) By Proposition 4.10, (4.2) holds if α is an admissible root. Therefore, it remains to show that x−βj ,1 v̄ = av̄ for all 7 < j < 20. (4.3) Assume first that supp(λ) ⊆ {1, 2, 3, 4, 5}. In this case α6 ∈ R(λ, 0) and (4.3) with j ∈ {8, 9, 11, 12} follows from the following relations x−β8,1 = [x−6 , x − β3,1 ], x−β9,1 = [x−6 , x − β4,1 ], x−β11,1 = [x−6 , x − β6,1 ], x−β12,1 = [x−6 , x − β7,1 ] together with the fact that β3, β4, β6, and β7 are admissible roots. Next, assume that we have proved (4.3) for j ∈ {10, 14, 15, 16}. Then, (4.3) for the remaining values of j follows from the following relations x−β13,1 = [x−6 , x − β10,1 ], x−β17,1 = [x−6 , x − β14,1 ], x−β18,1 = [x−6 , x − β15,1 ], x−β19,1 = [x−6 , x − β16,1 ]. A. Moura, F. Pereira 97 In order to prove (4.3) for j ∈ {10, 14, 15, 16}, it suffices to find elements Xj , Xj,1 ∈ UA(ñ −) such that Xj = x−βj , Xj,1 = x−βj ,1 , and Xj,1v = aj(q)Xjv + xjv (4.4) for some aj(q) ∈ A and xj ∈ UA(g) satisfying aj(1) = a and xj = 0. We prove the existence of such elements assuming ai+1 = aiq mi+mi+1+1 for all i < 5. (4.5) The case ai+1 = aiq −(mi+mi+1+1), i < 5, is proved similarly using part (b) of Proposition 2.10 instead of part (a). Let i0 = max{i ∈ I : mi 6= 0} (in the case ai+1 = aiq −(mi+mi+1+1), i < 5, we would use i0 = min{i ∈ I : mi 6= 0}). The relations Xj,1v = aj(q)Xjv+xjv of (4.4) are the quantized relations alluded to in the title of this subsection. Let λ′ be such that λ = λ′ωi0,ai0 ,mi0 . Let also v1, v2 be highest-ℓ-weight vectors of Vq(λ ′) and Vq(ωi0,ai0 ,mi0 ), respectively. By (4.5), Proposition 4.8, and Corollary 4.9, the assignment v 7→ v1 ⊗ v2 extends to an isomorphism V ∼= Uq(g̃)(v1 ⊗ v2) ⊆ Vq(λ ′)⊗ Vq(ωi0,ai0 ,mi0 ). Henceforth, we identify v with v1 ⊗ v2. We write down the proof of the existence of elements as in (4.4) for j = 16 assuming i0 = 5 (the other cases are proved similarly and the computations are simpler). Set X14 = [x−4 , [x − 3 , [x − 1 , x − 2 ]]], X16 = [x−5 , X14], and X16,1 = [x−5,1, X14]. Quite clearly, X16, X16,1 ∈ UA(g̃) satisfy the first two identities in (4.4). By Lemmas 1.3 and 1.4, modulo an element of the form xv with x ∈ UA(g̃)⊗ UA(g̃) such that x̄ = 0, we have X16v = x−5 X14(v1 ⊗ v2)−X14x − 5 (v1 ⊗ v2) = x−5 ((X14v1)⊗ v2)−X14(v1 ⊗ (x−5 v2)) = (x−5 X14v1)⊗ (k−1 5 v2) + (X14v1)⊗ (x−5 v2) − (X14v1)⊗ ((k1k2k3k4) −1x−5 v2)− v1 ⊗ (X14x − 5 v2) = q−m5(x−5 X14v1)⊗ v2 + (1− q−m5)(X14v1)⊗ (x−5 v2) − v1 ⊗ (X14x − 5 v2) while X16,1v = x−5,1X14(v1 ⊗ v2)−X14x − 5,1(v1 ⊗ v2) = x−5,1((X14v1)⊗ v2)−X14(v1 ⊗ (x−5,1v2)) = (x−5,1X14v1)⊗ (k5v2) + (X14v1)⊗ (x−5,1v2) − (X14v1)⊗ ((k1k2k3k4) −1x−5,1v2)− v1 ⊗ (X14x − 5,1v2) 98 Graded limits of minimal affinizations and beyond = qm5(x−5,1X14v1)⊗ v2 + (1− q−m5)(X14v1)⊗ (x−5,1v2) − v1 ⊗ (X14x − 5,1v2). Using Lemma 4.7 we get X16,1v = qm5(x−5,1X14v1)⊗ v2 + (1− q−m5)(X14v1)⊗ (a5q m5x−5 v2) − v1 ⊗ (X14(a5q m5v2)) = a5q m5X16v + qm5(x−5,1X14v1)⊗ v2 − a5(x − 5 X14v1)⊗ v2. Since a16(q) := a5q m5 satisfies a16(1) = a, in order to prove that X16 and X16,1 satisfy the last identity of (4.4), it suffices to show that qm5(x−5,1X14v1)⊗ v2 = a5(x − 5 X14v1)⊗ v2. (4.6) Notice that x+5,rX14v1 = 0 for all r ∈ Z and letW be the Uq(g̃5)-submodule of Vq(λ ′) generated byX14v1. Then, by Proposition 2.10(a),W is a highest- ℓ-weight module with highest ℓ-weight ω5,a4qm4 . It then follows from Lemma 4.7 that x−5,1X14v1 = a4q m4+1x−5 X14v1. This and (4.5) imply (4.6). The case supp(λ) ⊆ {1, 2, 3, 6} is dealt with similarly and the case supp(λ) ⊆ {3, 4, 5, 6} then follows using the symmetry of the Dynkin diagram. We omit the details. 4.5. Upper bounds In this subsection we prove (3.10). Let v ∈M(λ)λ be nonzero. Lemma 4.11. For every i ∈ I,m ∈ Z≥0, and α = ∑ j∈I ajαj ∈ R+ we have α ∈ R(i,m, ai). In particular: (a) R(1,m, 1) = R(5,m, 1) = R+. (b) R(6,m, 1) ⊇ R+ \ {β30} and R(6,m, 2) = R+. (c) R(2,m, 1) ⊇ R+ \ {β24, β26, β28, β29, β30} and R(2,m, 2) = R+. (d) R(4,m, 1) ⊇ R+ \ {β25, β27, β28, β29, β30} and R(4,m, 2) = R+. (e) R(3,m, 1) ⊇ R+ \ {βj : j ≥ 20}, R(3,m, 2) ⊇ R+ \ {β29, β30}, and R(3,m, 3) = R+. Proof. Statements (a)-(e) follow from the first statement by inspection of Table 1. Conversely, clearly items (a)-(e) together imply the first statement. The proof is analogous to that of [2, Proposition 1.2] (see also [24, Lemma 5.2.8]). We omit the details. A. Moura, F. Pereira 99 Observe that the above Lemma together with (3.5) imply x−αi,r v = x−βj ,r v = x−βk,s v = 0 for all i ∈ I, j < 20, k < 29, r ≥ 1, s ≥ 2 and R(λ, 3) = R+. Let R′(i,m, r) be the set on the right-hand-side of the inclusion symbol of the appropriate item of Lemma 4.11. It will follow from Section 4.6 below that, if m > 0, then R(i,m, r) = R′(i,m, r). (4.7) Set R′(λ, r) = ∩i∈IR ′(i,mi, r) and let r(λ) be the subspace of g[t] spanned by {x−α,1, x − β,2 : α ∈ R+\R′(λ, 1), β ∈ R+\R′(λ, 2)} which is clearly an abelian ideal of n−[t]. Since we are assuming m3 = 0, we have R′(λ, 2) = R+ and, therefore, r(λ) is the subspace of g[t] spanned by {x−α,1 : α ∈ R+\R′(λ, 1)}. Since R(λ, r) = R+ for all r ≥ 2 by (3.5), a straightforward application of the PBW Theorem implies M(λ) = U(n−[t])v = U(n−)U(r(λ))v. (4.8) Moreover, R(λ, 1) ⊇ R+ \ {β24, β25, β26, β27, β28, β29, β30}. (4.9) by Lemma 4.11 and, therefore, M(λ) = U(n−)U(x−β30,1 )U(x−β29,1 )U(x−β28,1 ) U(x−β27,1 )U(x−β26,1 )U(x−β25,1 )U(x−β24,1 )v. (4.10) We now apply Lemma 4.1 to prove that M(λ) = U(n−)U(x−β30,1 )U(x−β28,1 )U(x−β27,1 )U(x−β26,1 )U(x−β25,1 )U(x−β24,1 )v. (4.11) Indeed, let x = x−3 , y = x−β28,1 , z = x−β29,1 which generates a three- dimensional Heisenberg subalgebra of g[t]. Since xv = 0, it follows from Lemma 4.1 that (x−β28,1 )r(x−β29,1 )sv is a multiple of (x−3 ) s(x−β28,1 )r+sv for every r, s ∈ Z≥0. Since [x−3 , x − βj ,1 ] = 0 for all 24 ≤ j ≤ 30, j 6= 28, (4.11) follows. Given r = (r1, r2, r3, r4, r5, r6) ∈ Z6 ≥0, set xr = (x−β30,1 )r6(x−β28,1 )r5(x−β27,1 )r4(x−β26,1 )r3(x−β25,1 )r2(x−β24,1 )r1 so that (4.11) is equivalent to M(λ) = ∑ r∈Z6 ≥0 U(n−)xrv. (4.12) 100 Graded limits of minimal affinizations and beyond Recall the definition of wt(r) in Subsection 3.2 and use Table 2 to observe that xrv ∈M(λ)[gr(r)]wt(r). Consider the Heisenberg subalgebra of g[t] generated by {x−1 , x − β25,1 , x−β27,1 }. Since (x−1 ) m1+1v = 0 and [x−1 , x − βj ,1 ] = 0 for all 24 ≤ j ≤ 30, j 6= 25, it follows from Lemma 4.1 that we can restrict the sum of (4.12) to r ∈ Z6 ≥0 such that r4 ≤ m1. Similarly, by working with the Heisenberg subalgebra generated by {x−5 , x − β24,1 , x−β26,1 } we can assume r3 ≤ m5. Next, we show that we can restrict the sum of (4.12) to r ∈ Z6 ≥0 such that r1+r3+r5 ≤ m2 and r2+r4+r5 ≤ m4. By contradiction, assume this is not the case. It then follows from Lemma 2.4 that there exists r ∈ Z6 ≥0 satisfying either r1 + r3 + r5 > m2 or r2 + r4 + r5 > m4 and such that V (wt(r)) is an irreducible summand of M(λ). Moreover, the injectivity of wt : Z6 → P implies that the projection of xrv on this summand is non zero. Fix such r and suppose r1+r3+r5 > m2 (the other case follows from the symmetry of the Dynkin diagram). Let s = (r1, r2, r3, 0, r4 + r5, r6) and notice that (x+2 ) r4xsv = cxrv for some c ∈ C×. (4.13) This easily follows from the relations [x+2 , x − βj ,1 ] = 0 for all 24 ≤ j ≤ 30, j 6= 26, 28, [x+2 , x − β26,1 ] = x−β23,1 , [x+2 , x − β28,1 ] = x−β27,1 , [x−β23,1 , x−βj ,1 ] = 0 for all 24 ≤ j ≤ 30, and x−β23,1 v = 0. It follows from (4.13) that the projection of xsv on V (wt(r)) is non zero and, hence, V (wt(r))wt(s) 6= 0. We claim that this is a contradiction. Indeed, notice that wt(s)(h2) = (m2−r1−r3−r4−r5). Hence, σ2wt(s) = wt(s)− (m2 − r1 − r3 − r4 − r5)α2 is a weight of V (wt(r)). Here, σ2 ∈ W is the simple reflection associated with α2. Since wt(s) = wt(r)− r4α2, it follows that σ2wt(s) = wt(r) + (r1 + r3 + r5 −m2)α2 > wt(r), contradicting V (wt(r))σ2wt(s) 6= 0. So far we proved that the sum in (4.12) can be restricted to r ∈ Z6 ≥0 such that r4 ≤ m1, r3 ≤ m5, r1 + r3 + r5 ≤ m2, and r2 + r4 + r5 ≤ m4. Now, observe that, for such r, wt(r) ∈ P+ iff r ∈ A. Therefore, by Lemma 2.4, we must have a surjective homomorphism of g-modules ⊕ r ∈ Ar V (wt(r)) →M(λ)[r] for every r ∈ Z≥0 and (3.10) follows. A. Moura, F. Pereira 101 Remark. Let w ∈ T (λ)λ be nonzero and notice that, since T (λ) is a quotient of M(λ), equations (4.12) remain valid after replacing M(λ) by T (λ) on the left-hand-side and v by w on the right-hand-side. 4.6. The Kirillov-Reshetikhin case In this subsection we assume λ = miωi for some i ∈ I, i 6= 3, and prove (3.11) in this case. As mentioned earlier, for such λ, (3.12) (and hence (3.11)) follows from [18, 16] (see also [2]). However, in order to prove (3.11) for more general λ later, we will need further details about this case than just (3.11). Hence, we consider it separately. We split the proof in cases according to the value of i. We keep denoting by v a nonzero vector in M(λ)λ. Assume i = 1 or i = 5 and notice that Lemma 4.11 implies r(λ) = 0 in this case. Hence, M(λ) = U(n−)v and it follows that M(λ) is isomorphic to the pullback of V (λ) by the map g[t] → g, x ⊗ f(t) 7→ f(0)x. Since A = {λ} in this case, (3.11) follows. Now suppose i = 6. Notice that Ar = {(0, 0, 0, 0, 0, r)} for all 0 ≤ r ≤ m6 and Ar = ∅ otherwise. Since wt((0, 0, 0, 0, 0, r)) = (m6 − r)ω6, (3.11) becomes t(m6−r)ω6,r 6= 0 for all 0 ≤ r ≤ m6. (4.14) We begin proving this in the case m6 = 1 in which case we have T (λ) = M(λ) by definition. Observe that Homg(g⊗ V (ω6), V (ω6)) 6= 0 which is true since V (ω6) is isomorphic to the adjoint representation. Hence, we can apply the construction given by (3.15) with V0 = V (ω6) and V1 = V (0). One easily checks that the highest weight vector w0 of V0 satisfies the relations satisfied by v and, hence, the module V constructed in this way is a quotient of M(ω6). Since V [0] ∼= V (ω6) and V [1] ∼= V (0), (4.14) follows. Moreover, we clearly have x−β30,1 w0 6= 0 (otherwise the map p0 would be zero) and, hence, x−β30,1 v 6= 0. In particular, (4.7) holds for i = 6. For m6 > 1, let w ∈ M(ω6)ω6 be nonzero. Since T (λ) is generated by w⊗m6 ∈ M(ω6) ⊗m6 one easily checks that (x−β30,1 )rw⊗m6 6= 0 for all r ≤ m6. In particular, (x−β30,1 )rv 6= 0 iff r ≤ m6. (4.15) By the remark closing Subsection 4.5, T (λ) = ∑m6 r=0 U(n−)(x−β30,1 )rw⊗m6 . Hence, (x−β30,1 )rw⊗m6 must be a highest-weight vector in T (λ)[r] which implies (4.14). Next, let i = 2. The proof is parallel to the previous case. Namely, (3.11) becomes equivalent to t(m2−r)ω2+rω5,r 6= 0 for all 0 ≤ r ≤ m2. (4.16) 102 Graded limits of minimal affinizations and beyond Notice that x−β24,1 plays the role that x−β30,1 did in the case i = 6. If m2 = 1, we again use the construction given by (3.15) this time with V0 = V (ω2) and V1 = V (ω5). In particular, it follows that x−β24,1 v 6= 0. For m2 > 1, let w ∈M(ω2)ω2 be nonzero. As before, we conclude that (x−β24,1 )rw⊗m2 6= 0 for all 0 ≤ r ≤ m2. Equation (4.16) follows as in the previous case by using that T (λ) = ∑m2 r=0 U(n−)(x−β24,1 )rw⊗m2 . We now record the following lemma which, in particular, proves (4.7) for i = 2. Lemma 4.12. Let rj ∈ Z≥0, j = 1, . . . , 5, and w = (x−β24,1 )r1(x−β26,1 )r2(x−β28,1 )r3(xβ29,1) r4(x−β30,1 )r5v. Then w is a nonzero scalar multiple of (x−6 ) r5(x−3 ) r5+r4(x−4 ) r5+r4+r3(x−5 ) r5+r4+r3+r2(x−β24,1 )r1+r2+r3+r4+r5v. Moreover, w is nonzero iff r1 + · · ·+ r5 ≤ m2. In particular, R(2,m2, 1) = R′(2,m2, 1). Proof. The last statement follows immediately from the second. The first statement follows from straightforward successive applications of Lemma 4.1. Namely, we first consider the Heisenberg subalgebra generated by x = x−6 , y = x−β29,1 , and z = x−β30,1 together with the relation xv = 0 to get (x−β29,1 )r4(x−β30,1 )r5v = η(x−6 ) r5(x−β29,1 )r4+r5v for some nonzero scalar η. Since [x−6 , x − βj ,1 ] = 0 for j = 24, 26, 28, it follows that (x−β24,1 )r1(x−β26,1 )r2(x−β28,1 )r3(xβ29,1) r4(x−β30,1 )r5v = = η(x−6 ) r5(x−β24,1 )r1(x−β26,1 )r2(x−β28,1 )r3(xβ29,1) r4+r5v. By similarly considering the subalgebras generated by {x−3 , x − β28,1 , x−β29,1 }, {x−4 , x − β26,1 , x−β28,1 }, and {x−5 , x − β24,1 , x−β26,1 } in this order, the first statement follows. We have seen above that (x−β24,1 )rw⊗m2 6= 0 iff r ≤ m2. This implies (x−β24,1 )r1+r2+r3+r4+r5v 6= 0 iff r1 + · · · + r5 ≤ m2. Since x+5 (x − β24,1 )rv = (x−β24,1 )rx+5 v = 0 and h5(x − β24,1 )rv = rv, it follows that (x−5 ) s(x−β24,1 )rv 6= 0 for all 0 ≤ s ≤ r. In particular, (x−5 ) r5+r4+r3+r2(x−β24,1 )r1+r2+r3+r4+r5v 6= 0. The proof is completed proceeding similarly. The case i = 4 is obtained from the previous case by using the nontrivial Dynkin diagram automorphism of g. In particular we have: A. Moura, F. Pereira 103 Lemma 4.13. Let rj ∈ Z≥0, j = 1, . . . , 5, and w = (x−β25,1 )r1(x−β27,1 )r2(x−β28,1 )r3(xβ29,1) r4(x−β30,1 )r5v. Then w is a nonzero scalar multiple of (x−6 ) r5(x−3 ) r5+r4(x−2 ) r5+r4+r3(x−1 ) r5+r4+r3+r2(x−β25,1 )r1+r2+r3+r4+r5v. Moreover, w is nonzero iff r1 + · · ·+ r5 ≤ m4. In particular, R(4,m4, 1) = R′(4,m4, 1). 4.7. Lower bounds We now complete the proof of (3.11) for λ as in Theorem 3.14. In fact, we will carry out most of the proof assuming only that λ(h3) = 0. Recall the notation xr, r ∈ A, developed in Section 4.5. In addition, we shall use the following notation. Denote by vi,mi a nonzero vector in M(miωi)miωi and by vsi,mi the image of vi,mi in M(miωi)(s). By definition of the truncated module M(miωi)(s) we have M(miωi)(s)[r] = 0 if r > s. (4.17) Given s = (si)i∈I ∈ ZI ≥0, let Ts(λ) be the submodule of ⊗ i ∈ I M(miωi)(si) generated by vs := ⊗ i ∈ I vsii,mi . Since T (λ) is the submodule of ⊗ i ∈ I M(miωi) generated by v := ⊗ i ∈ I vi,mi , there exists a unique epimorphism from T (λ) onto Ts(λ) such that v 7→ vs . Let tsµ,r denote the multiplicity of V (µ) as an irreducible constituent of Ts(λ)[r]. Observe that, since \|Awt(r),gr(r)\| = 1 for all r ∈ A, in order to prove (3.11), it suffices to prove that for each r ∈ A there exists s ∈ ZI ≥0 such that tswt(r),gr(r) ≥ 1. (4.18) It will be convenient to write the tensor product ⊗ i ∈ I M(miωi) in the following order: M(m2ω2)⊗M(m4ω4)⊗M(m6ω6)⊗M(m1ω1)⊗M(m5ω5), where we already used that m3 = 0 and, hence, M(m3ω3) ∼= V (0) ∼= C. In particular, v = v2,m2 ⊗ v4,m4 ⊗ v6,m6 ⊗ v1,m1 ⊗ v5,m5 and similarly for vs , s ∈ ZI ≥0. To shorten notation we write w = v1,m1 ⊗ v5,m5 when convenient so that v = v2,m2 ⊗ v4,m4 ⊗ v6,m6 ⊗ w. Let {ej : j = 1, . . . , 6} be the canonical basis of Z6 ≥0. Given r ∈ Z, set Z6[r] = {r ∈ Z6 : gr(r) = r}, and observe that Z6[0] is a free Z- module having b := {(e1 − e5), (e2 − e5), (e5 − e3), (e5 − e4), (e5 − e6)} 104 Graded limits of minimal affinizations and beyond as an ordered Z-basis. Define bj ∈ b, j = 1, . . . , 5, by requiring that b = {b1, . . . ,b5} as an ordered set. Clearly, r, r′ ∈ Z6[r] iff r− r ′ ∈ Z6[0]. Given j = (j1, j2, j3, j4, j5) ∈ Z5 and s ∈ ZI ≥0 such that s2 ≤ m2, s4 ≤ m4, s6 ≤ m6, observe that ro = (s2, s4, 0, 0, 0, s6) ∈ As2+s4+s6 and set r j = ro− 5∑ l=1 jlbl = (s2− j1, s4− j2, j3, j4, j1+ j2− j3− j4− j5, s6+ j5). Thus, r ∈ Z6[s2+ s4+ s6] iff r = r j for some j ∈ Z5. For shortening some expressions, given j ∈ Z5, we may use the notation j0 = j1+j2−j3−j4−j5. Notice that r j ∈ A iff 0 ≤ j3 ≤ m5, 0 ≤ j4 ≤ m1, j1 ≤ s2, j2 ≤ s4, j0 ≥ 0, (4.19) j5 ≤ m6 − s6, j1 − j3 − j5 ≤ m4 − s4, j2 − j4 − j5 ≤ m2 − s2. Set A(s) = {r ∈ A : xrvs 6= 0} ∩ Z6[s2 + s4 + s6] and let B(s) be the set of tuples j ∈ Z5 ≥0 satisfying j3 ≤ j1 ≤ s2, j4 ≤ j2 ≤ s4, j3 ≤ m5, j4 ≤ m1, j0 ≥ 0, (4.20) j5 ≤ m6 − s6, j1 − j3 − j5 ≤ m4 − s4, j2 − j4 − j5 ≤ m2 − s2. In Subsection 4.8 we will show that r j ∈ A(s) ⇔ j ∈ B(s). (4.21) It follows from (4.21) that Ts(λ)[s2 + s4 + s6] = ∑ j∈B(s) U(n−)xr j vs . (4.22) For j,k ∈ B(s) we have wt(r j )− wt(r k ) = (k2 − j2)α1 + (k1 − j1)α5 + (k5 − j5)(α3 + α6)+ (4.23) + (k2 − j2 + j4 − k4)α2 + (k1 − j1 + j3 − k3)α4. In particular, wt(ro) is the unique maximal weight of Ts(λ)[s2 + s4 + s6] and, hence, tswt(ro ),s2+s4+s6 ≥ 1. (4.24) A. Moura, F. Pereira 105 Lemma 4.14. Let r ∈ A. Then, there exists s ∈ ZI ≥0 and j ∈ B(s) such that j5 = 0 and r = r j . In particular, r ∈ A(s). Proof. Let s1 = s3 = s5 = 0, s2 = r1 + r3 + r5, s4 = r2 + r4, and s6 = r6. As before, set ro = (s2, s4, 0, 0, 0, s6) and notice that ro ∈ A. One easily checks that r = rj where j = (r3 + r5, r4, r3, r4, 0). By (4.21), r ∈ A(s) iff j ∈ B(s). The checking of the latter is straightforward. The above lemma shows that it suffices to show (4.18) in the case that r = r j for some s ∈ ZI ≥0 and j ∈ B(s) such that j5 = 0. In this case, it follows from the proof of (4.21) (see the last line of Subsection 4.8) that xr j vs is a nonzero scalar multiple of v j := (x−β28,1 )j1−j3(x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 ⊗ ⊗ (x−β28,1 )j2−j4(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 ⊗ w′ where w′ = (x−β30,1 )s6vs66,m6 ⊗ w. Notice v j 6= 0 since j ∈ B(s). From now on we fix s ∈ ZI ≥0, write B = B(s), and set B0 = {j ∈ B(s) : j5 = 0}. Given k ∈ B, let B+ k = {j ∈ B(s) : wt(r k ) < wt(r j )} and Bk = B+ k ∪ {k}. It easily follows from (4.23) that k ∈ B0 ⇒ Bk ⊆ B0. (4.25) By Lemma 2.4, (4.22), and the injectivity of wt : A → P , (4.18) holds for r = rk iff v k /∈ V + k := ∑ j∈B+ k U(n−)v j . (4.26) Equivalently, (4.18) holds for r = rk iff we have an isomorphism of g-modules Vk := ∑ j∈Bk U(n−)v j ∼= ⊕ j∈Bk V (wt(r j )). (4.27) Given j ∈ B0, define the height of j to be ht(j) = ht(wt(ro)− wt(r j )) = 2(j1 + j2)− (j3 + j4) = j1 + j2 + j0. 106 Graded limits of minimal affinizations and beyond We prove (4.27) by induction on k = ht(k). Equation (4.24) implies that (4.27) holds for k = 0. Thus, assume k > 0 and, by induction hypothesis, that (4.27) holds for j ∈ B0 such that ht(j) < k. It follows from the induction hypothesis and (4.25) that dim((V + k )wt(r k )) = ∑ j∈B+ k dim(V (wt(r j ))wt(r k )). (4.28) We are left to show that dim((Vk)wt(r k )) = dim((V + k )wt(rk)) + 1. (4.29) Let J− = {1, 2}, J+ = {4, 5}, J = J− ∪ J+ ⊆ I so that g J± ∼= sl3 and g J ∼= sl3⊕sl3. By Proposition 2.2,U(g J± )V (wt(r j ))wt(r j ) ∼= V (wt(r j )J±) and similarly for J in place of J±. Moreover, we have isomorphisms of vector spaces V (wt(r j ))wt(r k ) ∼= V (wt(r j )J)wt(r k )J ∼= ∼= V (wt(r j )J−)wt(r k )J− ⊗ V (wt(r j )J+)wt(r k )J+ . (4.30) The first isomorphism above is clear and the second follows from Proposi- tion 2.3. If j ∈ Bk, it easily follows from (4.23) that k2 − j2 ≤ wt(r j )(h1) = m1 + s4 − j2 − j4, k2 − j2 + j4 − k4 ≤ wt(r j )(h2) = m2 − s2 − j2 + 2j4, k1 − j1 ≤ wt(r j )(h5) = m5 + s2 − j1 − j3, k1 − j1 + j3 − k3 ≤ wt(r j )(h4) = m4 − s4 − j1 + 2j3. Hence, we can use Lemma 4.3 to compute dim(V (wt(r j )J−)wt(r k )J− ) = min{k2 − j2, k2 − j2 + j4 − k4}+ 1 and (4.31) dim(V (wt(r j )J+)wt(r k )J+ ) = min{k1 − j1, k1 − j1 + j3 − k3}+ 1. A. Moura, F. Pereira 107 Plugging this in (4.28) we get dim((V + k )wt(r k )) = ∑ j∈B+ k (min{k2 − j2, k2 − j2 + j4 − k4}+ 1)× × (min{k1 − j1, k1 − j1 + j3 − k3}+ 1). (4.32) We will need the following notation. Given, i1, i2, . . . , il ∈ I, and a1, . . . , al ∈ Z≥0, set x a1,...,al i1,...,il = (x−i1) (a1) · · · (x−il ) (al). Also, given j ∈ Bk, set l−(j) = min{k2 − j2, k2 − j2 + j4 − k4}, l+(j) = min{k1 − j1, k1 − j1 + j3 − k3} so that (4.29) can be rewritten as dim((Vk)wt(r k )) = ∑ j∈Bk (l−(j) + 1)(l+(j) + 1). (4.33) It now follows from Lemma 4.4 and (4.22) that (4.33) holds iff the vectors x p5,p4(j),k1−j1−p5 5,4,5 x p1,p2(j),k2−j2−p1 1,2,1 vj are linearly independent (4.34) for j ∈ Bk, 0 ≤ p1 ≤ l−(j), 0 ≤ p5 ≤ l+(j). Here p2(j) = k2 − j2 + j4 − k4 and p4(j) = k1 − j1 + j3 − k3. We will prove (4.34) only for λ as in Theorem 3.14. However, let us develop for a little longer the general case. In particular, we will show that all the vectors in (4.34) are nonzero. Set p = (p1, p5),xj ,p = x p5,p4(j),k1−j1−p5 5,4,5 x p1,p2(j),k2−j2−p1 1,2,1 , and v j ,p = x j ,p v j . Thus, we want to show that the vectors v j ,p are lin- early independent for j and p as above. From now on, when now confusion arises, we simplify notation and write l− in place of l−(j), etc. Recall that v j = (x−β28,1 )j1−j3(x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 ⊗ ⊗ (x−β28,1 )j2−j4(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 ⊗ w′. To simplify the expression above, set v6 = (x−β30,1 )s6vs66,m6 , x 2 j = (x−β28,1 )j1−j3(x−β26,1 )j3(x−β24,1 )s2−j1 , 108 Graded limits of minimal affinizations and beyond and x 4 j = (x−β28,1 )j2−j4(x−β27,1 )j4(x−β25,1 )s4−j2 so that v j = x 2 j vs22,m2 ⊗ x 4 j vs44,m4 ⊗ v6 ⊗ w. (4.35) Also, using Lemma 4.12 we get x 2 j vs22,m2 = (x−4 ) j1−j3(x−5 ) j1(x−β24,1 )s2vs22,m2 and x 4 j vs44,m4 = (x−2 ) j2−j4(x−1 ) j2(x−β25,1 )s4vs44,m4 (4.36) up to nonzero scalar multiples. By applying the comultiplication one sees that v j ,p is equal to ∑ χ x d2,e2,f2 5,4,5 x a2,b2,c2 1,2,1 x 2 j vs22,m2 ⊗ x d4,e4,f4 5,4,5 x a4,b4,c4 1,2,1 x 4 j vs44,m4 ⊗ ⊗ v6 ⊗ x a1,b1,c1 1,2,1 v1,m1 ⊗ x d5,e5,f5 5,4,5 v5,m5 (4.37) where χ runs over the set of collections of nonnegative integers al, bl, cl, dl, el, fl satisfying a2 + a4 + a1 = p1, b2 + b4 + b1 = p2, c2 + c4 + c1 = k2 − j2 − p1, (4.38) d2 + d4 + d5 = p5, e2 + e4 + e5 = p4, f2 + f4 + f5 = k1 − j1 − p5. Above we also used that xa,b,c 1,2,1v5,m5 = x a,b,c 5,4,5v1,m1 = x a,b,c 1,2,1v6 = x a,b,c 5,4,5v6 = 0 whenever a + b + c > 0. We will need to study the summands on the right-hand-side of (4.37). Using Lemma 4.5 we see that x a1,b1,c1 1,2,1 v1,m1 6= 0 iff a1 + c1 ≤ m1 and b1 ≤ c1 and, in that case, x a1,b1,c1 1,2,1 v1,m1 = ηxb1,a1+c1 2,1 v1,m1 for some positive rational number η (depending on a1, b1, c1). Similarly, x d5,e5,f5 5,4,5 v5,m5 6= 0 iff d5 + f5 ≤ m5 and e5 ≤ f5 and, in that case, x d5,e5,f5 5,4,5 v5,m5 is a positive multiple of x e5,d5+f5 4,5 v5,m5 . Next, we study the factor x d2,e2,f2 5,4,5 x a2,b2,c2 1,2,1 x 2 j vs22,m2 = x a2,b2,c2 1,2,1 x d2,e2,f2 5,4,5 x 2 j vs22,m2 . Notice that x+5 (x − β24,1 )s2vs22,m2 = h4(x − β24,1 )s2vs22,m2 = 0, and h5(x − β24,1 )s2vs22,m2 = s2. Therefore, we can use Lemma 4.5 together with (4.36) to see that x d2,e2,f2 5,4,5 x 2 j vs22,m2 is a nonnegative rational multiple of x e2+j1−j3,j1+f2+d2 4,5 (x−β24,1 )s2vs22,m2 A. Moura, F. Pereira 109 and it is nonzero provided e2 ≤ j3 + f2 and j1 + d2 + f2 ≤ s2. Since d2 ≤ p5, f2 ≤ k1 − j1 − p5 by (4.38), and k1 ≤ s2, the latter is always satisfied. One easily checks that x+1 x 2 j vs22,m2 = h1x 2 j vs22,m2 = 0 which implies x d2,e2,f2 5,4,5 x a2,b2,c2 1,2,1 x 2 j vs22,m2 = 0 if c2 6= 0. Next, using the relations [x+2 , x − β24,1 ] = x−β21,1 , [x−β21,1 , x−β24,1 ] = 0, x−β21,1 vs22,m2 = 0, one sees that x+2 x 2 j vs22,m2 = 0. Since h2x 2 j vs22,m2 = (m2 − s2)x 2 j vs22,m2 , it follows from Lemma 4.5 that x a2,b2,c2 1,2,1 x 2 j vs22,m2 6= 0 iff c2 = 0 and a2 ≤ b2 ≤ m2 − s2. Since we anyway have b2 ≤ p2 = (k2 − k4) − (j2 − j4) ≤ m2 − s2, the relevant conditions are c2 = 0 and a2 ≤ b2. Therefore, we find that x d2,e2,f2 5,4,5 x a2,b2,c2 1,2,1 x 2 j vs22,m2 is a nonnegative rational multiple of x e2+j1−j3,j1+f2+d2 4,5 x a2,b2 1,2 (x−β24,1 )s2vs22,m2 which is nonzero iff a2 ≤ b2 and e2 ≤ j3 + f2. Similarly, we get that x d4,e4,f4 5,4,5 x a4,b4,c4 1,2,1 x 4 j vs44,m4 is a nonnegative rational multiple of x b4+j2−j4,j2+c4+a4 2,1 x d4,e4 5,4 (x−β25,1 )s4vs44,m4 which is nonzero iff d4 ≤ e4 and b4 ≤ j4 + c4. Therefore, the sum in (4.37) is a linear combination of the vectors x e′2,f ′ 2 4,5 x a2,b2 1,2 (x−β24,1 )s2vs22,m2 ⊗ x d4,e4 5,4 x b′4,c ′ 4 2,1 (x−β25,1 )s4vs44,m4 ⊗ ⊗ v6 ⊗ x b1,c ′ 1 2,1 v1,m1 ⊗ x e5,f ′ 5 4,5 v5,m5 (4.39) where c′1 = c1 + a1, b′4 = b4 + j2 − j4, c′4 = c4 + a4 + j2, 110 Graded limits of minimal affinizations and beyond f ′5 = f5 + d5, e′2 = e2 + j1 − j3, f ′2 = f2 + d2 + j1, with the numbers al, bl, . . . , fl satisfying (4.38) as well as a1 + c1 ≤ m1, b1 ≤ c1, a2 ≤ b2, b4 ≤ c4 + j4, c2 = 0, (4.40) d5 + f5 ≤ m5, e5 ≤ f5, d4 ≤ e4, e2 ≤ f2 + j3, f4 = 0. Notice that a2 = a1 = b1 = b4 = c1 = 0, a4 = p1, b2 = p2, c4 = k2 − j2 − p1, d4 = d5 = e5 = e2 = f5 = 0, d2 = p5, e4 = p4, f2 = k1 − j1 − p5, satisfy (4.38) and (4.40), which implies that the set of nonzero summands in (4.37) is nonempty. One easily sees that the vectors in (4.39), for distinct values of (a2, b1, b2, b ′ 4, c ′ 1, c ′ 4, d4, e ′ 2, e4, e5, f ′ 2, f ′ 5), are linearly independent by looking at the weights of their tensor factors. Since v j ,p is a linear combination of these vectors with positive rational coefficients, it follows that v j ,p 6= 0 for all choices of j and p. We now restrict ourselves to λ as in Theorem 3.14. To simplify notation, we rewrite the vectors in (4.39) as v a2,b2,e ′ 2,f ′ 2 2 ⊗ v b′4,c ′ 4,d4,e4 4 ⊗ v6 ⊗ v b1,c ′ 1 1 ⊗ v e5,f ′ 5 5 . (4.41) If {2, 4} * supp(λ), the argument reduces to one identical to the one used in the proof of [22, Proposition 5.7] (all the details can be found in [24, Lemma 5.3.9]). From now on we assume supp(λ) ⊆ {2, 4, 6} which is the remaining case to consider. In this case, we must have j3 = j4 = k3 = k4 = 0, p2 = l−, p4 = l+. In particular, (4.38) and (4.40) reduce to a1 = b1 = c1 = c2 = 0, a2 + a4 = p1, a2 ≤ b2, b2 + b4 = k2 − j2, c4 = k2 − j2 − p1, b4 ≤ c4, d5 = e5 = f5 = f4 = 0, d4 + d2 = p5, d4 ≤ e4, f2 = k1 − j1 − p5, e4 + e2 = k1 − j1, e2 ≤ f2. Therefore, v j ,p is a linear combination of vectors of the form va2,b2,k1−e4,k1−d4 2 ⊗ vk2−b2,k2−a2,d4,e4 4 ⊗ v6 with 0 ≤ a2 ≤ p1 ≤ b2 ≤ l−, 0 ≤ d4 ≤ p5 ≤ e4 ≤ l+. (4.42) Set va,b,d,e = va,b,k1−e,k1−d 2 ⊗ vk2−b,k2−a,d,e 4 ⊗ v6 (4.43) A. Moura, F. Pereira 111 and observe that the coefficient of va,b,d,e in v j ,p is nonzero iff j1 ≤ k1 − e, j2 ≤ k2 − b, a ≤ p1, d ≤ p5. To complete the proof, we now show by induction on n1 ∈ Z≥0 that the set {v j ,p : (k1 − j1) ≤ n1} is linearly independent. We prove this performing a further induction on n2 ∈ Z≥0 to show that the set {v j ,p : (k1 − j1) ≤ n1, (k2 − j2) ≤ n2} is linearly independent. Set S(n1, n2) = {(j,p) : k1 − j1 ≤ n1, k2 − j2 ≤ n2}, S[n1, n2) = {(j,p) : k1 − j1 = n1, k2 − j2 ≤ n2}, S(n1, n2] = {(j,p) : k1 − j1 ≤ n1, k2 − j2 = n2}, S[n1, n2] = {(j,p) : k1 − j1 = n1, k2 − j2 = n2}. The inductions clearly start when n1 = n2 = 0 since {v j ,p : (j,p) ∈ S(0, 0)} = {v k }. Assume now that n2 > 0 and, by induction hypothesis, that the set {v j ,p : (j,p) ∈ S(n1, n2 − 1)} is linearly independent. Let c j ,p ∈ C be such that ∑ (j,p)∈S(n1,n2) c j ,p v j ,p = 0. (4.44) By the induction hypothesis, it remains to show that c j ,p = 0 for all (j,p) ∈ S(n1, n2]. (4.45) Set S[n1, n2](m) = {(j,p) ∈ S[n1, n2] : (p1, p5) = (n2−r, n1−s), r+s ≤ m}. Observe that if (j,p) ∈ S(n1, n2) is such that the coefficient of vn2−r,n2,n1−s,n1 in v j ,p is nonzero, then (j,p) ∈ S[n1, n2] and (p1, p5) = (n2 − r′, n1 − s′), 0 ≤ r′ ≤ r, 0 ≤ s′ ≤ s. An easy induction on r + s ≥ 0 shows that c j ,p = 0 for all (j,p) ∈ S[n1, n2](r+ s). This implies c j ,p = 0 for all (j,p) ∈ S[n1, n2]. Similarly, if (j,p) ∈ S(n1, n2)\S[n1, n2] is such that the coefficient of vn2−r,n2,n1−1−s,n1−1 in v j ,p is nonzero, then (j,p) ∈ S[n1 − 1, n2] and (p1, p5) = (n2 − r′, n1 − 1− s′), 0 ≤ r′ ≤ r, 0 ≤ s′ ≤ s. Again, an easy induction on r + s ≥ 0 shows that c j ,p = 0 for all (j,p) ∈ S[n1 − 1, n2](r + s). Proceeding recursively in this way one proves c j ,p = 0 for all (j,p) ∈ S[n1 − j, n2], 0 ≤ j ≤ n1. Since S(n1, n2] = ∪jS[n1 − j, n2], (4.45) follows. 112 Graded limits of minimal affinizations and beyond The above paragraph proves the induction step on n2. It remains to show that the induction on n2 starts when n1 > 0. Thus, assume n1 > 0, n2 = 0 and, by induction hypothesis on n1, that {v j ,p : (j,p) ∈ S(n1 − 1, 0)} is linearly independent. Let c j ,p ∈ C be such that ∑ (j,p)∈S(n1,0) c j ,p v j ,p = 0. (4.46) By the induction hypothesis, it remains to show that c j ,p = 0 for all (j,p) ∈ S[n1, 0]. (4.47) The proof of (4.47) is similar to that of (4.45) and we omit the details. Remark. Observe that the above proof of (4.45) is based on finding values of a, b, d, e such that va,b,d,e appears with nonzero coefficient in v j ,p for exactly one value of of the pair (j,p) ∈ S(n1, n2) and so on. The difficult in adapting the above proof for proving (4.34) for all λ not supported in the trivalent node resides in the fact that, if {2, 4} ⊆ supp(λ) and either m1 6= 0 or m5 6= 0, one can give examples of (j,p) 6= (j′,p′) such that the summands of the form va,b,d,e with nonzero coefficients appearing in v j ,p are exactly the same as those appearing in v j ′ ,p′ . Hence, one would need to keep a very efficient control of the coefficients. 4.8. Proof of (4.21) By (4.19), in order to prove that r j ∈ A(s) ⇒ j ∈ B(s), it remains to show that xr j vs 6= 0 only if j3 ≤ j1, j4 ≤ j2, and j5 ≥ 0. It follows from Lemma 4.11 that xr j vs = (x−β30,1 )s6+j5(x−β28,1 )j0 ( (x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 ⊗ ⊗(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 ⊗ vs66,m6 ⊗ w ) Notice that if s2 − j1 + j3 > s2 we have (x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 = 0 by (4.17). In other words, r j ∈ A(s) only if j3 ≤ j1. Similarly, we must have j4 ≤ j2. Continuing the above computation we get that xr j vs = (x−β30,1 )s6+j5v′ where v′ is the vector A. Moura, F. Pereira 113 j0∑ k=0 ( j0 k ) (x−β28,1 )j0−k(x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 ⊗ ⊗ (x−β28,1 )k(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 ⊗ vs66,m6 ⊗ w. By (4.17), (x−β28,1 )j0−k(x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 = 0 if (j0 − k) + j3 + (s2 − j1) > s2. Hence, the summand corresponding to k in the above summation is nonzero only if j2 − j4 − j5 ≤ k. Similarly, (x−β28,1 )k(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 = 0 if k + j4 + (s4 − j2) > s4, i.e., if k > j2 − j4. Thus, the summand corresponding to k in the above sum- mation is nonzero only if j2− j4− j5 ≤ k ≤ j2− j4. In particular, we must have j5 ≥ 0. To complete the proof of (4.21), we need to show that j ∈ B(s) ⇒ xr j vs 6= 0. Set j− = max{0, j2 − j4 − j5} and j+ = min{j0, j2 − j4} and observe that j ∈ B(s) ⇒ j− ≤ j+. Given j− ≤ k ≤ j+, set vk = ( j0 k ) (x−β28,1 )j0−k(x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 ⊗ ⊗ (x−β28,1 )k(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 . Notice that Lemmas 4.12 and 4.13 imply that vk 6= 0. Continuing the above computation we see that xr j vs = s6+j5∑ l=0 j+∑ k=j− ( s6+j5 l ) (x−β30,1 )s6+j5−lvk ⊗ (x−β30,1 )lvs66,m6 ⊗ w = = ( s6+j5 s6 ) j+∑ k=j− (x−β30,1 )j5vk ⊗ (x−β30,1 )s6vs66,m6 ⊗ w. The second equality above is proved as follows. By (4.17), (x−β30,1 )s6+j5−lvk = 0 if (s6 + j5 − l) + (j0 − k) + j3 + (s2 − j1) + k + j4+(s4− j2) > s2+ s4, i.e., if l < s6. Similarly, (x−β30,1 )lvs66,m6 = 0 if l > s6. By (4.15), (x−β30,1 )s6vs66,m6 6= 0 and, therefore, it remains to show that (x−β30,1 )j5 j+∑ k=j− vk 6= 0. (4.48) Indeed, ( j0 k )−1 (x−β30,1 )j5vk is equal to j5∑ l=0 ( j5 l ) (x−β30,1 )l(x−β28,1 )j0−k(x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 ⊗ 114 Graded limits of minimal affinizations and beyond ⊗ (x−β30,1 )j5−l(x−β28,1 )k(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 . Making use of (4.17) once more we see that (x−β30,1 )j5vk = ( j0 k )( j5 k−j2+j4+j5 ) vk2 ⊗ vk4 where vk2 = (x−β30,1 )k−j2+j4+j5(x−β28,1 )j0−k(x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 and vk4 = (x−β30,1 )j2−j4−k(x−β28,1 )k(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 . Lemma 4.12 implies that vk2 6= 0 while Lemma 4.13 implies that vk4 6= 0. Observing that vk2 are weight vectors of distinct weight and similarly for vk4 , (4.48) follows. This completes the proof of (4.21). Notice also that, if j5 = 0, it follows from the computations above that xr j vs is a nonzero scalar multiple of (x−β28,1 )j1−j3(x−β26,1 )j3(x−β24,1 )s2−j1vs22,m2 ⊗ ⊗ (x−β28,1 )j2−j4(x−β27,1 )j4(x−β25,1 )s4−j2vs44,m4 ⊗ (x−β30,1 )s6vs66,m6 ⊗ w. References [1] V. Chari, Minimal affinizations of representations of quantum groups: the rank-2 case, Publ. Res. Inst. Math. Sci. 31 (1995), 873–911. [2] , On the fermionic formula and the Kirillov-Reshetikhin conjecture, Int. Math. Res. Not. 2001 (2001), 629–654. [3] , Braid group actions and tensor products, Internat. Math. Res. Notices 2002 (2002), 357–382. [4] V. Chari and J. Greenstein, A family of Koszul algebras arising from finite- dimensional representations of simple Lie algebras, Adv. Math. 220 (2009), no. 4, 1193–1221. [5] , Minimal affinizations as projective objects, Journal of Geometry and Physics, Vol. 61, No. 9., 1717–1732. doi:10.1016/j.geomphys.2011.06.011. [6] V. Chari and D. Hernandez, Beyond Kirillov-Reshetikhin modules, Contemp. Math. 506 (2010), 49–81. [7] V. Chari and M. Kleber, Symmetric Functions and Representations of Quantum Affine Algebras, Contemp. Math. 297 (2002), 27–45. [8] V. Chari and A. Moura, Characters and blocks for finite-dimensional represen- tations of quantum affine algebras, Internat. Math Res. Notices 2005 (2005), 257–298. [9] , The restricted Kirillov-Reshetikhin modules for the current and twisted current algebras, Comm. Math. Phys. 266 (2006), 431–454. A. Moura, F. Pereira 115 [10] , Kirillov–Reshetikhin modules associated to G2, Contemp. Math. 442 (2007) 41–59. [11] V. Chari and A. Pressley, A guide to quantum groups, Cambridge University Press (1994). [12] , Minimal affinizations of representations of quantum groups: the simply laced case, J. of Algebra 184 (1996), 1–30. [13] , Weyl modules for classical and quantum affine algebras, Represent. Theory 5 (2001), 191–223. [14] E. Frenkel and E. Mukhin, Combinatorics of q-characters of finite-dimensional representations of quantum affine algebras, Comm. Math. Phys. 216 (2001), 23–57. [15] E. Frenkel and N. Reshetikhin, The q-characters of representations of quantum affine algebras and deformations of W-algebras, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), Contemp. Math. 248 (1999), 163–205. [16] G. Hatayama, A. Kuniba, M. Okado, T. Takagi, and Y. Yamada, Remarks on the Fermionic Formula. Contemp. Math. 248 (1999), 243–291. [17] D. Hernandez, The Kirillov-Reshetikhin conjecture and solutions of T-systems, J. Reine Angew. Math. 596 (2006), 63–87. [18] , Kirillov-Reshetikhin conjecture : the general case, Int. Math. Res. Not. 2010 (2010), no. 1, 149–193. [19] J. Humphreys, Introduction to Lie algebras and representation theory, GTM9 Springer (1972). [20] D. Jakelić and A. Moura, Tensor products, characters, and blocks of finite- dimensional representations of quantum affine algebras at roots of unity, Int. Math. Res. Notices (2011) 2011 (18): 4147–4199. doi:10.1093/imrn/rnq250. [21] G. Lusztig, Introduction to quantum groups, Birkäuser (1993). [22] A. Moura, Restricted limits of minimal affinizations, Pacific J. Math 244 (2010), 359–397. [23] H. Nakajima, Quiver varieties and t-analogs of q-characters of quantum affine algebras, Ann. of Math. 160 (2004), 1057–1097. [24] F. Pereira, Caracteres de limites clбssicos de afinizaзхes minimais de tipo E6, M.Sc. Thesis, Unicamp (2010). Contact information A. Moura, F. Pereira Departamento de Matemática, Universidade Es- tadual de Campinas, Campinas - SP - Brazil, 13083-859 E-Mail: aamoura@ime.unicamp.br, fernandapereira@ime.unicamp.br Received by the editors: 20.08.2011 and in final form 02.10.2011.

Graded limits of minimal affinizations and beyond: the multiplicity free case for type E₆

Репозитарії

Схожі ресурси