Structure Constants of Diagonal Reduction Algebras of gl Type

We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra gln into gln⊕gln. Its representation theory is related to the theory of decompositions of tensor products of gln-modules.

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Datum:	2011
Hauptverfasser:	Khoroshkin, S., Ogievetsky, O.
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spelling	irk-123456789-1471862019-02-14T01:25:36Z Structure Constants of Diagonal Reduction Algebras of gl Type Khoroshkin, S. Ogievetsky, O. We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra gln into gln⊕gln. Its representation theory is related to the theory of decompositions of tensor products of gln-modules. 2011 Article Structure Constants of Diagonal Reduction Algebras of gl Type / S.Khoroshkin, O. Ogievetsky // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 10 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16S30; 17B35 DOI:10.3842/SIGMA.2011.064 http://dspace.nbuv.gov.ua/handle/123456789/147186 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra gln into gln⊕gln. Its representation theory is related to the theory of decompositions of tensor products of gln-modules.
format	Article
author	Khoroshkin, S. Ogievetsky, O.
spellingShingle	Khoroshkin, S. Ogievetsky, O. Structure Constants of Diagonal Reduction Algebras of gl Type Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Khoroshkin, S. Ogievetsky, O.
author_sort	Khoroshkin, S.
title	Structure Constants of Diagonal Reduction Algebras of gl Type
title_short	Structure Constants of Diagonal Reduction Algebras of gl Type
title_full	Structure Constants of Diagonal Reduction Algebras of gl Type
title_fullStr	Structure Constants of Diagonal Reduction Algebras of gl Type
title_full_unstemmed	Structure Constants of Diagonal Reduction Algebras of gl Type
title_sort	structure constants of diagonal reduction algebras of gl type
publisher	Інститут математики НАН України
publishDate	2011
url	http://dspace.nbuv.gov.ua/handle/123456789/147186
citation_txt	Structure Constants of Diagonal Reduction Algebras of gl Type / S.Khoroshkin, O. Ogievetsky // Symmetry, Integrability and Geometry: Methods and Applications. — 2011. — Т. 7. — Бібліогр.: 10 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT khoroshkins structureconstantsofdiagonalreductionalgebrasofgltype AT ogievetskyo structureconstantsofdiagonalreductionalgebrasofgltype
first_indexed	2025-07-11T01:34:04Z
last_indexed	2025-07-11T01:34:04Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 7 (2011), 064, 34 pages Structure Constants of Diagonal Reduction Algebras of gl Type Sergei KHOROSHKIN a,b and Oleg OGIEVETSKY c,d,e a) Institute of Theoretical and Experimental Physics, 117218 Moscow, Russia E-mail: khor@itep.ru b) Higher School of Economics, 20 Myasnitskaya Str., 101000 Moscow, Russia c) J.-V. Poncelet French-Russian Laboratory, UMI 2615 du CNRS, Independent University of Moscow, 11 B. Vlasievski per., 119002 Moscow, Russia E-mail: oleg.ogievetsky@gmail.com d) Centre de Physique Théorique1, Luminy, 13288 Marseille, France e) On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, 53 Leninsky Prospekt, 119991 Moscow, Russia Received January 14, 2011, in final form June 27, 2011; Published online July 09, 2011 doi:10.3842/SIGMA.2011.064 Abstract. We describe, in terms of generators and relations, the reduction algebra, related to the diagonal embedding of the Lie algebra gln into gln ⊕ gln. Its representation theory is related to the theory of decompositions of tensor products of gln-modules. Key words: reduction algebra; extremal projector; Zhelobenko operators 2010 Mathematics Subject Classification: 16S30; 17B35 Contents 1 Introduction 2 2 Notation 4 3 Reduction algebra Zn 5 4 Main results 7 4.1 New variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.2 Braid group action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4.3 Defining relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 4.4 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.5 sln . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4.6 Stabilization and cutting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5 Proofs 17 5.1 Tensor J . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 5.2 Braid group action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.3 Derivation of relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.4 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 6 Examples: sl3 and sl2 27 References 34 1Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud Toulon – Var; Laboratoire Affilié à la FRUMAM (FR 2291) mailto:khor@itep.ru mailto:oleg.ogievetsky@gmail.com http://dx.doi.org/10.3842/SIGMA.2011.064 2 S. Khoroshkin and O. Ogievetsky 1 Introduction This paper completes the work [7]: it contains a derivation of basic relations for the diagonal reduction algebras of gl type, their low dimensional examples and properties. Let g be a Lie algebra, k ⊂ g its reductive Lie subalgebra and V an irreducible finite- dimensional g-module, which decomposes, as an k-module, into a direct sum of irreducible k- modules Vi with certain multiplicities mi, V ≈ ∑ i Vi ⊗Wi. (1.1) Here Wi = Homk(Vi, V ) are the spaces of multiplicities, mi = dimWi. While the multiplici- ties mi present certain combinatorial data, the spaces Wi of multiplicities itself may exhibit a ‘hidden structure’ of modules over certain special algebras [4]. The well-known example is the Olshanski centralizer construction [9], where g = gln+m, k = glm and the spaces Wi carry the structure of irreducible Yangian Y (gln)-modules. In general, the multiplicity spaces Wi are irreducible modules over the centralizer U(g)k of k in the universal enveloping algebra U(g) [8]. However, these centralizers have a rather complicated algebraic structure and are hardly convenient for applications. Besides, under the above assumptions, the direct sum W = ⊕iWi becomes a module over the reduction (or Mickelsson) algebra. The reduction algebra is defined as follows. Suppose k is given with a triangular decomposition k = n− + h + n. (1.2) Denote by I+ the left ideal of A := U(g), generated by elements of n, I+ := An . Then the reduction algebra Sn(A), related to the pair (g, k), is defined as the quotient Norm(I+)/I+ of the normalizer of the ideal I+ over I+. It is equipped with a natural structure of the associative algebra. By definition, for any g-module V the space V n of vectors, annihilated by n, is a module over Sn(A). Since V is finite-dimensional, V n is isomorphic to ⊕iWi, so the latter space can be viewed as an Sn(A)-module as well; the zero-weight component of Sn(A), which contains a quotient of the centralizer U(g)k, preserves each multiplicity space Wi. The representation theory of the reduction algebra Sn(A) is closely related to the theory of branching rules g ↓ k for the restrictions of representations of g to k. The reduction algebra simplifies after the localization over the multiplicative set generated by elements hγ + k, where γ ranges through the set of roots of k, k ∈ Z, and hγ is the coroot corresponding to γ. Let U(h) be the localization of the universal enveloping algebra U(h) of the Cartan subalgebra h of k over the above multiplicative set. The localized reduction algebra Zn(A) is an algebra over the commutative ring U(h); the principal part of the defining relations is quadratic but the relations may contain linear or degree 0 terms, see [10, 6]. Besides, the reduction algebra admits another description as a (localized) double coset space n−A\A/An endowed with the multiplication map defined by means of the insertion of the extremal projector [6] of Asherova–Smirnov–Tolstoy [3]. The centralizer Ak is a subalgebra of Zn(A). It was shown in [7] that the general reduction algebra Zn(A) admits a presentation over U(h) such that the defining relations are ordering relations for the generators, in an arbitrary order, compatible with the natural partial order on h∗. The set of ordering relations for the general reduction algebra Zn(A) was shown in [7] to be “algorithmically efficient” in the sense that any expression in the algebra can be ordered with the help of this set. The structure constants of the reduction algebra are in principle determined with the help of the extremal projector P or the tensor J studied by Arnaudon, Buffenoir, Ragoucy and Roche [1]. However the explicit description of the algebra is hardly achievable directly. Structure Constants of Diagonal Reduction Algebras of gl Type 3 In the present paper, we are interested in the special restriction problem, when g is the direct sum of two copies of a reductive Lie algebra a and k is the diagonally embedded a. The resulting reduction algebra for the symmetric pair (a ⊕ a, a) we call diagonal reduction algebra DR(a) of a. The theory of branching rules for a ⊕ a ↓ a is the theory of decompositions of the tensor products of a-modules into a direct sum of irreducible a-modules. We restrict ourselves here to the Lie algebra a = gln of the general linear group. In this situation finite-dimensional irreducible modules over g are tensor products of two irreducible gln- modules, the decomposition (1.1) is the decomposition of the tensor product into the direct sum of irreducible gln-modules, and the multiplicities mi are the Littlewood–Richardson coefficients. The reduction algebra DR(gln) for brevity will be denoted further by Zn. In [7] we suggested a set R of relations for the algebra Zn. We demonstrated that the set R is equivalent, over U(h), to the set of the defining ordering relations provided that all relations from the set R are valid. The main goal of the present paper is the verification of all relations from the system R. There are two principal tools in our derivation. First, we use the braid group action by the Zhelobenko automorphisms of reduction algebras [10, 6]. Second, we employ the stabilization phenomenon, discovered in [7], for the multiplication rule and for the defining relations with respect to the standard embeddings gln ↪→ gln+1; stabilization provides a natural way of extending relations for Zn to relations for Zn+1 (Zn is not a subalgebra of Zn+1). We perform necessary calculations for low n (at most n = 4); the braid group action and the stabilization law allow to extend the results for general n. As an illustration, we write down the complete lists of defining relations in the form of ordering relations for the reduction algebras DR(sl3) and DR(sl2). Although for a finite n the task of deriving the set of defining (ordering) relations for DR(sln) is achievable in a finite time, it is useful to have the list of relations for small n in front of the eyes. We return to the stabilization and cut phenomena and make more precise statements con- cerning now the embedding of the Lie algebra gln⊕gl1 into the Lie algebra gln+1 (more generally, of gln⊕glm into gln+m). As a consequence we find that cutting preserves the centrality: the cut of a central element of the algebra Zn+m is central in the algebra Zn ⊗ Zm. We also show that, similarly to the Harish-Chandra map, the restriction of the cutting to the center is a homomor- phism. As an example, we derive the Casimir operators for the algebra DR(sl2) by cutting the Casimir operators for the algebra DR(sl3). The relations in the diagonal reduction algebra have a quadratic and a degree zero part. The algebra, defined by the homogeneous quadratic part of the relations, tends, in a quite simple regime, to a commutative algebra (the homogeneous algebra can be thus considered as a “dynamical” deformation of a commutative algebra; “dynamical” here means that the left and right multiplications by elements of the ring U(h) differ). This observation about the limit is used in the proof in [7] of the completeness of the set of derived relations over the field of fractions of U(h). We prove the completeness by establishing the equivalence between the set of derived relations and the set of ordering relations. The stabilization law enables one to give a definition of the reduction “algebra” Z∞ related to the diagonal embedding of the inductive limit gl∞ of gln into gl∞ ⊕ gl∞ (strictly speaking, Z∞ is not an algebra, some relations have an infinite number of terms). We also discuss the diagonal reduction algebra for the special linear Lie algebra sln; it is a direct tensor factor in Zn. Such a precise description, as the one we give for Zn, is known for a few examples of the reduction algebras: the most known is related to the embedding of gln to gln+1 [10]. Its repre- sentation theory was used for the derivation of precise formulas for the action of the generators of gln on the Gelfand–Zetlin basic vectors [2]. The reduction algebra for the pair (gln, gln+1) is based on the root embedding gln ⊂ gln+1 of Lie algebras. In contrast to this example, the 4 S. Khoroshkin and O. Ogievetsky diagonal reduction algebra DR(a) is based on the diagonal embedding of a into a⊕ a, which is not a root embedding of reductive Lie algebras. 2 Notation Let Eij , i, j = 1 , . . . , n, be the standard generators of the Lie algebra gln, with the commutation relations [Eij , Ekl] = δjkEil − δilEkj , where δjk is the Kronecker symbol. We shall also use the root notation Hα, Eα, E−α, . . . for elements of gln. Let E(1) ij and E(2) ij , i, j = 1 , . . . , n, be the standard generators of the two copies of the Lie algebra gln in g := gln ⊕ gln, [E(a) ij , E (b) kl ] = δab ( δjkE (a) il − δilE (a) kj ) . Set eij := E(1) ij + E(2) ij , Eij := E(1) ij − E (2) ij . The elements eij span the diagonally embedded Lie algebra k ' gln, while Eij form an adjoint k-module p. The Lie algebra k and the space p constitute a symmetric pair, that is, [k, k] ⊂ k, [k, p] ⊂ p, and [p, p] ⊂ k: [eij , ekl] = δjkeil − δilekj , [eij , Ekl] = δjkEil − δilEkj , [Eij , Ekl] = δjkeil − δilekj . In the sequel, ha means the element eaa of the Cartan subalgebra h of the subalgebra k ∈ gln⊕gln and hab the element eaa − ebb. Let {εa} be the basis of h∗ dual to the basis {ha} of h, εa(hb) = δab. We shall use as well the root notation hα, eα, e−α for elements of k, and Hα, Eα, E−α for elements of p. The Lie subalgebra n in the triangular decomposition (1.2) is spanned by the root vectors eij with i < j and the Lie subalgebra n− by the root vectors eij with i > j. Let b+ and b− be the corresponding Borel subalgebras, b+ = h ⊕ n and b− = h ⊕ n−. Denote by ∆+ and ∆− the sets of positive and negative roots in the root system ∆ = ∆+ ∪ ∆− of k: ∆+ consists of roots εi − εj with i < j and ∆− consists of roots εi − εj with i > j. Let Q be the root lattice, Q := {γ ∈ h∗ \| γ = ∑ α∈∆+,nα∈Z nαα}. It contains the positive cone Q+, Q+ := { γ ∈ h∗ \| γ = ∑ α∈∆+,nα∈Z, nα≥0 nαα } . For λ, µ ∈ h∗, the notation λ > µ (2.1) means that the difference λ− µ belongs to Q+, λ− µ ∈ Q+. This is a partial order in h∗. We fix the following action of the cover of the symmetric group Sn (the Weyl group of the diagonal k) on the Lie algebra gln ⊕ gln by automorphisms σ́i(x) := Adexp(ei,i+1)Adexp(−ei+1,i)Adexp(ei,i+1)(x), so that σ́i(ekl) = (−1)δik+δileσi(k)σi(l), σ́i(Ekl) = (−1)δik+δilEσi(k)σi(l). Structure Constants of Diagonal Reduction Algebras of gl Type 5 Here σi = (i, i+ 1) is an elementary transposition in the symmetric group. We extend naturally the above action of the cover of Sn to the action by automorphisms on the associative algebra A ≡ An := U(gln)⊗U(gln). The restriction of this action to h coincides with the natural action σ(hk) = hσ(k), σ ∈ Sn, of the Weyl group on the Cartan subalgebra. Besides, we use the shifted action of Sn on the polynomial algebra U(h) (and its localizations) by automorphisms; the shifted action is defined by σ ◦ hk := hσ(k) + k − σ(k), k = 1, . . . , n; σ ∈ Sn. (2.2) It becomes the usual action for the variables h̊k := hk − k, h̊ij := h̊i − h̊j ; (2.3) by (2.2) for any σ ∈ Sn we have σ ◦ h̊k = h̊σ(k), σ ◦ h̊ij = h̊σ(i)σ(j). It will be sometimes convenient to denote the commutator [a, b] of two elements a and b of an associative algebra by â(b) := [a, b]. (2.4) 3 Reduction algebra Zn In this section we recall the definition of the reduction algebras, in particular the diagonal reduction algebras of the gl type. We introduce the order for which the ordering relations for the algebra Zn will be discussed. The formulas for the Zhelobenko automorphisms for the algebra Zn are given; some basic facts about the standard involution, anti-involution and central elements for the algebra Zn are presented at the end of the section. 1. Let U(h) and Ā be the rings of fractions of the algebras U(h) and A with respect to the multiplicative set, generated by elements hij + l, l ∈ Z, 1 ≤ i < j ≤ n. Define Zn to be the double coset space of Ā by its left ideal I+ := Ān, generated by elements of n, and the right ideal I− := n−Ā, generated by elements of n−, Zn := Ā/(I+ + I− ). The space Zn is an associative algebra with respect to the multiplication map a � b := aPb. (3.1) Here P is the extremal projector [3] for the diagonal gln. It is an element of a certain extension of the algebra U(gln) satisfying the relations eijP = Peji = 0 for all i and j such that 1 ≤ i < j ≤ n. The algebra Zn is a particular example of a reduction algebra; in our context, Zn is defined by the coproduct (the diagonal inclusion) U(gln)→ A. 2. The main structure theorems for the reduction algebras are given in [7, Section 2]. In the sequel we choose a weight linear basis {pK} of p (p is the k-invariant complement to k in g, g = k+p) and equip it with a total order ≺. The total order ≺ will be compatible with the partial order < on h∗, see (2.1), in the sense that µK < µL ⇒ pK ≺ pL. We shall sometimes write I ≺ J instead of pI ≺ pJ . For an arbitrary element a ∈ Ā let ã be its image in the reduction algebra; in particular, p̃K is the image in the reduction algebra of the basic vector pK ∈ p. 6 S. Khoroshkin and O. Ogievetsky 3. In our situation we choose the set of vectors Eij , i, j = 1, . . . , n, as a basis of the space p. The weight of Eij is εi− εj . The compatibility of a total order ≺ with the partial order < on h∗ means the condition Eij ≺ Ekl if i− j > k − l. The order in each subset {Eij \|i− j = a} with a fixed a can be chosen arbitrarily. For instance, we can set Eij ≺ Ekl if i− j > k − l or i− j = k − l and i > k. (3.2) Denote the images of the elements Eij in Zn by zij . We use also the notation ti for the elements zii and tij := ti− tj for the elements zii− zjj . The order (3.2) induces as well the order on the generators zij of the algebra Zn: zij ≺ zkl ⇔ Eij ≺ Ekl. The statement (d) in the paper [7, Section 2] implies an existence of structure constants B(ab),(cd),(ij),(kl) ∈ U(h) and D(ab),(cd) ∈ U(h) such that for any a, b, c, d = 1, . . . , n we have zab � zcd = ∑ i,j,k,l:zij�zkl B(ab),(cd),(ij),(kl)zij � zkl + D(ab),(cd). (3.3) In particular, the algebra Zn (in general, the reduction algebra related to a symmetric pair (k, p), g := k + p) is Z2-graded; the degree of zab is 1 and the degree of any element from U(h) is 0. The relations (3.3) together with the weight conditions [h, zab] = (εa − εb)(h)zab are the defining relations for the algebra Zn. Note that the denominators of the structure constants B(ab),(cd),(ij),(kl) and D(ab),(cd) are pro- ducts of linear factors of the form h̊ij + `, i < j, where ` ≥ −1 is an integer, see [7]. 4. The algebra Zn can be equipped with the action of Zhelobenko automorphisms [6]. Denote by q̌i the Zhelobenko automorphism q̌i : Zn → Zn corresponding to the transposition σi ∈ Sn. It is defined as follows [6]. First we define a map q̌i : A→ Ā/I+ by q̌i(x) := ∑ k≥0 (−1)k k! êki,i+1(σ́i(x))eki+1,i k∏ a=1 (hi,i+1 − a+ 1)−1 mod I+ . (3.4) Here êi,i+1 stands for the adjoint action of the element ei,i+1, see (2.4). The operator q̌i has the property q̌i(hx) = (σi ◦ h)q̌i(x) (3.5) for any x ∈ A and h ∈ h; σ ◦ h is defined in (2.2). With the help of (3.5), the map q̌i can be extended to the map (denoted by the same symbol) q̌i : Ā → Ā/I− by setting q̌i(a(h)x) = (σi ◦ a(h))q̌i(x) for any x ∈ A and a(h) ∈ U(h). One can further prove that q̌i(I+) = 0 and q̌i(I− ) ⊂ (I− + I+)/I+ , so that q̌i can be viewed as a linear operator q̌i : Zn → Zn. Due to [6], this is an algebra automorphism, satisfying (3.5). The operators q̌i satisfy the braid group relations [10]: q̌iq̌i+1q̌i = q̌i+1q̌iq̌i+1, q̌iq̌j = q̌j q̌i, \|i− j\| > 1, Structure Constants of Diagonal Reduction Algebras of gl Type 7 and the inversion relation [6]: q̌2 i (x) = 1 hi,i+1 + 1 σ́2 i (x)(hi,i+1 + 1), x ∈ Zn. (3.6) In particular, q̌2 i (x) = x if x is of zero weight. 5. The Chevalley anti-involution ε in U(gln ⊕ gln), ε(eij) := eji, ε(Eij) := Eji, induces the anti-involution ε in the algebra Zn: ε(zij) = zji, ε(hk) = hk. (3.7) Besides, the outer automorphism of the Dynkin diagram of gln induces the involutive automor- phism ω of Zn, ω(zij) = (−1)i+j+1zj′i′ , ω(hk) = −hk′ , (3.8) where i′ = n+ 1− i. The operations ε and ω commute, εω = ωε. Central elements of the subalgebra U(gln) ⊗ 1 ⊂ A, generated by n Casimir operators of degrees 1 , . . . , n, as well as central elements of the subalgebra 1⊗U(gln) ⊂ A project to central elements of the algebra Zn. In particular, central elements of degree 1 project to central elements I(n,h) := h1 + · · ·+ hn (3.9) and I(n,t) := t1 + · · ·+ tn (3.10) of the algebra Zn. The difference of central elements of degree two projects to the central element n∑ i=1 (hi − 2i)ti (3.11) of the algebra Zn. The images of other Casimir operators are more complicated. 4 Main results This section contains the principal results of the paper. We first give preliminary information on the new basis in which the defining relations for the algebra Zn can be written down in an economical fashion. The braid group action on the new generators is then explicitly given in Subsection 4.2. The complete set of the defining relations for the algebra Zn is written down in Subsection 4.3. The regime for which both the set of the derived defining relations and the set of the defining ordering relation have a controllable “limiting behavior” is introduced in Subsection 4.4. Subsection 4.5 deals with the diagonal reduction algebra for sln; the quadratic Casimir operator for DR(sln) as well as for the diagonal reduction algebra for an arbitrary semi-simple Lie algebra k is given there. Subsection 4.6 is devoted to the stabilization and cut phenomena with respect to the embedding of the Lie algebra gln⊕glm into the Lie algebra gln+m; the theorem about the behavior of the centers of the diagonal reduction algebra under the cutting is proved there. 8 S. Khoroshkin and O. Ogievetsky 4.1 New variables We shall use the following elements of U(h): Aij := h̊ij h̊ij − 1 , A′ij := h̊ij − 1 h̊ij , Bij := h̊ij − 1 h̊ij − 2 , B′ij := h̊ij − 2 h̊ij − 1 , C ′ij := h̊ij − 3 h̊ij − 2 , the variables h̊ij are defined in (2.3). Note that AijA ′ ij = BijB ′ ij = 1. Define elements t̊1, . . . , t̊n ∈ Zn by t̊1 := t1, t̊2 := q̌1(t1), t̊3 := q̌2q̌1(t1), . . . , t̊n := q̌n−1 · · · q̌2q̌1(t1). Using (3.4) we find the relations q̌i(ti) = − 1 h̊i,i+1 − 1 ti + h̊i,i+1 h̊i,i+1 − 1 ti+1, q̌i(ti+1) = h̊i,i+1 h̊i,i+1 − 1 ti − 1 h̊i,i+1 − 1 ti+1, q̌i(tk) = tk, k 6= i, i+ 1, (4.1) which can be used to convert the definition (4.1) into a linear over the ring U(h) change of variables: t̊l = tl l−1∏ j=1 Ajl − l−1∑ k=1 tk 1 h̊kl − 1 k−1∏ j=1 Ajl, tl = t̊l l−1∏ j=1 A′jl + l−1∑ k=1 t̊k 1 h̊kl l−1∏ j=1 j 6=k A′jk. (4.2) For example, t̊2 = − 1 h̊12 − 1 t1 + h̊12 h̊12 − 1 t2, t2 = 1 h̊12 t̊1 + h̊12 − 1 h̊12 t̊2, t̊3 = − 1 h̊13 − 1 t1 − h̊13 (̊h13 − 1)(̊h23 − 1) t2 + h̊13̊h23 (̊h13 − 1)(̊h23 − 1) t3, t3 = h̊12 + 1 h̊12̊h13 t̊1 + h̊12 − 1 h̊12̊h23 t̊2 + (̊h13 − 1)(̊h23 − 1) h̊13̊h23 t̊3. In terms of the new variables t̊’s, the linear in t central element (3.10) reads ∑ ti = ∑ t̊i ∏ a:a6=i h̊ia + 1 h̊ia . 4.2 Braid group action Since q̌2 i (x) = x for any element x of zero weight, the braid group acts as its symmetric group quotient on the space of weight 0 elements. It follows from (4.1) and q̌i(t1) = t1 for all i > 1 that q̌σ (̊ti) = t̊σ(i) (4.3) for any σ ∈ Sn. Structure Constants of Diagonal Reduction Algebras of gl Type 9 The action of the Zhelobenko automorphisms, see Section 3, on the generators zkl looks as follows: q̌i(zik) = −zi+1,kAi,i+1, q̌i(zki) = −zk,i+1, k 6= i, i+ 1, q̌i(zi+1,k) = zi,k, q̌i(zk,i+1) = zk,iAi,i+1, k 6= i, i+ 1, (4.4) q̌i(zi,i+1) = −zi+1,iAi,i+1Bi,i+1, q̌i(zi+1,i) = −zi,i+1, q̌i(zj,k) = zj,k, j, k 6= i, i+ 1. Denote i′ = n + 1 − i, as before. The braid group action (4.4) is compatible with the anti- involution ε and the involution ω (note that ω(̊hij) = h̊j′i′), see (3.7) and (3.8), in the following sense: εq̌i = q̌−1 i ε, (4.5) ωq̌i = q̌i′−1ω. (4.6) Let w0 be the longest element of the Weyl group of gln, the symmetric group Sn. Similarly to the squares of the transformations corresponding to the simple roots, see (3.6), the action of q̌2 w0 is the conjugation by a certain element of U(h). Lemma 1. We have q̌2 w0 (x) = S−1xS, (4.7) where S = ∏ i,j:i<j h̊ij . (4.8) The proof shows that the formula (4.7) works for an arbitrary reductive Lie algebra, with S = ∏ α∈∆+ h̊α. Proposition 2. The action of q̌w0 on generators reads q̌w0(zij) = (−1)i+jzi′j′ ∏ a:a<i′ Aai′ ∏ b:b>j′ Aj′b, (4.9) q̌w0 (̊ti) = t̊i′ . (4.10) The proofs of Lemma 1 and Proposition 2 are in Section 5. 4.3 Defining relations To save space we omit in this section the symbol � for the multiplication in the algebra Zn. It should not lead to any confusion since no other multiplication is used in this section. Each relation which we will derive will be of a certain weight, equal to a sum of two roots. From general considerations the upper estimate for the number of terms in a quadratic relation of weight λ = α+ β is the number \|λ\| of quadratic combinations zα′zβ′ with α′ + β′ = λ. There are several types, excluding the trivial one, λ = 2(εi − εj), \|λ\| = 1: 1. λ = ±(2εi − εj − εk), where i, j and k are pairwise distinct. Then \|λ\| = 2. 2. λ = εi − εj + εk − εl with pairwise distinct i, j, k and l. Then \|λ\| = 4. 3. λ = εi−εj , i 6= j. For zα′zβ′ , there are 2(n−2) possibilities (subtype 3a) with α′ = εi−εk, β′ = εk − εj or α′ = εk − εj , β′ = εi − εk with k 6= i, j and 2n possibilities (subtype 3b) with α′ = 0, β′ = εi − εj or α′ = εi − εj , β′ = 0. Thus \|λ\| = 4(n− 1). 10 S. Khoroshkin and O. Ogievetsky 4. λ = 0. There are n2 possibilities (subtype 4a) with α′ = 0, β′ = 0 and n(n−1) possibilities (subtype 4b) with α′ = εi − εj , β′ = εj − εi, i 6= j. Here \|λ\| = n(2n− 1). Below we write down relations for each type (and subtype) separately. The relations of the types 1 and 2 have a simple form in terms of the original generators zij . To write the relations of the types 3 and 4, it is convenient to renormalize the generators zij with i 6= j. Namely, we set z̊ij = zij i−1∏ k=1 Aki. (4.11) In terms of the generators z̊ij , the formulas (4.4) for the action of the automorphisms q̌i translate as follows: q̌i(̊zik) = −z̊i+1,k, q̌i(̊zi+1,k) = z̊i,kAi+1,i, k 6= i, i+ 1, q̌i(̊zki) = −z̊k,i+1, q̌i(̊zk,i+1) = z̊k,iAi,i+1 = A′i+1,iz̊k,i, k 6= i, i+ 1, q̌i(̊zi,i+1) = −A′i+1,iz̊i+1,i, q̌i(̊zi+1,i) = −z̊i,i+1Ai+1,i, q̌i(̊zj,k) = z̊j,k, j, k 6= i, i+ 1. 1. The relations of the type 1 are: zijzik = zikzijAkj , zjizki = zkizjiA ′ kj , for j < k, i 6= j, k. (4.12) 2. Denote Dijkl := ( 1 h̊ik − 1 h̊jl ) . Then, for any four pairwise different indices i, j, k and l, we have the following relations of the type 2: [zij , zkl] = zkjzilDijkl, i < k, j < l, zijzkl − zklzijA′jlA′lj = zkjzilDijkl, i < k, j > l. (4.13) 3a. Let i 6= k 6= l 6= i. Denote E̊ikl := − ( (̊ti − t̊k) h̊il + 1 h̊ikh̊il + (̊tk − t̊l) h̊il − 1 h̊kl̊hil ) z̊il + ∑ a:a6=i,k,l z̊alz̊ia Bai h̊ka + 1 . With this notation the first group of the relations of the type 3 is: z̊ikz̊klA ′ ik − z̊klz̊ikBki = E̊ikl, i < k < l, z̊ikz̊klA ′ ikA ′ lkBlk − z̊klz̊ikBki = E̊ikl, i < l < k, z̊ikz̊klAki − z̊klz̊ikBki = E̊ikl, k < i < l, (4.14) z̊ikz̊klAkiAliB ′ li − z̊klz̊ikBki = E̊ikl, k < l < i, z̊ikz̊klA ′ ikA ′ lkBlkAliB ′ li − z̊klz̊ikBki = E̊ikl, l < i < k, z̊ikz̊klAkiA ′ lkBlkAliB ′ li − z̊klz̊ikBki = E̊ikl, l < k < i. The relations (4.14) can be written in a more compact way with the help of both systems, zij and z̊ij , of generators. Let now Eikl := − ( (̊ti − t̊k) h̊il + 1 h̊ikh̊il + (̊tk − t̊l) h̊il − 1 h̊kl̊hil ) zil + ∑ a:a6=i,k,l z̊alzia Bai h̊ka + 1 . Structure Constants of Diagonal Reduction Algebras of gl Type 11 Then zikz̊klA ′ ik − z̊klzikBki = Eikl, k < l, zikz̊klA ′ ikA ′ lkBlk − z̊klzikBki = Eikl, l < k. (4.15) Moreover, after an extra redefinition: z̊kl˚ = z̊klBlk for k > l, the left hand side of the second line in (4.15) becomes, up to a common factor, the same as the left hand side of the first line, namely, it reads (zik z̊kl˚ A′ik − z̊kl˚ zikBki)A ′ lk. 3b. Let i 6= j 6= k 6= i. The second group of relations of the type 3 reads: z̊ij t̊i = t̊iz̊ijC ′ ji − t̊j z̊ij 1 h̊ij + 2 − ∑ a:a6=i,j z̊aj z̊ia 1 h̊ia + 2 , z̊ij t̊j = −t̊iz̊ij C ′ji h̊ij − 1 + t̊j z̊ijAijA ′ jiBji + ∑ a:a6=i,j z̊aj z̊iaAijA ′ jiBai̊hja + 1, (4.16) z̊ij t̊k = t̊iz̊ij (̊hij + 3)Bji (̊h2 ik − 1)(̊hjk − 1) + t̊j z̊ij (̊hij + 1)Bji (̊hik − 1)(̊hjk − 1)2 + t̊kz̊ijAikAkiAjkB ′ jk − z̊kj z̊ik (̊hij + 1)Bki (̊hik − 1)(̊hjk − 1) − ∑ a:a6=i,j,k z̊aj z̊ia h̊ij + 1 (̊hik − 1)(̊hjk − 1) Bai h̊ka + 1 . 4a. The relations of the weight zero (the type 4) are also divided into 2 groups. This is the first group of the relations: [̊ti, t̊j ] = 0. (4.17) As follows from the proof, the relations (4.17) hold for the diagonal reduction algebra for an arbitrary reductive Lie algebra: the images of the generators, corresponding to the Cartan subalgebra, commute. 4b. Finally, the second group of the relations of the type 4 is [̊zij , z̊ji] = h̊ij − 1 h̊ij (̊ti − t̊j)2 + ∑ a:a6=i,j ( 1 h̊ja + 1 z̊aiz̊ia − 1 h̊ia + 1 z̊aj z̊ja ) , (4.18) where i 6= j. Main statement. Denote by R the system (4.12), (4.13), (4.14), (4.16), (4.17) and (4.18) of the relations. Theorem 3. The relations R are the defining relations for the weight generators zij and ti of the algebra Zn. In particular, the set (3.3) of ordering relations follows over U(h) from (and is equivalent to) R. The derivation of the system R of the relations is given in Section 5. The validity in Zn of relations from the set R, together with the results from [7], completes the proof of Theorem 3 (Section 5.4). 4.4 Limit Let R≺ be the set of ordering relations (3.3). Denote by R0 the homogeneous (quadratic) part of the system R and by R≺0 the homogeneous part of the system R≺. 1. Placing coefficients from U(h) in all relations from R0 to the same side (to the right, for example) from the monomials p̃L � p̃M , one can give arbitrary numerical values to the variables hα (α’s are roots of k). 12 S. Khoroshkin and O. Ogievetsky The structure of the extremal projector P or the recurrence relation (5.4) implies that the system R0 admits, for an arbitrary reductive Lie algebra, the limit at hαi = cih, h → ∞ (αi ranges through the set of simple positive roots of k and ci are generic positive constants). Moreover, this homogeneous algebra becomes the usual commutative (polynomial) algebra in this limit; so this limiting behavior of the system R0, used in the proof, generalizes to a wider class of reduction algebras, related to a pair (g, k) as in the introduction. 2. The limiting procedure from paragraph 1 establishes the bijection between the set of relations and the set of unordered pairs (L,M), where L,M are indices of basic vectors of p. The proof in [7] shows that over D(h) the system R can be rewritten in the form of ordering relations for an arbitrary order on the set { p̃L} of generators. Here D(h) is the field of fractions of the ring U(h). By definition, the relations from R≺ are labeled by pairs (L,M) with L > M . The above bijection induces therefore a bijection between the sets R and R≺. 4.5 sln 1. Denote the subalgebra of Zn, generated by two central elements (3.9) and (3.10), by Yn; the algebra Yn is isomorphic to Z1. Since the extremal projector for sln is the same as for gln, the diagonal reduction algebra DR(sln) for sln is naturally a subalgebra of Zn. The subalgebra DR(sln) is complementary to Yn in the sense that Zn = Yn ⊗DR(sln). The algebra DR(sln) is generated by zij , i, j = 1, . . . , n, i 6= j, and ti,i+1 := ti − ti+1, i = 1, . . . , n−1 (and the Cartan subalgebra h, generated by hi,i+1, of the diagonally embedded sln). The elements ti,i+1 form a basis in the space of “traceless” combinations ∑ cmtm (traceless means that ∑ cm = 0), cm ∈ U(h). 2. The action of the braid group restricts onto the traceless subspace: q̌i(ti−1,i) = ti−1,i + h̊i,i+1 h̊i,i+1 − 1 ti,i+1, q̌i(ti+1,i+2) = h̊i,i+1 h̊i,i+1 − 1 ti,i+1 + ti+1,i+2, q̌i(ti,i+1) = − h̊i,i+1 + 1 h̊i,i+1 − 1 ti,i+1, q̌i(tk,k+1) = tk,k+1, k 6= i− 1, i, i+ 1. The traceless subspace with respect to the generators ti and the traceless subspace with respect to the generators t̊i (that is, the space of linear combinations ∑ cmt̊m, cm ∈ U(h), with∑ cm = 0) coincide. Indeed, in the expression of tl as a linear combination of t̊k’s (the second line in (4.2)), we find, calculating residues and the value at infinity, that the sum of the coefficients is 1, l−1∏ j=1 A′jl + l−1∑ k=1 1 h̊kl l−1∏ j=1 j 6=k A′jk = 1. Therefore, in the decomposition of the difference ti− tj as a linear combination of t̊k’s, the sum of the coefficients vanishes, so it is traceless with respect to t̊k’s; tl,l+1 is a linear combination of t̊12, t̊23, . . . , t̊l,l+1 (and vice versa). It should be however noted that in contrast to (4.2), the coefficients in these combinations do not factorize into a product of linear monomials, the lowest example is t̊34: t̊12 = h̊12 h̊12 − 1 t12, t̊23 = h̊23 h̊13 − 1 ( − 1 h̊12 − 1 t12 + h̊13 h̊23 − 1 t23 ) , Structure Constants of Diagonal Reduction Algebras of gl Type 13 t̊34 = h̊34 h̊14 − 1 ( − 1 h̊13 − 1 t12 − h̊14(̊h13 − 1) + h̊23(̊h24 − 1) (̊h13 − 1)(̊h23 − 1)(̊h24 − 1) t23 + h̊14̊h24 (̊h24 − 1)(̊h34 − 1) t34 ) . 3. One can directly see that the commutations between zij and the differences tk − tl close. The renormalization (4.11) is compatible with the sl-condition and, as we have seen, the set {ti,i+1} of generators can be replaced by the set {̊ti,i+1}. Therefore, one can work with the generators z̊ij , i, j = 1, . . . , n, i 6= j, and t̊i,i+1 := ti − ti+1, i = 1, . . . , n − 1. A direct look at the relations (4.12), (4.13), (4.14), (4.16), (4.17) and (4.18) shows that the only non-trivial verification concerns the relations (4.16); one has to check here the following assertion: when z̊ moves through t̊i,i+1, only traceless combinations of t̊l’s appear in the right hand side. Write a relation from the list (4.16) in the form z̊ij t̊l = ∑ m χ (i,j,l,m) m t̊mz̊ij + · · · , χ(i,j,l,m) m ∈ U(h), where dots stand for terms with z̊z̊. The assertion follows from the direct observation that for all i, j and l the sum of the coefficients χ (i,j,l,m) m is 1, ∑ m χ (i,j,l,m) m = 1. 4. With the help of the central elements (3.9), (3.10) and (3.11) one can build a unique linear in t’s traceless combination: n∑ i=1 (hi − 2i)ti − ( 1 n n∑ i=1 hi − n− 1 ) n∑ j=1 tj . It clearly depends only on the differences hi − hj and belongs therefore to the center of the subalgebra DR(sln). One can write this central element in the form n−1∑ u,v=1 Cuvhu,u+1tv,v+1 + n−1∑ v=1 (n− v)vtv,v+1 = n−1∑ u,v=1 Cuv (̊hu,u+1 + 1)tv,v+1, (4.19) where Cuv is the inverse Cartan matrix of sln. In general, let k be a semi-simple Lie algebra of rank r with the Cartan matrix aij . Let bij be the symmetrized Cartan matrix and ( , ) the scalar product on h∗ induced by the invariant non-degenerate bilinear form on k, so that aij = dibij , bij = (αi, αj), di = 2/(αi, αi). For each i = 1, . . . , r let α∨i be the coroot vector corresponding to the simple root αi, so that αj(α ∨ i ) = aij . Let dij be the matrix, inverse to cij = dibijdj . Let ρ ∈ h∗ be the half-sum of all positive roots. Write ρ = 1 2 r∑ i=1 niαi, where ni are nonnegative integers. Let tαi be the images of Hαi = α∨i (1)−α∨i (2) in the diagonal reduction algebra DR(k) and hαi = α∨i (1)+α∨i (2) be the coroot vectors of the diagonally embedded Lie algebra k. The generalization of the central element (4.19) to the reduction algebra DR(k) reads r∑ i,i=1 dijhαitαj + r∑ i=1 ni(αi, αi)tαi . 14 S. Khoroshkin and O. Ogievetsky 4.6 Stabilization and cutting In [7] we discovered the stabilization and cut phenomena which are heavily used in our derivation of the set of defining relations for the diagonal reduction algebras of gl-type. The consideration in [7] uses the standard (by the first coordinates) embedding of gln into gln+1. In this subsection we shall make several more precise statements about the stabilization and cut considering now the embedding of gln ⊕ gl1 into gln+1 (more generally, gln ⊕ glm into gln+m). These precisions are needed to establish the behavior of the center of the diagonal reduction algebra: namely we shall see that cutting preserves the centrality. Notation: h in this subsection denotes the Cartan subalgebra of gln+m. Consider an embedding of gln ⊕ glm into gln+m, given by an assignment eij 7→ eij , i, j = 1, . . . , n, and eab 7→ en+a,n+b, a, b = 1, . . . ,m, where ekl in the source are the generators of gln ⊕ glm and target ekl are in gln+m. This rule together with the similar rule Eij 7→ Eij and Eab 7→ En+a,n+b defines an embedding of the Lie algebra (gln ⊕ glm)⊕ (gln ⊕ glm) into the Lie algebra gln+m⊕gln+m and of the enveloping algebras An⊗Am = U(gln⊕gln)⊗U(glm⊕glm) into An+m = U(gln+m ⊕ gln+m). This embedding clearly maps nilpotent subalgebras of gln ⊕ glm to the corresponding nilpotent subalgebras of gln+m and thus defines an embedding ιn,m : Zn ⊗ Zm → Zn+m of the corresponding double coset spaces. However, the map ιn,m is not a homomorphism of algebras. This is because the multiplication maps are defined with the help of projectors, which are different for gln ⊕ glm and gln+m. However, as we will explain now we can control certain differences between the two multi- plication maps. Let Vn,m be the left ideal of the algebra Zn+m generated by elements zia with i = 1, . . . , n and a = n+ 1, . . . , n+m; let V′n,m be the right ideal of the algebra Zn+m generated by elements zai with i = 1, . . . , n and a = n+ 1, . . . , n+m. Write any element λ ∈ Q+ (the positive cone of the root lattice of gln+m) in the form λ = ∑n+m k=1 λkεk. The element λ can be presented as a sum λ = λ′ + λ′′, (4.20) where λ′ is an element of the root lattice of gln⊕ glm, and λ′′ is proportional to the simple root εn − εn+1: λ′ = ∑n+m k=1 λ′kεk with ∑n k=1 λ ′ k = ∑n+m k=n+1 λ ′ k = 0 and λ′′ = c(εn − εn+1). Lemma 4. The left ideal Vn,m ⊂ Zn+m consists of images in Zn+m of sums ∑ iaXiaEia with Xia ∈ Ān+m, i = 1 , . . . , n and a = n+ 1, . . . , n+m. The right ideal Vn,m ⊂ Zn+m consists of images in Zn+m of sums ∑ aiEaiYai with Yai ∈ Ān+m, i = 1 , . . . , n and a = n+ 1 , . . . , n+m. Proof. Present the projector P for the Lie algebra gln+m as a sum of terms ξe−γ1 · · · e−γteγ′1 · · · eγ′t′ , (4.21) where ξ ∈ U(h), γ1, . . . , γt and γ′1, . . . , γ ′ t′ are positive roots of gln+m. For any λ ∈ Q+ denote by Pλ the sum of above elements with γ1 + · · ·+ γt = γ′1 + · · ·+ γ′t′ = λ. Then P = ∑ λ∈Q+ Pλ. For any X,Y ∈ Ā define the element X �λ Y as the image of XPλY in the reduction algebra. We have X � Y = ∑ λ∈Q+ X �λ Y . For any X ∈ Ān+m, i = 1 , . . . , n and a = n+ 1 , . . . , n+m consider the product X �λ zia. The product X �λzin is zero if λ′′ 6= 0 (the component λ′′ is defined by (4.20)). Indeed, in this case in each summand of Pλ one of eγ′ k′ is equal to some ejb, j = 1 , . . . , n and b = n+1 , . . . , n+m. Choose an ordered basis of n+ which ends by all such ejb (ordered arbitrarily); any element of U(n+) can be written as a sum of ordered monomials, that is, monomials in which all such ejb stand on the right. Since [ejb, Eia] = 0 for any i, j = 1 , . . . , n and a, b = n + 1 , . . . , n + m, the product eγ′ k′ Eia belongs to the left ideal I+ and thus X �λ zia = 0 in Zn+m. Structure Constants of Diagonal Reduction Algebras of gl Type 15 If λ′′ = 0 then generators of n+ in monomials entering the decomposition of Pλ are among the elements eij , 1 ≤ i < j ≤ n, and eab, n + 1 ≤ a < b ≤ n + m and thus their adjoint action leaves the space, spanned by all Eia, i = 1 , . . . , n, a = n + 1 , . . . , n + m invariant, so X �λ zia can be presented as an image of the sum ∑ jbXjbEjb with Xjb ∈ Ān+m, j = 1 , . . . , n, b = n+ 1 , . . . , n+m. Thus, the left ideal, generated by all zia is contained in the vector space of images in Zn+m of sums ∑ jbXjbEjb. Moreover, for any X ∈ Ān+m the element X �zia is the image of XEia+ ∑ j,b: j<i, b>a X (jb)Ejb for some X(jb) and the double induction on i and a proves the inverse inclusion. The second part of lemma is proved similarly. � Corollary 5. We have the following decomposition of the free left (and right) U(h)-modules: Zn+m = In,m ⊕U(h) · ιn,m(Zn ⊗ Zm), (4.22) where In,m := Vn,m + V′n,m. Proof. The double coset space Zn+m is a free left and right U(h)-module with a basis consisting of images of ordered monomials on elements Eij , i, j = 1 , . . . , n + m; recall that we always use orders compatible with the partial order < on h∗, see (c) in Section 3, paragraph 2. We can choose an order for which all ordered monomials are of the form XY Z, where X is a monomial on Eai with i = 1 , . . . , n and a = n+1 , . . . , n+m, Z is a monomial on Eia with i = 1 , . . . , n and a = n+1 , . . . , n+m while Y is a monomial on Eij with i, j = 1 , . . . , n or i, j = n+1 , . . . , n+m. Then we apply the lemma above. � For a moment denote for each k > 0 the multiplication map in Zk by �(k) : Zk ⊗ Zk → Zk (instead of the default notation �, see (3.1)); denote also for each k, l > 0 by �(k,l) the multiplication map �(k) ⊗ �(l) in Zk ⊗ Zl. Proposition 6. For any x, y ∈ Zn ⊗ Zm we have ιn,m(x) �(n+m) ιn,m(y) = ιn,m(x �(n,m) y) + z, where z is some element of Jn,m := Vn,m ∩V′n,m. Let hn and hm be the Cartan subalgebras of gln and glm, respectively. Denote the space Zn ⊗U(hn) U(h) ⊗U(hm) Zm by U(h) · (Zn ⊗ Zm). The composition law �(n,m) naturally extends to the space U(h) · (Zn ⊗ Zm) equipping it with an associative algebra structure (we keep the same symbol �(n,m) for the extended composition law in U(h) · (Zn ⊗ Zm)). Also, the map ιn,m admits a natural extension to a map ιn,m : U(h) · (Zn ⊗ Zm) → Zn+m denoted by the same symbol and defined by the rule ιn,m(ϕx) := ϕ ιn,m(x) for any ϕ ∈ U(h) and x ∈ Zn ⊗ Zm. The statement of Proposition 6 remains valid for this extension as well, that is, one can take x, y ∈ U(h) · (Zn ⊗ Zm) in the formulation. Proof of Proposition 6. Denote by Pn,m := Pn⊗Pm the projector for the Lie algebra gln⊕glm. It is sufficient to prove the following statement. Suppose X and Y are (non-commutative) polynomials in Eij with i, j = 1 , . . . , n Then the product of x and y in Zn+m coincides with the image in Zn+m of X Pn,mY modulo the left ideal Vn,m and modulo the right ideal V′n,m). Due to the structure of the projector the condition λ′′ = 0, see (4.20), implies that the product X �λ Y related to gln ⊕ glm coincides with product X �λ Y related to gln+m. Let now λ′′ 6= 0. Then each monomial eγ′1 · · · eγ′t′ in the decomposition of Pλ, see (4.21), contains generators eia with i ∈ {1 , . . . , n} and a ∈ {n+1 , . . . , n+m}; these eia can be assumed to be right factors of the corresponding monomial (like in the proof of Lemma 4). The commutator 16 S. Khoroshkin and O. Ogievetsky of any such generator eia with every factor in Y is a linear combination of the elements Ejb with j ∈ {1 , . . . , n} and b ∈ {n+ 1 , . . . , n+m}. Moving the resulting Ejb to the right we see that the product X �λ Y is the image in Zn+m of an element of the form ∑ s XsYs where each Ys belongs to the left ideal of Ān+m generated by Ejb with j ∈ {1 , . . . , n} and b ∈ {n+ 1 , . . . , n+m} (one can say more: each Ys can be written in a form ∑ j,b Y (jb) s Ejb where each Y (jb) s ∈ Ān+m does not involve generators Eck with k ∈ {1 , . . . , n} and c ∈ {n + 1 , . . . , n + m}; we don’t need this stronger form). Thus, due to Lemma 4, X �λ Y ∈ Vn,m. Similarly, each Xs participating in the sum ∑ sXsYs, see above, belongs to the right ideal of Ān+m generated by the elements Ebj with j ∈ {1 , . . . , n} and b ∈ {n + 1 , . . . , n + m}. So, again by Lemma 4, X �λ Y ∈ V′n,m. � Suppose that we have a relation∑ k ak �(n,m) bk = 0, (4.23) where all ak and bk are elements of Zn⊗Zm. Then, due to Proposition 6, we have the following relation in Zn+m:∑ k āk �(m+n) b̄k = z, (4.24) where āk = ιn,m(ak), b̄k = ιn,m(bk) and z ∈ Jn,m = Vn,m ∩V′n,m. On the other hand, suppose we have the following relation in Zn+m:∑ k āk �(m+n) b̄k = u, (4.25) where all ak and bk are elements of Zn ⊗ Zm, āk = ιn,m(ak), b̄k = ιn,m(bk), and u ∈ In,m = Vn,m + V′n,m. Then the elements ak and bk satisfy the relation (4.23) and u ∈ Jn,m. Indeed, suppose that the relation (4.25) is satisfied and ∑ k ak �(n,m) bk = v for some v ∈ Zn ⊗ Zm. It follows from Proposition 6 that ∑ k āk�(m+n) b̄k−v̄ belongs to Jn,m; here v̄ = ιn,m(v). Then (4.25) implies that v̄ ∈ In,m and thus v̄ = 0 due to Corollary 5. Thus v = 0, since the map ιn,m is an inclusion, and u ∈ Jn,m. We refer to the implication (4.23) ⇒ (4.24) as stabilization. Call cutting the (almost inverse) implication (4.25) ⇒ (4.23) which can be understood as a procedure of getting relations in Zn ⊗ Zm from relations in Zn+m; we say that (4.23) is the cut of (4.25). Clearly all relations in Zn ⊗ Zm can be obtained by cutting appropriate relations in Zn+m. Let πn,m : Zn+m → U(h) · (Zn ⊗ Zm) be the composition of the projection π̄n,m of Zn+m onto ιn,m(U(h) · Zn ⊗ Zm) = U(h) · ιn,m(Zn ⊗ Zm) along In,m, see (4.22), and of the inverse to the inclusion ιn,m: πn,m = ι−1 n,m ◦ π̄n,m. We have the following consequence of Proposition 6 and Corollary 5. Proposition 7. Let x be a central element of Zn+m. Then πn,m(x) is a central element of U(h) · (Zn ⊗ Zm). Proof. Denote X = πn,m(x). Then, by definition, x = ιn,m(X) + z, where z ∈ In,m. Since x is central, it is of zero weight; so X and z are of zero weight as well. Thus each monomial entering the decomposition of z contains both types of generators, Eai and Eia, where i ∈ {1 , . . . , n} and Structure Constants of Diagonal Reduction Algebras of gl Type 17 a ∈ {n + 1 , . . . , n + m}, which implies that z ∈ Jn,m = Vn,m ∩ V′n,m. Take any Y ∈ Zn ⊗ Zm. We now prove that X �(n,m) Y − Y �(n,m) X = 0. Denote y = ιn,m(Y ). Due to Proposition 6, ιn,m(X �(n,m) Y − Y �(n,m) X) = (x− z) �(m+n) y − y �(m+n) (x− z) + z′, (4.26) where z′ ∈ Jn,m = Vn,m ∩ V′n,m. Since x is central in Zn+m, the right hand side of (4.26) is equal to y �(m+n) z − z �(m+n) y + z′, which is an element of In,m = Vn,m ⊕V′n,m since z, z′ ∈ Jn,m. On the other hand, the left hand side of (4.26) belongs to U(h) · ιn,m(Zn ⊗ Zm). Thus, by Corollary 5, both sides of (4.26) are equal to zero and X �(n,m) Y − Y �(n,m) X = 0 since the map ιn,m is injective. � The map πn,m obeys properties similar to those of the Harish-Chandra map U(g)h → U(h) (U(g)h is the space of elements of zero weight). For instance, its restriction to the center of Zn+m is a homomorphism. More precisely, if x is a central element of Zn+m, then πn,m(x �(m+n) y) = πn,m(x) �(n,m) πn,m(y) and πn,m(y �(m+n) x) = πn,m(y) �(n,m) πn,m(x) (4.27) for any y ∈ Zn+m. Indeed, let X = πn,m(x), Y = πn,m(y). Then x = ιn,m(X)− z, y = ιn,m(Y )− u, where u ∈ In,m while, as it was noted in the proof of Proposition 7, z ∈ Jn,m. Moreover, it is clear that z can be written in the form z = ∑ a z ′ aza, where za ∈ Vn,m and z′a ∈ V′n,m (for instance, use the order as in the proof of Corollary 5). Then (dropping for brevity the multiplication symbol �(m+n)) we have ιn,m(X)ιn,m(Y ) = (x+ z)(y + u) = ( x+ ∑ a z′aza ) (y + z̃′ + z̃) = xy + ∑ a z′aza(y + z̃′ + z̃) + xz̃ + z̃′x ≡ xy mod In,m. (4.28) Here z̃ ∈ Vn,m and z̃′ ∈ V′n,m. In the last equality we used the centrality of x. Due to Proposi- tion 6, (4.28) is precisely equivalent to the fist part of (4.27). The second part of (4.27) is proved similarly. 5 Proofs 5.1 Tensor J The multiplication map � in Zn (we return to the original notation) is given by the prescrip- tion (3.1), as in any reduction algebra. It can be formally expanded into a series over the root lattice of certain bilinear maps as follows. Set U(b±) := U(h)⊗U(h) U(b±), U 12 (b) := U(b−)⊗U(h) U(b+). All these are associative algebras. Besides, both algebras U(b±) are U(h)-bimodules. The alge- bra U 12 (b) admits three commuting actions of U(h). Two of them are given by the assignments X(Y ⊗ Z) := XY ⊗ Z, (Y ⊗ Z)X := Y ⊗ ZX, 18 S. Khoroshkin and O. Ogievetsky for any X ∈ U(h), Y ∈ U(b−) and Z ∈ U(b+). The third action associates to any X ∈ U(h), Y ⊗ Z ∈ U 12 (b) the element Y X ⊗ Z = Y ⊗XZ ∈ U 12 (b). Present the projector P in an ordered form: P = ∑ γ,i F̀γ,iÈγ,iH̀γ,i = ∑ γ,i H̀γ,iF̀γ,iÈγ,i, (5.1) the summation is over γ ∈ Q+ and i ∈ Z≥0; every F̀γ,i is an element of U(n−) of the weight −γ, every Èγ,i is an element of U(n+) of the weight γ and H̀γ,i ∈ U(h). Let J be the following element of U 12 (b): J := ∑ γ,i F̀γ,i ⊗ Èγ,iH̀γ,i = ∑ γ,i H̀γ,iF̀γ,i ⊗ Èγ,i, γ ∈ Q+, i ∈ Z≥0. Due to the PBW theorem in U(gln) the tensor J is uniquely defined by the projector P ; it is of total weight zero: hJ = Jh for any h ∈ h. We have the weight decomposition of J with respect to the adjoint action of h in the second tensor factor of U 12 (b): J = ⊕ λ∈Q+ Jλ, where Jλ consists of all the terms, corresponding to F̀λ,iÈλ,iH̀λ,i in (5.1) (contributing to λ ∈ Q+ in the summation), Jλ := ∑ i F̀λ,i ⊗ Èλ,iH̀λ,i. By definition of J, the multiplication � in the double coset space Zn can be described by the relation a � b = m ((a⊗ 1)J(1⊗ b)) , (5.2) where m( ∑ i ci⊗di) is the image in Zn of the element ∑ i cidi. Moreover, in (5.2) we can replace all products Èγ,ib in the second tensor factor by the adjoint action of Èγ,i on b (in fact, for Èγ,i = eγm · · · eγ1 , we can replace Èγ,ib by [Èγ,i, b] or by êγm · · · êγ1(b), see (2.4)) and likewise all products aF̀γ,i in the first tensor factor by the opposite adjoint action of F̀γ,i on a. We have a decomposition of the product � into a sum over Q+: a � b = ∑ λ∈Q+ a �λ b, where a �λ b := m ((a⊗ 1)Jλ(1⊗ b)) . (5.3) If a and b are weight elements of Zn of weights ν(a) and ν(b), then the product a �λ b is the image in Zn of the sum ∑ i aibi, where the weight of each bi is ν(b)+λ, and the weight of each ai is ν(a)− λ. The tensor J satisfies the Arnaudon–Buffenoir–Ragoucy–Roche (ABRR) difference equa- tion [1], see also [5] for the translation of the results of [1] to the language of reduction algebras. To describe the equation, let ϑ = 1 2 ∑n k=1 h̊ 2 k ∈ U(h); for any positive root γ ∈ ∆+, denote by Tγ the following linear operator on the vector space U 12 (b): Tγ(X ⊗ Y ) := Xe−γ ⊗ eγY. The ABRR equation means the relation [1, 5]: [1⊗ ϑ, J] = − ∑ γ∈∆+ Tγ(J). Structure Constants of Diagonal Reduction Algebras of gl Type 19 This relation is equivalent to the following system of recurrence relations for the weight compo- nents Jλ: Jλ · (̊ hλ + (λ, λ) 2 ) = − ∑ γ∈∆+ Tγ (Jλ−γ) , (5.4) where h̊λ := ∑ k λkh̊k for λ = ∑ k λkεk. The recurrence relations (5.4) together with the initial condition J0 = 1⊗ 1 uniquely determine all weight components Jλ. It should be noted that the recurrence relations (5.4) provides less information about the structure of the denominators (from U(h)) of the summands of the extremal projector P than the information implied by the product formula (see [3]) for the extremal projector. Using (5.4) we get in particular: Jα = −(̊hα + 1)−1e−a ⊗ eα, α = εi − εi+1, (5.5) Jα+β = (̊hα+β + 1)−1 ( −e−α−β ⊗ eα+β + (̊hα + 1)−1e−αe−β ⊗ eβeα + (̊hβ + 1)−1e−βe−α ⊗ eαeβ ) , α = εi−1 − εi, β = εi − εi+1, (5.6) Jεi−εj+εk−εl = Jεi−εj · Jεk−εl , i < j < k < l. (5.7) 5.2 Braid group action The proof of the relations (4.1) and (4.4) consists of the following arguments, valid for any reduction algebra. Let α be any simple root of gln, α = εi − εi+1 and gα the corresponding sl2 subalgebra of gln. It is spanned by the elements eα = ei,i+1, e−α = ei+1,i and hα = hi − hi+1. Let σ́α = σ́i be the corresponding automorphism of the algebra A and q̌α = q̌i the Zhelobenko automorphism of Zn. Assume that Y ∈ A belongs, with respect to the adjoint action of gα, to an irreducible finite-dimensional gα-module of dimension 2j + 1, j ∈ {0, 1/2, 1, . . .}. Assume further that Y is homogeneous, of weight 2m, [hα, Y ] = 2mY . Identify Y with its image in Zn. Then q̌α(Y ) coincides with the image in Zn of the element j+m∏ i=1 (hα + i+ 1) · σ́α(Y ) · j+m∏ i=1 (hα − i+ 1)−1. This can be checked directly using [6, Proposition 6.5]. In the realization of irreducible sl2-modules as the spaces of homogeneous polynomials in two variables u and v, eα 7→ u ∂ ∂v , hα 7→ u ∂ ∂u − v ∂ ∂v and e−α 7→ v ∂ ∂u , the operator σ́α becomes (σ́αf)(u, v) = f(−v, u), or, in the basis \|j, k〉 := xj+kyj−k (j labels the representation; k = 0, 1, . . . , 2j), σ́α : \|j,−j + k〉 7→ (−1)k\|j, j − k〉. Proof of Lemma 1, Subsection 4.2. To see this, write a reduced expression for q̌w0 , q̌w0 = q̌αi1 · · · q̌αiM with αi1 , . . . , αiM simple roots. Then q̌w0 = q̌αiM · · · q̌αi1 as well. Writing, for q̌2 w0 , the second expression after the first one, we get squares of q̌αis ’s (which are conjugations by h̊−1 αis ’s; they thus commute) one after another. Moving these conjugations to the left through the remaining q̌’s, we produce, exactly like in the construction of a system of all positive roots from a reduced expression for the longest element of the Weyl group of a reductive Lie group, the conjugation by the product (4.8) over all positive roots. � 20 S. Khoroshkin and O. Ogievetsky Proof of Proposition 2, Subsection 4.2. Only formula (4.9) needs a proof (formula (4.10) is a particular case of (4.3)). For a moment, denote the longest element of the symmetric group Sn by q̌ (n) w0 . Let ψj := q̌j q̌j−1 · · · q̌1 (the product in the descending order). We have q̌ (n+1) w0 = q̌ (n) w0 ψn and q̌ (n+1) w0 = ψ1ψ2 · · ·ψn (the product in the ascending order). For j < n it follows from (4.4) that ψj(zn+1,1) = (−1)jzn+1,j+1 (say, by induction on j). So, ψn(zn+1,1) = qnψn−1(zn+1,j+1) = (−1)n−1qn(zn+1,n) = (−1)nzn,n+1, again by (4.4). Next, ψkψk+1 · · ·ψn−1(zn,n+1) = zk,n+1 by induction on n − k and again (4.4). Thus, q̌w0(zn+1,1) = (−1)nz1,n+1, (5.8) establishing (4.9) for i = n + 1 and j = 1. We now prove (4.9) for i > j (positions below the main diagonal) by induction backwards on the height i− j of a negative root; the formula (5.8) serves as the induction base. Assume that (4.9) is verified for a given level i−j and i−j−1 > 0 (so that the positions (i, j + 1) and (i − 1, j) are still under the main diagonal). By (4.4), zi,j+1 = −q̌j(zij), therefore q̌w0(zi,j+1) = −q̌w0(q̌j(zij)) = −q̌j′−1(q̌w0(zij)) = (−1)i+j+1q̌j′−1 zi′j′ ∏ a:a<i′ Aai′ ∏ b:b>j′ Aj′b  = (−1)i+j+1zi′,j′−1Aj′−1,j′ ∏ a:a<i′ Aai′ ∏ b:b>j′ Aj′−1,b = (−1)i+j+1zi′,(j+1)′ ∏ a:a<i′ Aai′ ∏ b:b>(j+1)′ A(j+1)′,b. In the second equality we used the identity q̌w0 q̌j = q̌j′−1q̌w0 in the braid group; the third equality is the induction assumption; in the fourth equality we used that i′ 6= j′ − 1 (since i− j− 1 > 0) and then (4.4); in the fifth equality we replaced j′− 1 by (j+ 1)′. The calculation for q̌w0(zi−1,j) is similar; it uses zi−1,j = q̌i−1(zij). The proof of the formula (4.9) for positions below the main diagonal is finished. The proof of (4.9) for i < j (positions above the main diagonal) follows now from Lem- ma 1. � 5.3 Derivation of relations The set of defining relations in Zn divides into several different types, see Section 4.3. We prove the necessary amount of relations of each type and get the rest by applying the transformations from the braid group as well as the anti-involution ε, see (3.7). We never use the automorphism ω, defined in (3.8), in the derivation of relations. However, the involution ω is compatible with our set of relations in the sense explained in Section 5.4. In the following we denote by the symbol ≡ the equalities of elements from Ā modulo the sum (I− + I+) of two ideals I− and I+ defined in the beginning of Section 3. Moreover, for any two elements X and Y of the algebra Ā we may regard the expressions X �Y and X �λY as the sums of elements from Ā defined in (5.2) and (5.3). The sum X �λ Y is finite. By the construction, all but a finite number of terms in the product X �Y belong to (I− + I+). Unlike to the system of notation adopted in Section 3, our proof of each relation in Zn will use equalities in Ā taken modulo (I− + I+). Structure Constants of Diagonal Reduction Algebras of gl Type 21 We also use the notation Hi for the element Eii ∈ A and Hij = Hi −Hj = Eii − Ejj . 1. We first prove in Zn the relation z12 � z13 = z13 � z12 h̊23 h̊23 + 1 . (5.9) Elements z12 and z13 are images in Zn of E12 and E13. Consider the product E12 �λ E13. Since the adjoint action of gln preserves the space p, see Section 2, this product is the sum of such monomials EijEkl, with coefficients in U(h), that (i): the weight εk − εl of Ekl is equal to the weight ε1 − ε3 of E13 plus λ ∈ Q+, while (ii): the weight εi − εj of Eij is equal to the weight ε1 − ε2 of E12 minus λ. Assume that E12 �λ E13 6= 0. By (i), λ = −ε1 + ε3 + εk − εl and it can be positive only if k = 1 and l ≥ 3. So, the condition (i) implies that either λ = 0 or λ = ε3− εl with l > 3. The possibility λ = ε3− εl, l > 3, is excluded by the condition (ii). Therefore, λ = 0 and E12 � E13 ≡ E12E13. (5.10) Similarly, for λ ∈ Q+, which can non-trivially contribute to the product E13 �E12, the analogue of the condition (i) on the weight λ gives the restriction λ = 0 or λ = ε2−εk, k > 2; the analogue of the condition (ii) further restricts λ: λ = 0 or λ = ε2 − ε3, so we have E13 �ε2−ε3 E12 ≡ −E13e32e23 1 h̊23 + 1 E12 ≡ −E13e32e23E12 1 h̊23 ≡ E12E13 1 h̊23 , since Jε2−ε3 = −e32 ⊗ e23(̊h23 + 1)−1 as it follows from the ABRR equation, see (5.5), or from the precise explicit expression for the projector P , see [3]. Thus, since E12 and E13 commute in the universal enveloping algebra E13 � E12 ≡ E13E12 + E13 �ε2−ε3 E12 = E12E13 ( 1 + 1 h̊23 ) , (5.11) Comparing (5.10) and (5.11) we find (5.9). Applying to (5.9) the anti-involution ε, see (3.7), we get the relation z21 � z31 = z31 � z21 h̊23 + 1 h̊23 . (5.12) The rest of the relations (4.12) are obtained from (5.9) and (5.12) by applying different transformations q̌w from the Weyl group. 2. Now we prove in Zn the relation z13 � z24 − z24 � z13 = ( 1 h̊12 − 1 h̊34 ) z23 � z14. (5.13) We begin by the proof of this relation in Z4. We proceed in the same manner as for the derivation of the relation (5.9), E13 � E24 ≡ E13E24 + E13 �ε1−ε2 E24 ≡ E13E24 − E13e21e12 1 h̊12 + 1 E24 ≡ E13E24 + E23E14 1 h̊12 , E24 � E13 ≡ E24E13 + E24 �ε3−ε4 E13 ≡ E24E13 − E24e43e34 1 h̊34 + 1 E13 ≡ E13E24 + E23E14 1 h̊34 , 22 S. Khoroshkin and O. Ogievetsky E23 � E14 ≡ E23E14. Combining the three latter equalities we obtain (5.13) in Z4. The difference of the left and right hand sides of (5.13) in Zn is a linear combination of monomials in zij of the total weight ε1 +ε2−ε3−ε4. The weight is non-trivial, so the monomials can be only quadratic. Due to the stabilization phenomenon, each monomial should contain zij with i > 4 or j > 4, but, by the weight arguments, there is no such non-zero possibility, which completes the proof of the relation (5.13) in Zn. The rest of relations (4.13) is then obtained by applications of the transformations from the braid group. 3a. We continue and derive in Z4 the relation (we remind the notation tij := zii − zjj , see Section 3, and Hij = Eii − Ejj): z23 � z12 − z12 � z23 = t12 � z13 1 h̊12 + t23 � z13 1 h̊23 − z43 � z14 h̊34 + 1 h̊34̊h24 . (5.14) Using (5.5)–(5.7), we calculate, to obtain the result for Z4: E12 � E23 ≡ E12E23 + E12 �ε1−ε2 E23 + E12 �ε1−ε2+ε3−ε4 E23 ≡ E12E23 −H12E13 1 h̊12 , (5.15) E23 � E12 ≡ E23E12 + E23 �ε2−ε3 E12 + E23 �ε2−ε4 E12 ≡ E23E12 +H23E13 1 h̊23 − E43E14 (̊h23 − 1) h̊23̊h24 , (5.16) H12 � E13 ≡ H12E13 +H12 �ε3−ε4 E13 ≡ H12E13, (5.17) H23 � E13 ≡ H23E13 +H23 �ε3−ε4 E13 ≡ H23E13 + E43E14 1 h̊34 , (5.18) E43 � E14 ≡ E43E14. (5.19) Combining the above equalities and taking into account that [E12, E23] = e13 ≡ 0, we get (5.14) in Z4. We could apply here the stability arguments (as we shall do in the sequel) but we give some more details at this point to give a flavor of how such derivations of relations work. For the same, as (5.15)–(5.19), calculations for Zn, the analogues of the conditions (i) and (ii), see paragraph 1 of this subsection, restrict λ to be of the form ε1 − ε2 + ε3 − εk, k ≥ 3 for (5.15); ε2 − εk, k ≥ 2 for (5.16); ε3 − εk, k ≥ 3 for (5.17) and (5.18); ε4 − εk, k ≥ 4 for (5.19). It follows, for, say, n = 5, that the right hand sides of (5.15)–(5.19) might be modified only by an addition of the term proportional to E53E15; and this will be compensated by an addition of the term, proportional to z53 � z15 to the right hand side of (5.14), since E53 � E15 ≡ E53E15; the proportionality coefficient is uniquely defined. This pattern clearly continues and we conclude that there is a relation in Zn of the form z23 � z12 − z12 � z23 = t12 � z13 1 h̊12 + t23 � z13 1 h̊23 − ∑ k>3 zk3 � z1kXk, (5.20) with certain, uniquely defined, coefficients Xk ∈ U(h), k = 4 , . . . , n, and already known X4 = (̊h34 +1)̊h−1 34 h̊ −1 24 . To find X5, . . . , Xn, we apply to (5.20) the automorphisms q̌k, k = 4 , . . . , n−1, which leave invariant the left hand side and the first two terms in the right hand side of (5.20). The uniqueness of the relation of the form (5.20), together with the equality q̌k(zk3 � z1k) = zk+1,3 �z1,k+1(̊hk,k+1 +1)̊h−1 k,k+1, imply the recurrence relation Xk+1 = q̌k(Xk) · (̊hk,k+1 +1)̊h−1 k,k+1 Structure Constants of Diagonal Reduction Algebras of gl Type 23 and we find Xk = 1 h̊2k k−1∏ j=3 h̊jk + 1 h̊jk . After the renormalization (4.11) and the change of variables (4.2), the derived relation be- comes one of the relations in the first line of (4.14). Applying the transformations from the braid group, we obtain the rest of the relations from the list (4.14). 3b. We have the following equalities in Z3: z12 � t1 = t1 � z12 h̊12 + 2 h̊12 + 1 − t2 � z12 1 h̊12 + 1 − z32 � z13 h̊23 + 1 h̊23(̊h13 + 1) , (5.21) z12 � t2 = −t1z12 1 h̊12 + 1 + t2 � z12 h̊12 + 2 h̊12 + 1 + z32 � z13 h̊13 + 2 h̊23(̊h13 + 1) , (5.22) and the equality in Z4: z12 � t4 = t4 � z12 − z42 � z14 h̊12 + 1 (̊h14 + 1)̊h24 . (5.23) Equalities (5.21) and (5.22) are the results of the following calculations for Z3, using (5.5)–(5.7), and of the commutativity [H1, E12] = e12 ≡ 0, [H2, E12] = −e12 ≡ 0: E12 �H1 ≡ E12H1 +H12E12 1 h̊12 + 1 − E32E13 h̊12 (̊h12 + 1)(̊h13 + 1) , E12 �H2 ≡ E12H2 −H12E12 1 h̊12 + 1 − E32E13 1 (̊h12 + 1)(̊h13 + 1) , H1 � E12 ≡ H1E12, H2 � E12 ≡ H2E12 − E32E13 1 h̊23 , E32 � E13 ≡ E32E13. The derivation of (5.23) can be done with the help of the following calculations for Z4: E12 �H4 ≡ E12H4 + E42E14 1 h̊14 + 1 , (5.24) H4 � E12 ≡ H4E12 + E42E14 1 h̊24 , E42 � E14 ≡ E42E14. We shall make a comment about the line (5.24) only. Here one might expect, by the ana- logues of the conditions (i) and (ii), see paragraph 1 of this subsection, non-trivial contri- butions to E12 � H4 from the weights 0, ε1 − ε2, ε1 − ε3 and ε1 − ε4. So we need, in ad- dition to (5.5)–(5.7), some information about Jε1−ε4 . It follows from the ABRR equation that Jε1−ε4 (̊h14 + 1) = −Tε1−ε2(Jε2−ε4) − Tε1−ε3(Jε3−ε4) − Tε1−ε4(J0) − Tε2−ε3(Jε1−ε2+ε3−ε4) − Tε2−ε4(Jε1−ε2) − Tε3−ε4(Jε1−ε3). Since e13 and e12 commute with H4, the parts Tε2−ε4(Jε1−ε2) and Tε3−ε4(Jε1−ε3) do not contribute; e42 and e43 commute with E12, so the parts Tε1−ε2(Jε2−ε4) and Tε1−ε3(Jε3−ε4) do not contribute either; Jε1−ε2+ε3−ε4 = Jε3−ε4Jε1−ε2 does not contribute again since e12 commute with H4. Thus the only contribution is from Tε1−ε4(J0) and we quickly arrive at (5.24). Applying the automorphism q̌3 of the algebra Z4 to the relation (5.23), see (4.1) and (4.4), we find z12 � ( t3̊h34 − t4 ) = ( t3̊h34 − t4 ) � z12 − z32 � z13 h̊34(̊h12 + 1) (̊h13 + 1)̊h23 . 24 S. Khoroshkin and O. Ogievetsky We then add (5.23) to this relation and obtain the following relation in Z4: z12 � t3 = t3 � z12 − z32 � z13 (̊h12 + 1) (̊h13 + 1)̊h23 − z42 � z14 h̊12 + 1 (̊h14 + 1)̊h24̊h34 . (5.25) The stabilization arguments for (5.21), (5.22) and (5.25) imply the existence of the following relations in Zn: z12 � t1 = t1 � z12 h̊12 + 2 h̊12 + 1 − t2 � z12 1 h̊12 + 1 + ∑ k>2 zk2 � z1kX (1) k , (5.26) z12 � t2 = −t1 � z12 1 h̊12 + 1 + t2 � z12 h̊12 + 2 h̊12 + 1 + ∑ k>2 zk2 � z1kX (2) k , (5.27) z12 � t3 = t3 � z12 − z32 � z13 (̊h12 + 1) (̊h13 + 1)̊h23 + ∑ k>3 zk2 � z1kX (3) k , (5.28) where all X (i) k belong to U(h) and the initial X (i) k are known: X (1) 3 = − h̊23 + 1 h̊23(̊h13 + 1) , X (2) 3 = h̊13 + 2 h̊23(̊h13 + 1) , X (3) 4 = − h̊12 + 1 (̊h14 + 1)̊h24̊h34 . By the braid group transformation laws, X (i) k+1 = q̌k ( X (i) k ) · (̊hk,k+1 + 1)̊h−1 k,k+1 with k > 2 for i = 1, 2 and k > 3 for i = 3, so that X (1) k = − 1 h̊1k + 1 k−1∏ j=2 h̊jk + 1 h̊jk , X (2) k = h̊1k + 2 (̊h1k + 1)̊h2k k−1∏ j=3 h̊jk + 1 h̊jk , X (3) k = − h̊12 + 1 (̊h1k + 1)̊h2kh̊3k k−1∏ j=4 h̊jk + 1 h̊jk . After the renormalization (4.11) and the change of variables (4.2), the relations (5.26)–(5.28) turn into the relations (4.16) for i = 1, j = 2 and k = 3. Applying the transformations from the braid group, we obtain the rest of the relations from the list (4.16). 4a. We now prove the relations (4.17) using the arguments similar to [10, Subsection 6.1.2]. Consider the products Hk�λHl and Hl�λHk with λ 6= 0. These products are linear combinations, over U(h), of monomials akl;~γ := Hke−γ1 · · · e−γmeγm · · · eγ1Hl and alk;~γ := Hle−γ1 · · · e−γmeγm · · · eγ1Hk, respectively; here m ≥ 0 and ~γ := {γ1, . . . , γm}. By construction, the coefficient, from U(h), of the monomial akl;~γ in Hk �λHl equals the coefficient of alk;~γ in Hl �λHk. The expressions akl;~γ and alk;~γ are both equal in Zn to (γ1, εk)(γ1, εl)E−γ1e−γ2 · · · e−γmeγm · · · eγ2Eγ1 . Thus Hk �λ Hl ≡ Hl �λ Hk for any λ 6= 0. In the zero weight part �0 of the product � we have the equality HkHl = HlHk as well. Therefore, Hk �Hl ≡ Hl �Hk. 4b. The last group (4.18) of relations is left. Like above, we first explicitly derive the following relation in Z3: z12 � z21 = h12 − t12 � t12 1 h̊12 − 1 + z21 � z12 (̊h12 − 1)(̊h12 + 2) h̊12(̊h12 + 1) Structure Constants of Diagonal Reduction Algebras of gl Type 25 + z31 � z13 (̊h12 − 1)(̊h13 + 2) h̊12̊h23(̊h13 + 1) − z32 � z23 h̊23 + 2 (̊h23 + 1)̊h13 (5.29) (the first term in the right hand side is h12, without hat). The relation (5.29) is a corollary of the following calculations for Z3, together with the commutation relation [E12, E21] = h12, E12 � E21 ≡ E12E21 −H2 12 1 (̊h12 − 1) + E21E12 2 (̊h12 − 1)̊h12 − E32E23 h̊12 − 2 (̊h12 − 1)̊h13 + E31E13 h̊12 − 2 (̊h12 − 1)̊h12̊h13 , (5.30) H12 �H12 ≡ H2 12 − E21E12 4 h̊12 + 1 − E32E23 1 h̊23 + 1 + E31E13 ( −1 + 1 h̊23 + 1 + 4 h̊12 + 1 ) 1 h̊13 + 1 , (5.31) E21 � E12 ≡ E21E12 − E31E13 1 h̊23 , (5.32) E32 � E23 ≡ E32E23 − E31E13 1 h̊12 , (5.33) E31 � E13 ≡ E31E13. (5.34) Here only the calculation of E12 � E21 deserves a little explanation; by the analogues of the conditions (i) and (ii), see paragraph 1 of this subsection, non-trivial contributions to E12 �E21 from the weights 0, ε1−ε2, 2(ε1−ε2), ε1−ε3 and 2ε1−ε2−ε3 are possible. By the ABRR equation, J2(ε1−ε2)(2̊h12 + 4) = −Tε1−ε2(Jε1−ε2) and J2ε1−ε2−ε3(2̊h1 − h̊2 − h̊3 + 3) = −Tε1−ε2(Jε1−ε3) − Tε1−ε3(Jε1−ε2)− Tε2−ε3(J2(ε1−ε2)). We leave further details to the reader. By the stabilization law in Z4 we have a relation z12 � z21 = h12 − t12 � t12 1 h̊12 − 1 + ∑ 1≤i<j≤n zji � zijXij , Xij ∈ U(h) (5.35) with n = 4, which differs from (5.29) by a presence of terms z43 � z34, z42 � z24, z41 � z14, with coefficients in U(h). Consider in Z4 the products z12�z21, t12�t12 and zji�zij , 1 ≤ i < j ≤ 4. The weights (ε3 − ε4) − (εi − εj) do not belong to the cone Q+ if 1 ≤ i < j < 4. Thus in the decomposition Eji � Eij ≡ ∑ k<l ElkEklakl, akl ∈ U(h), 1 ≤ i < j < 4, the term with E43E34 has a zero coefficient, a34 = 0. The same statement holds for the products E41 �E14 and E42 �E24 since the weights (ε3 − ε4)− (εi − ε4) do not belong to Q+ for i = 1, 2. Consider the product E12 � E21. Here the term with E43E34 is equal to E12 �ε1−ε2+ε3−ε4 E21. By (5.7), Jε1−ε2+ε3−ε4 = e43e21 ⊗ e12e34(̊h12 + 1)−1(̊h34 + 1)−1 and E12 �ε1−ε2+ε3−ε4 E21 = E12e43e21e12e34E21 1 (̊h12 − 1)(̊h34 + 1) ≡ 0, since [e34, E21] = 0 (and [E12, e43] = 0). In the similar manner, the term with E43E34 in H12�H12 equals H12 �ε3−ε4 H12 and vanishes since [e34, H12] = 0. 26 S. Khoroshkin and O. Ogievetsky On the other hand, the product E43 �E34 definitely contains E43E34 = E43 �0 E34. We thus conclude that the term z43 � z34 is absent in (5.35), that is X34 = 0. For n > 4, again by the stabilization law, we have a unique relation of the form (5.35). By uniqueness, it is invariant with respect to the transformations q̌3, q̌4, . . . , q̌n−1 which do not change the product z12 � z21. Since X34 = 0, we find, applying q̌4, q̌5, . . . , q̌n−1, that X3j = 0, j > 3, wherefrom we further conclude, applying q̌3, q̌4, . . . , q̌j−2, that Xij = 0, 2 < i < j. We get finally the following relation in Zn: z12 � z21 = h12 − t12 � t12 1 h̊12 − 1 + ∑ k=2 ,...,n zk1 � z1kX1k + ∑ k=3 ,...,n zk2 � z2kX2k (5.36) with known X12 = (̊h12 − 1)(̊h12 + 2) h̊12(̊h12 + 1) , X13 = (̊h12 − 1)(̊h13 + 2) h̊12̊h23(̊h13 + 1) , X23 = − h̊23 + 2 (̊h23 + 1)̊h13 . Applying to (5.36) the transformations q̌3, q̌4, . . . , q̌n−1 we find by uniqueness Xi,k+1 = h̊k,k+1 + 1 h̊k,k+1 · q̌k(Xik), i = 1, 2; k = 3, 4, . . . , n− 1, and thus X1k = (̊h12 − 1)(̊h1k + 2) h̊12̊h2k (̊h1k + 1) · k−1∏ a=3 h̊ak + 1 h̊ak , X2k = − h̊2k + 2 (̊h2k + 1)̊h1k · k−1∏ a=3 h̊ak + 1 h̊ak for k > 2. After the renormalization (4.11) and the change of variables (4.2), the relations (5.36) with the obtained X1k and X2k turns into the relation (4.18) for i = 1 and j = 2. Applying the transformations from the braid group, we obtain the rest of the relations from the list (4.18). 5.4 Proof of Theorem 3 For the proof of Theorem 3 we just apply the results of [7], which state that the system R is the system of defining relations once it is satisfied in the algebra Zn. Remark 1. An attentive look shows that the system R is closed under the anti-involution ε; that is, ε transforms any relation from R into a linear over U(h) combination of relations from R. Moreover, R and ε(R) are equivalent over U(h). Indeed, all relations in Section (5.3) were derived in three steps: first we derive a relation in Zn with n ≤ 4; next by the stabilization principle we extend the derived relation to Zn with arbitrary n; and then we find the whole list of relations of a given (sub)type by applying the braid group transformations (products of the generators q̌i). Due to (3.6) one could use q̌−1 i instead of q̌i equivalently over U(h). A straightforward calculation establishes the equivalence of the extended to arbitrary n lists R and ε(R) over U(h) for Zn with n ≤ 4 (this verification is lengthy for some relations). Then with the help of (4.6) we finish the check of the equivalence of R and ε(R) over U(h) for Zn with arbitrary n. Similar arguments establish the equivalence of R and ω(R) over U(h); here ω is the invo- lution defined in (3.8). In [7] this equivalence was obtained differently, as a by-product of the equivalence, over U(h), of the system R and the system (3.3) of ordering relations. Structure Constants of Diagonal Reduction Algebras of gl Type 27 6 Examples: sl3 and sl2 In this section we write down the complete list of ordering relations for the diagonal reduction algebras DR(sl3) and DR(sl2). For completeness we include the formulas for the action of the braid group generators and the expressions for the central elements. We first give the list of relations for sl3. It is straightforward to give the list for sl2 directly; we comment however on how the list of relations and the expressions for the central elements for sl(2) can be obtained by the cut procedure. The list of relations for gl3 follows immediately from the list for sl3. 1. Relations for DR(sl3). We write the ordering relations for the natural set of genera- tors zij , without redefinitions. We use here the following notation for sl3: zα := z12, zβ := z23, zα+β := z13, z−α := z21, z−β := z23, z−α−β := z31, tα := t12, tβ := t23, hα := h12, hβ := h23. The relations are given for the following order �̀ (this order was used in the proof in [7] of the completeness of the set of relations): zα+β �̀ zα �̀ zβ �̀ tβ �̀ tα �̀ z−β �̀ z−α �̀ z−α−β. (6.1) Due to the established in Theorem 3 and remarks in Section 4.4 bijection between the set R and the set R≺ of defining relations, one can divide the ordering relations into the types, in the same way as we divided the defining relations from the list R. The relations of type 1 are immediately rewritten as ordering relations: zα+β � zα = zα � zα+β hβ + 2 hβ + 1 , (6.2) zα+β � zβ = zβ � zα+β hα + 2 hα + 1 , (6.3) zα � z−β = z−β � zα hα + hβ + 3 hα + hβ + 2 , (6.4) zβ � z−α = z−α � zβ hα + hβ + 3 hα + hβ + 2 , (6.5) z−α � z−α−β = z−α−β � z−α hβ + 2 hβ + 1 , (6.6) z−β � z−α−β = z−α−β � z−β hα + 2 hα + 1 . (6.7) The relations of type 2 are absent for sl3. The ordering relations corresponding to the relations of type 3 we collect according to their weights. For each weight there is one relation of subtype (3a) and two relations of subtype (3b). Weight α+ β: zα � zβ = −tα � zα+β 1 hα + 1 − tβ � zα+β 1 hβ + 1 + zβ � zα, (6.8) zα+β � tα = tα � zα+β hαhβ + h2 β + 2hα + 6hβ + 9 (hβ + 2)(hα + hβ + 3) + tβ � zα+β h2 β + hα + 6hβ + 9 (hβ + 1)(hβ + 2)(hα + hβ + 3) − zβ � zα hα + 2hβ + 6 (hα + 2)(hβ + 2) , (6.9) zα+β � tβ = tα � zα+β hβ (hβ + 2)(hα + hβ + 3) + tβ � zα+β hβ ( hαhβ + h2 β + 3hα + 7hβ + 11 ) (hβ + 1)(hβ + 2)(hα + hβ + 3) 28 S. Khoroshkin and O. Ogievetsky + zβ � zα 2hα + hβ + 6 (hα + 2)(hβ + 2) . (6.10) Weight α: zα � tα = tα � zα hα + 4 hα + 2 − z−β � zα+β hα + 2hβ + 6 (hβ + 1)(hα + hβ + 3) , (6.11) zα � tβ = −tα � zα 1 hα + 2 + tβ � zα + z−β � zα+β 2hα + hβ + 6 (hβ + 1)(hα + hβ + 3) , (6.12) zα+β � z−β = −tα � zα hβ − 1 hβ(hα + hβ + 2) − tβ � zα hα + 2hβ + 2 hβ(hα + hβ + 2) + z−β � zα+β (hβ + 2)(hβ − 1) hβ(hβ + 1) . (6.13) Weight β: zβ � tα = tα � zβ − tβ � zβ 1 hβ + 2 − z−α � zα+β hα + 2hβ + 6 (hα + 1)(hα + hβ + 3) , (6.14) zβ � tβ = tβ � zβ hβ + 4 hβ + 2 + z−α � zα+β 2hα + hβ + 6 (hα + 1)(hα + hβ + 3) , (6.15) zα+β � z−α = tα � zβ 2hα + hβ + 2 hα(hα + hβ + 2) + tβ � zβ hα − 1 hα(hα + hβ + 2) + z−α � zα+β (hα + 2)(hα − 1) hα(hα + 1) . (6.16) Weight −β: tα � z−β = z−β � tα − z−β � tβ 1 hβ − z−α−β � zα hα + 2hβ + 3 (hα + 2)(hα + hβ + 2) , (6.17) tβ � z−β = z−β � tβ hβ + 2 hβ + z−α−β � zα 2hα + hβ + 6 (hα + 2)(hα + hβ + 2) , (6.18) zα � z−α−β = z−β � tα 2hα + hβ + 2 (hα + 1)(hα + hβ + 1) + z−β � tβ hα (hα + 1)(hα + hβ + 1) + z−α−β � zα hα(hα + 3) (hα + 1)(hα + 2) . (6.19) Weight −α: tα � z−α = z−α � tα hα + 2 hα − z−α−β � zβ hα + 2hβ + 6 (hβ + 2)(hα + hβ + 2) , (6.20) tβ � z−α = −z−α � tα 1 hα + z−α � tβ + z−α−β � zβ 2hα + hβ + 3 (hβ + 2)(hα + hβ + 2) , (6.21) zβ � z−α−β = −z−α � tα hβ (hβ + 1)(hα + hβ + 1) − z−α � tβ hα + 2hβ + 2 (hβ + 1)(hα + hβ + 1) + z−α−β � zβ hβ(hβ + 3) (hβ + 1)(hβ + 2) . (6.22) Weight −α− β: z−β � z−α = −z−α−β � tα 1 hα − z−α−β � tβ 1 hβ + z−α � z−β, (6.23) Structure Constants of Diagonal Reduction Algebras of gl Type 29 tα � z−α−β = z−α−β � tα hαhβ + h2 β + hα + 3hβ + 3 (hβ + 1)(hα + hβ + 1) + z−α−β � tβ h2 β + hα + 4hβ + 3 hβ(hβ + 1)(hα + hβ + 1) − z−α � z−β hα + 2hβ + 3 (hα + 1)(hβ + 1) , (6.24) tβ � z−α−β = z−α−β � tα hβ − 1 (hβ + 1)(hα + hβ + 1) + z−α−β � tβ (hβ − 1)(hαhβ + h2 β + 2hα + 4hβ + 3) hβ(hβ + 1)(hα + hβ + 1) + z−α � z−β 2hα + hβ + 3 (hα + 1)(hβ + 1) . (6.25) Finally, we rewrite the relations of the type 4, that is, of weight 0, in the form of ordering relations. In addition to the general commutativity relation (subtype (4a)) tβ � tα = tα � tβ, (6.26) we have three relations of subtype (4b): zα � z−α = hα − tα � tα 1 hα + z−α � zα hα(hα + 3) (hα + 1)(hα + 2) − z−β � zβ hβ + 3 (hβ + 2)(hα + hβ + 2) + z−α−β � zα+β hα(hα + hβ + 4) (hα + 1)(hβ + 1)(hα + hβ + 3) , (6.27) zβ � z−β = hβ − tβ � tβ 1 hβ − z−α � zα hα + 3 (hα + 2)(hα + hβ + 2) + z−β � zβ hβ(hβ + 3) (hβ + 1)(hβ + 2) + z−α−β � zα+β hβ(hα + hβ + 4) (hα + 1)(hβ + 1)(hα + hβ + 3) , (6.28) zα+β � z−α−β = hαhβ(hα + hβ + 2) (hα + 1)(hβ + 1) − ( tα � tα hβ hβ + 1 + 2tα � tβ + tβ � tβ hα hα + 1 ) 1 hα + hβ + 1 (6.29) − z−α � zα hα(hα + 3) (hα + 1)(hα + 2)(hβ + 1) − z−β � zβ hβ(hβ + 3) (hβ + 1)(hβ + 2)(hα + 1) + z−α−β � zα+β hαhβ(hα + hβ + 4)(̊h2 αh̊β + h̊αh̊ 2 β + h̊2 α + h̊αh̊β + h̊2 β) (hα + 1)2(hβ + 1)2(hα + hβ + 2)(hα + hβ + 3) , where in one factor in the numerator of the last coefficient we returned to the notation h̊α = hα + 1 and h̊β = hβ + 1 to make the expression fit into the line. 2. Relations for DR(gl3). The ordering relations for the reduction algebra DR(gl3) are easily restored from the list (6.2)–(6.29): the gl(3) generators t1, t2 and t3, with tα = t1− t2 and tβ = t2 − t3, can be written as t1 = 1 3 (2tα + tβ + I(3,t)), t2 = 1 3 (−tα + tβ + I(3,t)), t1 = 1 3 (−tα − 2tβ + I(3,t)), where I(3,t) is the image of the central generator of gl(3), I(3,t) = t1 + t2 + t3. Since I(3,t) is central, one immediately writes relations for DR(gl3). We illustrate it on the example of relations between the generator zα and the gl(3) generators t1, t2 and t3: zαt1 = t1zα hα + 3 hα + 2 − t2zα 1 hα + 2 − z−βzα+β hβ + 2 (hβ + 1)(hα + hβ + 3) , 30 S. Khoroshkin and O. Ogievetsky zαt2 = −t1zα 1 hα + 2 + t2zα hα + 3 hα + 2 + z−βzα+β hα + hβ + 4 (hβ + 1)(hα + hβ + 3) , zαt3 = t3zα − z−βzα+β hα + 2 (hβ + 1)(hα + hβ + 3) . 3. Braid group action. There are two braid group generators, q̌α and q̌β, for the diagonal reduction algebra DR(sl3). Given the action of q̌α, the action of q̌β on DR(sl3) can be recon- structed by using the automorphism ω, see (3.8), arising from the outer automorphism of the root system of sl3, which exchanges the roots α and β, q̌β = ωq̌αω −1. The action of the automorphism ω on the Cartan subalgebra 〈hα, hβ〉 of the diagonal Lie alge- bra sl3 and on the generators of the reduction algebra DR(sl3) reads hα ↔ hβ, tα ↔ tβ, zα ↔ zβ, z−α ↔ z−β, zα+β ↔ −zα+β, z−α−β ↔ −z−α−β. The action of the braid group generator q̌α on the Cartan subalgebra 〈hα, hβ〉 of the diagonal Lie algebra sl3 reads: q̌α(hα) = −hα − 2, q̌α(hβ) = hα + hβ + 1. (6.30) This action reduces to the standard action of the Weyl group for the shifted generators h̊α = hα + 1 and h̊β = hβ + 1. The action of q̌α on the zero weight generators {tα, tβ} of the diagonal reduction algebra DR(sl3) is given by: q̌α(tα) = −tα hα + 2 hα , q̌α(tβ) = tα hα + 1 hα + tβ. (6.31) Finally, the action of q̌α on the rest of the generators is q̌α(zα) = −z−α hα + 1 hα − 1 , q̌α(z−α) = −zα, q̌α(zβ) = zα+β, q̌α(zα+β) = −zβ hα + 1 hα , (6.32) q̌α(z−α−β) = −z−β, q̌α(z−β) = z−α−β hα + 1 hα . The set of ordering relations (6.2)–(6.29) is covariant with respect to the braid group gener- ated by q̌α and q̌β. “Covariant” means that the elements of the braid group map a relation to a linear over U(h) combination of relations. For example, the operator q̌α, up to multiplicative factors from U(h), transforms the relation (6.27) into itself and permutes the relations (6.28) and (6.29). Due to the choice (6.1) of the order, the set of relations (6.2)–(6.29) is invariant with respect to the anti-involution ε. The set of relations (6.2)–(6.29) is covariant under the involution ω as well. 4. Central elements of DR(sl3). The degree 1 and degree 2 (in generators zij) central elements of the reduction algebra DR(sl3) are: C{DR(sl3),1} = tα(2hα + hβ + 6) + tβ(hα + 2hβ + 6), C{DR(sl3),2} = 1 3 (tα � tα + tβ � tβ + tα � tβ + h2 α + h2 β + hαhβ) Structure Constants of Diagonal Reduction Algebras of gl Type 31 + z−α � zα hα + 3 hα + 2 + z−β � zβ hβ + 3 hβ + 2 + z−α−β � zα+β hα + hβ + 4 hα + hβ + 3 ( 1 + 1 hα + 1 + 1 hβ + 1 ) + 2(hα + hβ). Both Casimir operators, C{DR(sl3),1} and C{DR(sl3),2} arise from the quadratic Casimir opera- tor C{sl3,2} of the Lie algebra sl3, whose ordered form is C{sl3,2} = (E−αEα + E−βEβ + E−α−βEα+β) + 1 3 (H2 α +H2 β +HαHβ) +Hα +Hβ. The operator C{DR(sl3),1} is the image of C{sl3,2} ⊗ 1 − 1 ⊗ C{sl3,2} and the operator C{DR(sl3),2} is the image of C{sl3,2} ⊗ 1 + 1⊗ C{sl3,2}. We calculate C{sl3,2} ⊗ 1 + 1⊗ C{sl3,2} and replace the multiplication by the product �. Here one needs, in addition to (5.30)–(5.34), the expression for H23 �H23 which is obtained by applying the involution ω to (5.31) and the equality (in the notation of Section 5.3): H12 �H23 ≡ H12H23 + E21E12 2 h̊12 + 1 + E32E23 2 h̊23 + 1 − E31E13 ( 1 + 2 h̊12 + 1 + 2 h̊23 + 1 ) 1 h̊13 + 1 . The central elements C{DR(sl3),1} and C{DR(sl3),2} are invariant with respect to the braid group: q̌α ( C{DR(sl3),i}) = C{DR(sl3),i}, q̌β ( C{DR(sl3),i}) = C{DR(sl3),i}, i = 1, 2. The central elements C{DR(sl3),1} and C{DR(sl3),2} are invariant with respect to the anti-involution ε and the involution ω as well. 5. Diagonal reduction algebra DR(sl2). For the diagonal reduction algebra of sl2 we use the following notation: z+ := zα, z− := z−α, t := tα, h := hα. The cut provides the following description of the algebra Z2 with generators z+, z− and t: z+ � t = t � z+ h+ 4 h+ 2 , (6.33) z+ � z− = h− t � t1 h + z− � z+ h(h+ 3) (h+ 1)(h+ 2) , (6.34) t � z− = z− � t h+ 2 h . (6.35) The Casimir operators for DR(sl2) are C{DR(sl2),1} := (h+ 2)t, (6.36) C{DR(sl2),2} := z− � z+ (h+ 3) (h+ 2) + t � t1 4 + h(h+ 4) 4 . (6.37) Both operators, C{DR(sl2),1} and C{DR(sl2),2} arise from the quadratic Casimir operator C{sl2,2} of the Lie algebra sl2, C{sl2,2} = E−E+ + 1 4 H(H+ 2), 32 S. Khoroshkin and O. Ogievetsky C{DR(sl2),1} is the image of C{sl2,2} ⊗ 1− 1⊗ C{sl2,2} and C{DR(sl2),2} is the image of C{sl2,2} ⊗ 1 + 1⊗ C{sl2,2}. The Casimir operators can be obtained by the cutting also, as explained in Subsection 4.6, see Proposition 7. One replaces the sl3 generators by the gl3 generators in the Casimir operators for sl3 then cuts and rewrites, using the notation (3.9) and (3.10), the result according to the gl2 formulas t (2) 1 = 1 2 ( t+ I(2,t) ) , t (2) 2 = 1 2 ( − t+ I(2,t) ) , t = t (2) 1 − t (2) 2 , I(2,t) = t (2) 1 + t (2) 2 . The cut of C{DR(sl3),1} is 3 2 C{DR(sl2),1} + 1 2 I(2,t) � ( I(2,h) + 6 ) − t(3) 3 � ( I(2,h) + 6 ) − I(2,t)h3 + 2t (3) 3 h3 (6.38) and the cut of C{DR(sl3),2} is C{DR(sl2),2} + 1 12 I(2,t) � I(2,t) + 1 12 I(2,h) � ( I(2,h) + 12 ) − 1 3 I(2,t) � t(3) 3 − ( 1 3 I(2,h) + 2 ) h3 + 1 3 ( t (3) 3 � t (3) 3 + h2 3 ) . (6.39) As expected, the coefficients of (t (3) 3 )�ihj3 for all i and j in the expressions (6.38) and (6.39) are central elements of the algebra Z2. Due to (6.30), (6.31) and (6.32), the action of the braid group generator reads q̌(h) = −h− 2, q̌(t) = −th+ 2 h , q̌(z+) = −z− h+ 1 h− 1 , q̌(z−) = −z+. (6.40) It preserves the commutation relations of DR(sl2). The Casimir operators (6.36) and (6.37) are invariant under the transformation (6.40) and under the anti-involution ε. It should be noted that q̌ can be included in a family of more general automorphisms of the reduction algebra DR(sl2). Lemma 8. The most general automorphism of the reduction algebra DR(sl2) transforming the weights of elements in the same way as the operator q̌ and linear over U(h) in the genera- tors z+, z− and t is h 7→ −h− 2, t 7→ βt h+ 2 h , z+ 7→ z− 1 (h− 1)γ(h) , z− 7→ z+(h+ 3)γ(h+ 2), (6.41) where β = ±1 is a constant and γ(h) is an arbitrary function. Proof. We are looking for an invertible transformation which preserves the relations (6.33)– (6.35) and has the form h 7→ f1(h), t 7→ tf2(h), z+ 7→ z−f3(h), z− 7→ z+f4(h) (6.42) with f1(h), f2(h), f3(h), f4(h) ∈ U(h). Applying the transformation (6.42) to the relations (6.33) and (6.35), we find (after simplifications) the conditions: (h+ 2) ( f1(h) + 4 ) f2(h− 2)− x ( f1(h) + 2 ) f2(h) = 0, (6.43) (h+ 4) ( f1(h) + 2 ) f2(h)− (x+ 2)f1(h)f2(h+ 2) = 0. (6.44) Structure Constants of Diagonal Reduction Algebras of gl Type 33 Replacing h by h − 2 in the second equation and then excluding f2 from the system (6.43), (6.44), we obtain the difference equation f1(h)− f1(h− 2) + 2 = 0, whose general solution in U(h) is f1(h) = −h+ c, (6.45) where c is a constant. Applying the transformation (6.42) to the relation (6.34) and collecting the free term and the terms with t � t and z− � z+, we find (after simplifications) 1 + h ( f1(h) + 3 ) G(h)( f1(h) + 2 )( f1(h) + 1 ) = 0, (6.46) f2(h)2 f1(h) + f1(h) ( f1(h) + 3 ) G(h) h ( f1(h) + 2 )( f1(h) + 1 ) = 0, (6.47) G(h+ 2) + h(h+ 3)f1(h) ( f1(h) + 3 ) (h+ 1)(h+ 2) ( f1(h) + 2 )( f1(h) + 1 )G(h) = 0, (6.48) where G(h) := f3(h)f4(h− 2). Excluding G from the system (6.46), (6.47), we obtain f1(h)2 = h2f2(h)2 or f2(h) = β f1(h) h (6.49) with β2 = 1. The substitution of (6.45) and (6.49) into (6.43) leads to c = −2 and it then follows from (6.46) that G(h) = h+ 1 h− 1 . The remaining relation (6.48) is now automatically satisfied. The proof is finished. � The Casimir operator C{DR(sl2),2} is invariant under the general automorphism (6.41). The Casimir operator C{DR(sl2),1} is invariant under the automorphism (6.41) iff β = −1. The map q̌ defined by (6.40) is a particular choice of (6.41), corresponding to β = −1 and γ(h) = − 1 h+1 . The map (6.40) is not an involution (but it squares to the identity on the weight zero subspace of the algebra). However, the general map (6.41) squares to the identity on the whole algebra iff the function γ is odd, γ(−h) = −γ(h). Acknowledgments We thank Löıc Poulain d’Andecy for computer calculations for the algebra Z4. We thank Elena Ogievetskaya for the help in preparation of the manuscript. The present work was partly done during visits of S.K. to CPT and CIRM in Marseille. The authors thank both Institutes. S.K. was supported by the RFBR grant 11-01-00962, joint CNRS-RFBR grant 09-01-93106- NCNIL, and by Federal Agency for Science and Innovations of Russian Federation under contract 14.740.11.0347. 34 S. Khoroshkin and O. Ogievetsky References [1] Arnaudon D., Buffenoir E., Ragoucy E., Roche Ph., Universal solutions of quantum dynamical Yang–Baxter equations, Lett. Math. Phys. 44 (1998), 201–214, q-alg/9712037. [2] Asherova R.M., Smirnov Yu.F., Tolstoy V.N., Projection operators for simple Lie groups. II. General scheme for construction of lowering operators. The groups SU(n), Teoret. Mat. Fiz. 15 (1973), 107–119 (English transl.: Theoret. and Math. Phys. 15 (1973), 392–401). [3] Asherova R.M., Smirnov Yu.F., Tolstoy V.N., Description of a certain class of projection operators for complex semi-simple Lie algebras, Mat. Zametki 26 (1979), 15–25 (English transl.: Math. Notes 26 (1979), 499–504). [4] Cherednik I., Quantum groups as hidden symmetries of classical representation theory, in Differential Ge- ometric Methods in Theoretical Physics (Chester, 1988), Editor A.I. Solomon, World Sci. Publ., Teaneck, NJ, 1989, 47–54. [5] Khoroshkin S., An extremal projector and dynamical twist, Teoret. Mat. Fiz. 139 (2004), 158–176 (English transl.: Theoret. and Math. Phys. 139 (2004), 582–597). [6] Khoroshkin S., Ogievetsky O., Mickelsson algebras and Zhelobenko operators, J. Algebra 319 (2008), 2113– 2165, math.QA/0606259. [7] Khoroshkin S., Ogievetsky O., Diagonal reduction algebras of gl type, Funktsional. Anal. i Prilozhen. 44 (2010), no. 3, 27–49 (English transl.: Funct. Anal. Appl. 44 (2010), 182–198), arXiv:0912.4055. [8] Lepowsky J., McCollum G.W., On the determination of irreducible modules by restriction to a subalgebra, Trans. Amer. Math. Soc. 176 (1973), 45–47. [9] Olshanski G., Extension of the algebra U(g) for infinite-dimensional classical Lie algebras g, and the Yangians Y (gl(m)), Dokl. Akad. Nauk SSSR 297 (1987), 1050–1054 (English transl.: Soviet Math. Dokl. 36 (1988), 569–573). [10] Zhelobenko D., Representations of reductive Lie algebras, Nauka, Moscow, 1994 (in Russian). http://dx.doi.org/10.1023/A:1007498022373 http://arxiv.org/abs/q-alg/9712037 http://dx.doi.org/10.1007/BF01028268 http://dx.doi.org/10.1007/BF01140268 http://dx.doi.org/10.1023/B:TAMP.0000022749.42512.fd http://dx.doi.org/10.1016/j.jalgebra.2007.04.020 http://arxiv.org/abs/math.QA/0606259 http://dx.doi.org/10.1007/s10688-010-0023-0 http://arxiv.org/abs/0912.4055 http://www.ams.org/leavingmsn?url=http://dx.doi.org/10.2307/1996195 1 Introduction 2 Notation 3 Reduction algebra Zn 4 Main results 4.1 New variables 4.2 Braid group action 4.3 Defining relations 4.4 Limit 4.5 sln 4.6 Stabilization and cutting 5 Proofs 5.1 Tensor J 5.2 Braid group action 5.3 Derivation of relations 5.4 Proof of Theorem 3 6 Examples: sl3 and sl2 References

Structure Constants of Diagonal Reduction Algebras of gl Type

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