Differential Calculus on h-Deformed Spaces

We construct the rings of generalized differential operators on the h-deformed vector space of gl-type. In contrast to the q-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of h-deformed differential operators Diffh,σ(n) is labeled by...

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Datum:	2017
Hauptverfasser:	Herlemont, B., Ogievetsky, O.
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Veröffentlicht:	Інститут математики НАН України 2017
Schriftenreihe:	Symmetry, Integrability and Geometry: Methods and Applications
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spelling	irk-123456789-1492742019-02-20T01:23:41Z Differential Calculus on h-Deformed Spaces Herlemont, B. Ogievetsky, O. We construct the rings of generalized differential operators on the h-deformed vector space of gl-type. In contrast to the q-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of h-deformed differential operators Diffh,σ(n) is labeled by a rational function σ in n variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings Diffh,σ(n). 2017 Article Differential Calculus on h-Deformed Spaces / B. Herlemont, O. Ogievetsky // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 16S30; 16S32; 16T25; 13B30; 17B10; 39A14 DOI:10.3842/SIGMA.2017.082 http://dspace.nbuv.gov.ua/handle/123456789/149274 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 construct the rings of generalized differential operators on the h-deformed vector space of gl-type. In contrast to the q-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of h-deformed differential operators Diffh,σ(n) is labeled by a rational function σ in n variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings Diffh,σ(n).
format	Article
author	Herlemont, B. Ogievetsky, O.
spellingShingle	Herlemont, B. Ogievetsky, O. Differential Calculus on h-Deformed Spaces Symmetry, Integrability and Geometry: Methods and Applications
author_facet	Herlemont, B. Ogievetsky, O.
author_sort	Herlemont, B.
title	Differential Calculus on h-Deformed Spaces
title_short	Differential Calculus on h-Deformed Spaces
title_full	Differential Calculus on h-Deformed Spaces
title_fullStr	Differential Calculus on h-Deformed Spaces
title_full_unstemmed	Differential Calculus on h-Deformed Spaces
title_sort	differential calculus on h-deformed spaces
publisher	Інститут математики НАН України
publishDate	2017
url	http://dspace.nbuv.gov.ua/handle/123456789/149274
citation_txt	Differential Calculus on h-Deformed Spaces / B. Herlemont, O. Ogievetsky // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 22 назв. — англ.
series	Symmetry, Integrability and Geometry: Methods and Applications
work_keys_str_mv	AT herlemontb differentialcalculusonhdeformedspaces AT ogievetskyo differentialcalculusonhdeformedspaces
first_indexed	2025-07-12T21:13:30Z
last_indexed	2025-07-12T21:13:30Z
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fulltext	Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 082, 28 pages Differential Calculus on h-Deformed Spaces Basile HERLEMONT † and Oleg OGIEVETSKY †‡§ † Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France E-mail: herlemont.basile@hotmail.fr, oleg@cpt.univ-mrs.fr ‡ Kazan Federal University, Kremlevskaya 17, Kazan 420008, Russia § On leave of absence from P.N. Lebedev Physical Institute, Leninsky Pr. 53, 117924 Moscow, Russia Received April 18, 2017, in final form October 17, 2017; Published online October 24, 2017 https://doi.org/10.3842/SIGMA.2017.082 Abstract. We construct the rings of generalized differential operators on the h-deformed vector space of gl-type. In contrast to the q-deformed vector space, where the ring of differential operators is unique up to an isomorphism, the general ring of h-deformed differential operators Diffh,σ(n) is labeled by a rational function σ in n variables, satisfying an over-determined system of finite-difference equations. We obtain the general solution of the system and describe some properties of the rings Diffh,σ(n). Key words: differential operators; Yang–Baxter equation; reduction algebras; universal en- veloping algebra; representation theory; Poincaré–Birkhoff–Witt property; rings of fractions 2010 Mathematics Subject Classification: 16S30; 16S32; 16T25; 13B30; 17B10; 39A14 1 Introduction As the coordinate rings of q-deformed vector spaces, the coordinate rings of h-deformed vector spaces are defined with the help of a solution of the dynamical Yang–Baxter equation. The coordinate rings of h-deformed vector spaces appeared in several contexts. In [4] it was observed that such coordinate rings generate the Clebsch–Gordan coefficients for GL(2). These coordinate rings appear in the study of the cotangent bundle to a quantum group [1] and in the study of zero-modes in the WZNW model [1, 5, 7]. The coordinate rings of h-deformed vector spaces appear naturally in the theory of reduction algebras. The reduction algebras [9, 14, 17, 22] are designed to study the decompositions of representations of an associative algebra B with respect to its subalgebra B′. Let B′ be the universal enveloping algebra of a reductive Lie algebra g. Let M be a g-module and B the universal enveloping algebra of the semi-direct product of g with the abelian Lie algebra formed by N copies of M . Then the corresponding reduction algebra is precisely the coordinate ring of N copies of h-deformed vector spaces. We restrict our attention to the case g = gl(n). Let V be the tautological gl(n)-module and V ∗ its dual. We denote by V (n,N) the reduction algebra related to N copies of V and by V ∗(n,N) the reduction algebra related to N copies of V ∗. In this article we develop the differential calculus on the h-deformed vector spaces of gl-type as it is done in [19] for the q-deformed spaces. Formulated differently, we study the consistent, in the sense, explained in Section 3.2.1, pairings between the rings V (n,N) and V ∗(n,N ′). A consistent pairing allows to construct a flat deformation of the reduction algebra, related to N copies of V and N ′ copies of V ∗. We show that for N > 1 or N ′ > 1 the pairing is essentially This paper is a contribution to the Special Issue on Recent Advances in Quantum Integrable Systems. The full collection is available at http://www.emis.de/journals/SIGMA/RAQIS2016.html mailto:herlemont.basile@hotmail.fr mailto:oleg@cpt.univ-mrs.fr https://doi.org/10.3842/SIGMA.2017.082 http://www.emis.de/journals/SIGMA/RAQIS2016.html 2 B. Herlemont and O. Ogievetsky unique. However it turns out that for N = N ′ = 1 the result is surprisingly different from that for q-deformed vector spaces. The consistency leads to an over-determined system of finite- difference equations for a certain rational function σ, which we call “potential”, in n variables. The solution space W can be described as follows. Let K be the ground ring of characteristic 0 and K[t] the space of univariate polynomials over K. Then W is isomorphic to K[t]n modulo the (n − 1)-dimensional subspace spanned by n-tuples (tj , . . . , tj) for j = 0, 1, . . . , n − 2. Thus for each σ ∈ W we have a ring Diffh,σ(n) of generalized h-deformed differential operators. The polynomial solutions σ are linear combinations of complete symmetric polynomials; they correspond to the diagonal of K[t]n. The ring Diffh,σ(n) admits the action of the so-called Zhelobenko automorphisms if and only if the potential σ is polynomial. In Section 2 we give the definition of the coordinate rings of h-deformed vector spaces of gl-type. Section 3 starts with the description of two different known pairings between h-deformed vector spaces, that is, two different flat deformations of the reduction algebra related to V ⊕V ∗. The first deformation is the ring Diffh(n) which is the reduction algebra, with respect to gln, of the classical ring of polynomial differential operators. The second ring is related to the reduction algebra, with respect to gln, of the algebra U(gln+1). These two examples motivate our study. Then, in Section 3, we formulate the main question and results. We present the system of the finite-difference equations resulting from the Poincaré–Birkhoff–Witt property of the ring of generalized h-deformed differential operators. We obtain the general solution of the system and establish the existence of the potential. We give a characterization of polynomial potentials. We describe the centers of the rings Diffh,σ(n) and construct an isomorphism between a certain ring of fractions of the ring Diffh,σ(n) and a certain ring of fractions of the Weyl algebra. We describe a family of the lowest weight representations and calculate the values of central elements on them. We establish the uniqueness of the deformation in the situation when we have several copies of V or V ∗. Section 4 contains the proofs of the statements from Section 3. Notation. We denote by Sn the symmetric group on n letters. The symbol si stands for the transposition (i, i+ 1). Let h(n) be the abelian Lie algebra with generators h̃i, i = 1, . . . , n, and U(n) its universal enveloping algebra. Set h̃ij = h̃i − h̃j ∈ h(n). We define Ū(n) to be the ring of fractions of the commutative ring U(n) with respect to the multiplicative set of denominators, generated by the elements ( h̃ij + a )−1 , a ∈ Z, i, j = 1, . . . , n, i 6= j. Let ψi := ∏ k : k>i h̃ik, ψ ′ i := ∏ k : k<i h̃ik and χi := ψiψ ′ i, i = 1, . . . , n. (1.1) Let εj , j = 1, . . . , n, be the elementary translations of the generators of U(n), εj : h̃i 7→ h̃i + δji . For an element p ∈ Ū(n) we denote εj(p) by p[εj ]. We shall use the finite-difference operators ∆j defined by ∆jf := f − f [−εj ]. We denote by eL, L = 0, . . . , n, the elementary symmetric polynomials in the variables h̃1, . . . , h̃n, and by e(t) the generating function of the polynomials eL, eL = ∑ i1<···<iL h̃i1 · · · h̃iL , e(t) = n∑ L=0 eLt L = n∏ i=1 ( 1 + h̃it ) . We denote by R ∈ EndŪ(n) ( Ū(n)n ⊗Ū(n) Ū(n)n ) the standard solution of the dynamical Differential Calculus on h-Deformed Spaces 3 Yang–Baxter equation∑ a,b,u Rij ab Rbk ur[−εa] Rau mn = ∑ a,b,u Rjk ab[−εi] Ria mu Rub nr[−εm] of type A. The nonzero components of the operator R are Rij ij = 1 h̃ij , i 6= j, and Rij ji =  h̃2 ij − 1 h̃2 ij , i < j, 1, i ≥ j. (1.2) We shall need the following properties of R: Rij kl[εi + εj ] = Rij kl, i, j, k, l = 1, . . . , n, (1.3) Rij kl = 0 if (i, j) 6= (k, l) or (l, k), (1.4) R2 = Id. (1.5) We denote by Ψ ∈ EndŪ(n) ( Ū(n)n⊗Ū(n) Ū(n)n ) the dynamical version of the skew inverse of the operator R, defined by∑ k,l Ψik jl Rml nk [εm] = δinδ m j . (1.6) The nonzero components of the operator Ψ are, see [13], Ψij ij = Q+ i Q−j 1 h̃ij + 1 , Ψij ji =  1, i < j,( h̃ij − 1 )2 h̃ij ( h̃ij − 2 ) , i > j, (1.7) where Q±i = χi[±εi] χi . 2 Coordinate rings of h-deformed vector spaces Let F(n,N) be the ring with the generators xiα, i = 1, . . . , n, α = 1, . . . , N , and h̃i, i = 1, . . . , n, with the defining relations h̃ih̃j = h̃j h̃i, i, j = 1, . . . , n, (2.1) h̃ix jα = xjα ( h̃i + δji ) , i, j = 1, . . . , n, α = 1, . . . , N. (2.2) We shall say that an element f ∈ F(n,N) has an h(n)-weight ω ∈ h(n)∗ if h̃if = f ( h̃i + ω ( h̃i )) , i = 1, . . . , n. (2.3) The ring U(n) is naturally the subring of F(n,N). Let F̄(n,N) := Ū(n) ⊗U(n) F(n,N). The coordinate ring V(n,N) of N copies of the h-deformed vector space is the factor-ring of F̄(n,N) by the relations xiαxjβ = ∑ k,l Rij kl x kβxlα, i, j = 1, . . . , n, α, β = 1, . . . , N. (2.4) 4 B. Herlemont and O. Ogievetsky The ring V(n,N) is the reduction algebra, with respect to gln, of the semi-direct product of gln and the abelian Lie algebra V ⊕V ⊕· · ·⊕V (N times) where V is the (tautological) n-dimensional gln-module. According to the general theory of reduction algebras [9, 12, 22], V(n,N) is a free left (or right) Ū(n)-module; the ring V(n,N) has the following Poincaré–Birkhoff–Witt property: given an arbitrary order on the set xiα, i = 1, . . . , n, α = 1, . . . , N , the set of all ordered monomials in xiα is a basis of the left Ū(n)-module V(n,N). (2.5) Moreover, if {Rkl ij}ni,j,k,l=1 is an arbitrary array of functions in h̃i, i = 1, . . . , n, then the Poincaré– Birkhoff–Witt property of the algebra defined by the relations (2.4), together with the weight prescriptions (2.2), implies that R satisfies the dynamical Yang–Baxter equation when N ≥ 3. Similarly, let F∗(n,N) be the ring with the generators ∂̄iα, i = 1, . . . , n, α = 1, . . . , N , and h̃i, i = 1, . . . , n, with the defining relations (2.1) and h̃i∂̄jα = ∂̄jα ( h̃i − δji ) , i, j = 1, . . . , n, α = 1, . . . , N. (2.6) Let F̄ ∗ (n,N) := Ū(n)⊗U(n) F∗(n,N). The h(n)-weights are defined by the same equation (2.3). The coordinate ring V∗(n,N) of N copies of the “dual” h-deformed vector space is the factor-ring of F̄ ∗ (n,N) by the relations ∂̄lα∂̄kβ = ∑ i,j ∂̄jβ ∂̄iα Rij kl, k, l = 1, . . . , n, α, β = 1, . . . , N. (2.7) Again, the ring V∗(n,N) is the reduction algebra, with respect to gln, of the semi-direct product of gln and the abelian Lie algebra V ∗ ⊕ V ∗ ⊕ · · · ⊕ V ∗ (N times) where V ∗ is the gln-module, dual to V . The ring V∗(n,N) is a free left (or right) Ū(n)-module; it has a similar to V(n,N) Poincaré–Birkhoff–Witt property: given an arbitrary order on the set ∂̄iα , i = 1, . . . , n, α = 1, . . . , N , the set of all ordered monomials in ∂̄iα is a basis of the left Ū(n)-module V∗(n,N). (2.8) Again, the Poincaré–Birkhoff–Witt property of the algebra defined by the relations (2.7), to- gether with the weight prescriptions (2.6), implies that R satisfies the dynamical Yang–Baxter equation when N ≥ 3. For N = 1 we shall write V(n) and V∗(n) instead of V(n, 1) and V∗(n, 1). 3 Generalized rings of h-deformed differential operators 3.1 Two examples Before presenting the main question we consider two examples. 1. We denote by Wn the algebra of polynomial differential operators in n variables. It is the algebra with the generators Xj , Dj , j = 1, . . . , n, and the defining relations XiXj = XjXi, DiDj = DjDi, DiX j = δji +XjDi, i, j = 1, . . . , n. The map, defined on the set {eij}ni,j=1 of the standard generators of gln by eij 7→ XiDj , extends to a homomorphism U(gln)→Wn. The reduction algebra of Wn⊗U(gln) with respect to the diagonal embedding of U(gln) was denoted by Diffh(n) in [13]. It is generated, over Ū(n), by the images xi and ∂i, i = 1, . . . , n, of the generators Xi and Di. Let ∂̄i := ∂i ψi ψi[−εi] , Differential Calculus on h-Deformed Spaces 5 where the elements ψi are defined in (1.1). Then xi∂̄j = ∑ k,l ∂̄k Rki lj x l − δijσ (Diff) i , (3.1) where σ (Diff) i = 1, i = 1, . . . , n. The h(n)-weights of the generators are given by (2.2) and (2.6). Moreover, the set of the defining relations, over Ū(n), for the generators xi and ∂̄i, i = 1, . . . , n, consists of (2.4), (2.7) (with N = 1) and (3.1) (see [13, Proposition 3.3]). The algebra Diffh(n,N), formed by N copies of the algebra Diffh(n), was used in [8] for the study of the representation theory of Yangians, and in [13] for the R-matrix description of the diagonal reduction algebra of gln (we refer to [10, 11] for generalities on the diagonal reduction algebras of gl type). 2. Identifying each n × n matrix a with the larger matrix ( a 0 0 0 ) gives an embedding of gln into gln+1. The resulting reduction algebra RU(gln+1) gln , or simply Rgln+1 gln , was denoted by AZn in [22]. It is generated, over Ū(n), by the elements xi, yi, i = 1, . . . , n, and h̃n+1 = z − (n+ 1), where xi and yi are the images of the standard generators ei,n+1 and en+1,i of U(gln+1) and z is the image of the standard generator en+1,n+1. Let ∂̄i := yi ψi ψi[−εi] , where the elements ψi are defined in (1.1) (they depend on h̃1, . . . , h̃n only). The h(n)-weights of the generators are given by (2.2) and (2.6) while h̃n+1x i = xi ( h̃n+1 − 1 ) , h̃n+1∂̄i = ∂̄i ( h̃n+1 + 1 ) , i = 1, . . . , n. The set of the remaining defining relations consists of (2.4), (2.7) (with N = 1) and xi∂̄j = ∑ k,l ∂̄k Rki lj x l − δijσ (AZ) i , (3.2) where σ (AZ) i = −h̃i + h̃n+1 + 1, i = 1, . . . , n. The algebra AZn was used in [18] for the study of Harish-Chandra modules and in [20] for the construction of the Gelfand–Tsetlin bases [6]. The algebra AZn has a central element h̃1 + · · ·+ h̃n + h̃n+1. (3.3) In the factor-algebra AZn of AZn by the ideal, generated by the element (3.3), the relation (3.2) is replaced by xi∂̄j = ∑ k,l ∂̄k Rki lj x l − δijσ (AZ) i , (3.4) with σ (AZ) i = −h̃i − n∑ k=1 h̃k + 1, i = 1, . . . , n. 6 B. Herlemont and O. Ogievetsky 3.2 Main question and results 3.2.1 Main question Both rings, Diffh(n) and AZn satisfy the Poincaré–Birkhoff–Witt property. The only difference between these rings is in the form of the zero-order terms σ (Diff) i and σ (AZ) i in the cross- commutation relations (3.1) and (3.4) (compare to the ring of q-differential operators [19] where the zero-order term is essentially – up to redefinitions – unique). It is therefore natural to inves- tigate possible generalizations of the rings Diffh(n) and AZn. More precisely, given n elements σ1, . . . , σn of Ū(n), we let Diffh(σ1, . . . , σn) be the ring, over Ū(n), with the generators xi and ∂̄i, i = 1, . . . , n, subject to the defining relations (2.4), (2.7) (with N = 1) and the oscillator-like relations xi∂̄j = ∑ k,l ∂̄k Rki lj x l − δijσi. (3.5) The weight prescriptions for the generators are given by (2.2) and (2.6). The diagonal form of the zero-order term (the Kronecker symbol δij in the right hand side of (3.5)) is dictated by the h(n)-weight considerations. We shall study conditions under which the ring Diffh(σ1, . . . , σn) satisfies the Poincaré– Birkhoff–Witt property. More specifically, since the rings V(n) and V∗(n) both satisfy the Poincaré–Birkhoff–Witt property, our aim is to study conditions under which Diffh(σ1, . . . , σn) is isomorphic, as a Ū(n)-module, to V∗(n)⊗Ū(n) V(n). The assignment deg ( xi ) = deg ( ∂̄i ) = 1, i = 1, . . . , n, (3.6) defines the structure of a filtered algebra on Diffh(σ1, . . . , σn). The associated graded algebra is the homogeneous algebra Diffh(0, . . . , 0). This homogeneous algebra has the desired Poincaré– Birkhoff–Witt property because it is the reduction algebra, with respect to gln, of the semi-direct product of gln and the abelian Lie algebra V ⊕ V ∗. The standard argument shows that the ring Diffh(σ1, . . . , σn) can be viewed as a deforma- tion of the homogeneous ring Diffh(0, . . . , 0): for the generating set { x′i, ∂̄i } , where x′i = ~xi, all defining relations are the same except (3.5) in which σi gets replaced by ~σi; one can con- sider ~ as the deformation parameter. Thus our aim is to study the conditions under which this deformation is flat. 3.2.2 Poincaré–Birkhoff–Witt property It turns out that the Poincaré–Birkhoff–Witt property is equivalent to the system of finite- difference equations for the elements σ1, . . . , σn ∈ Ū(n). Proposition 3.1. The ring Diffh(σ1, . . . , σn) satisf ies the Poincaré–Birkhoff–Witt property if and only if the elements σ1, . . . , σn ∈ Ū(n) satisfy the following linear system of finite-difference equations h̃ij∆jσi = σi − σj , i, j = 1, . . . , n. (3.7) We postpone the proof to Section 4.1. Differential Calculus on h-Deformed Spaces 7 3.2.3 ∆-system The system (3.7) is closely related to the following linear system of finite-difference equations for one element σ ∈ Ū(n): ∆i∆j ( h̃ijσ ) = 0, i, j = 1, . . . , n. (3.8) We shall call it the “∆-system”. The ∆-system can be written in the form h̃ij∆j∆iσ = ∆iσ −∆jσ, i, j = 1, . . . , n. We describe the most general solution of the system (3.8). Definition 3.2. Let Wj , j = 1, . . . , n, be the vector space of the elements of Ū(n) of the form π ( h̃j ) χj where π ( h̃j ) is a univariate polynomial in h̃j , and χj is defined in (1.1). Let W be the sum of the vector spaces Wj , j = 1, . . . , n. Theorem 3.3. An element σ ∈ Ū(n) satisfies the system (3.8) if and only if σ ∈ W. The proof is in Section 4.2. The sum ∑ Wj is not direct. Definition 3.4. Let H be the K-vector space formed by linear combinations of the complete symmetric polynomials HL, L = 0, 1, 2, . . . , in the variables h̃1, . . . , h̃n, HL = ∑ i1≤···≤iL h̃i1 · · · h̃iL . Lemma 3.5. (i) Let L ∈ Z≥0. We have n∑ j=1 h̃Lj χj = { 0, L = 0, 1, . . . , n− 2, HL−n+1, L ≥ n− 1. (3.9) (ii) The space H is a subspace of W. Moreover, an element σ ∈ U(n) satisfies the system (3.8) if and only if σ ∈ H, that is, H =W ∩U(n). (3.10) The symmetric group Sn acts on the ring Ū(n) and on the space W by permutations of the variables h̃1, . . . , h̃n. We have H =WSn , (3.11) where W Sn denotes the subspace of Sn-invariants in W. (iii) Select j ∈ {1, . . . , n}. Then we have a direct sum decomposition W = ⊕ k : k 6=j Wk ⊕H. (3.12) The proof is in Section 4.2. Let t be an auxiliary indeterminate. We have a linear map of vector spaces K[t]n → W defined by (π1, . . . , πn) 7→ n∑ j=1 πj ( h̃j ) χj . It follows from Lemma 3.5 that this map is surjective and its kernel is the vector subspace of K[t]n spanned by n-tuples (tj , . . . , tj) for j = 0, 1, . . . , n − 2. The image of the diagonal in K[t]n, formed by n-tuples (π, . . . , π), is the space H. 8 B. Herlemont and O. Ogievetsky 3.2.4 Potential We shall give a general solution of the system (3.7). Proposition 3.6. Assume that the elements σ1, . . . , σn ∈ Ū(n) satisfy the system (3.7). Then there exists an element σ ∈ Ū(n) such that σi = ∆iσ, i = 1, . . . , n. We shall call the element σ the “potential” and write Diffh,σ(n) instead of Diffh(σ1, . . . , σn) if σi = ∆iσ, i = 1, . . . , n. According to Proposition 3.1, the ring Diffh,σ(n) satisfies the Poincaré–Birkhoff–Witt prop- erty iff the potential σ satisfies the ∆-system (3.8). In Section 4.4 we give two proofs of Proposition 3.6. In the first proof we directly describe the space of solutions of the system (3.7). As a by-product of this description we find that the potential exists and moreover belongs to the space W. The second proof uses a partial information contained in the system (3.7) and establishes only the existence of a potential and does not immediately produce the general solution of the system (3.7). Given the existence of a potential, the general solution is then obtained by Theorem 3.3. Let H′ be the K-vector space formed by linear combinations of the complete symmetric polynomials HL, L = 1, 2, . . . , and let W ′ = ⊕ k : k 6=1 Wk ⊕H′. (3.13) The potential σ is defined up to an additive constant, and it will be sometimes useful to uniquely define σ by requiring that σ ∈ W ′. 3.2.5 A characterization of polynomial potentials The polynomial potentials σ ∈ W can be characterized in different terms. The rings Diffh(n) and AZn admit the action of Zhelobenko automorphisms q̌1, . . . , q̌n−1 [9, 21]. Their action on the generators xi and ∂̄i, i = 1, . . . , n, is given by (see [13]) q̌i ( xi ) = −xi+1 h̃i,i+1 h̃i,i+1 − 1 , q̌i ( xi+1 ) = xi, q̌i ( xj ) = xj , j 6= i, i+ 1, q̌i(∂̄i) = − h̃i,i+1 − 1 h̃i,i+1 ∂̄i+1, q̌i ( ∂̄i+1 ) = ∂̄i, q̌i ( ∂̄j ) = ∂̄j , j 6= i, i+ 1, q̌i ( h̃j ) = h̃si(j). (3.14) Lemma 3.7. The ring Diffh,σ(n) admits the action of Zhelobenko automorphisms if and only if σ is a polynomial, σ ∈ H. The proof is in Section 4.5. In the examples discussed in Section 3.1, the ring Diffh(n) corresponds to σ = H1 and the ring AZn corresponds to σ = −H2 = − ∑ i,j : i≤j h̃ih̃j , ∆iH2 = h̃i + n∑ k=1 h̃k − 1. The question of constructing an associative algebra which contains U(gln) and whose reduction with respect to gln is Diffh,σ(n) for σ = Hk, k > 2, will be discussed elsewhere. Differential Calculus on h-Deformed Spaces 9 3.2.6 Center In [16] we have described the center of the ring Diffh(n). The center of the ring Diffh,σ(n), σ ∈ W, admits a similar description. Let Γi := ∂̄ix i for i = 1, . . . , n. Let c(t) = ∑ i e(t) 1 + h̃it Γi − ρ(t) = n∑ k=1 ckt k−1, (3.15) where t is an auxiliary variable and ρ(t) a polynomial of degree n − 1 in t with coefficients in Ū(n). Proposition 3.8. (i) Let σ ∈ W and σj = ∆jσ, j = 1, . . . , n. The elements c1, . . . , cn are central in the ring Diffh,σ(n) if and only if the polynomial ρ satisfies the system of finite-difference equations ∆jρ(t) = e(t) 1 + h̃jt σj . (3.16) (ii) For an arbitrary σ ∈ W the system (3.16) admits a solution. Since the system (3.16) is linear, it is sufficient to present a solution for an element σ ∈ Wk for each k = 1, . . . , n, that is, for σ = A ( h̃k ) χk , where A is a univariate polynomial. (3.17) The solution of the system (3.16) for the element σ of the form (3.17) is, up to an additive constant from K, ρ(t) = e(t) 1 + h̃kt σ. (iii) The center of the ring Diffh,σ(n) is isomorphic to the polynomial ring K[t1, . . . , tn]; the isomorphism is given by tj 7→ cj, j = 1, . . . , n. The proof is in Section 4.6. 3.2.7 Rings of fractions In [16] we have established an isomorphism between certain rings of fractions of the ring Diffh(n) and the Weyl algebra Wn. It turns out that when we pass to the analogous ring of fractions of the ring Diffh,σ(n), we loose the information about the potential σ. Thus we obtain the isomorphism with the same, as for the ring Diffh(n), ring of fractions of the Weyl algebra Wn. We denote, as for the ring Diffh(n), by S−1 x Diffh,σ(n) the localization of the ring Diffh,σ(n) with respect to the multiplicative set Sx generated by xj , j = 1, . . . , n. Lemma 3.9. Let σ and σ′ be two elements of the space W ′, see (3.13). (i) The rings S−1 x Diffh,σ(n) and S−1 x Diffh,σ′(n) are isomorphic. (ii) However, the rings Diffh,σ(n) and Diffh,σ′(n) are isomorphic, as filtered rings over Ū(n) (where the filtration is defined by (3.6)), if and only if σ = γσ′ for some γ ∈ K∗. The proof is in Section 4.7. 10 B. Herlemont and O. Ogievetsky 3.2.8 Lowest weight representations The ring Diffh,σ(n) has an n-parametric family of lowest weight representations, similar to the lowest weight representations of the ring Diffh(n), see [16]. We recall the definition. Let Dn be an Ū(n)-subring of Diffh,σ(n) generated by {∂̄i}ni=1. Let ~λ := {λ1, . . . , λn} be a sequence, of length n, of complex numbers such that λi − λj /∈ Z for all i, j = 1, . . . , n, i 6= j. Denote by M~λ the one-dimensional K-vector space with the basis vector \| 〉. The formulas h̃i : \| 〉 7→ λi\| 〉, ∂̄i : \| 〉 7→ 0, i = 1, . . . , n, (3.18) define the Dn-module structure on M~λ . The lowest weight representation of lowest weight ~λ is the induced representation Ind Diffh,σ(n) Dn M~λ . We describe the values of the central polynomial c(t), see (3.15), on the lowest weight repre- sentations. Proposition 3.10. The element c(t) acts on Ind Diffh,σ(n) Dn M~λ by the multiplication on the scalar −ρ(t)[−ε], where ε = ε1 + · · ·+ εn. (3.19) The proof is in Section 4.8. 3.2.9 Several copies The coexistence of several copies imposes much more severe restrictions on the flatness of the deformation. Namely, let L be the ring with the generators xiα, i = 1, . . . , n, α = 1, . . . , N ′, and ∂̄jβ, j = 1, . . . , n, β = 1, . . . , N subject to the following defining relations. The h(n)- weights of the generators are given by (2.2) and (2.6). The generators xiα satisfy the rela- tions (2.4). The generators ∂̄jβ satisfy the relations (2.7). We impose the general oscillator- like cross-commutation relations, compatible with the h(n)-weights, between the generators xiα and ∂̄jβ: xiα∂̄jβ = ∑ k,l ∂̄kβ Rki lj x lα − δijσiαβ, i, j = 1, . . . , n, α = 1, . . . , N ′, β = 1, . . . , N, with some σiαβ ∈ Ū(n). Lemma 3.11. Assume that at least one of the numbers N and N ′ is bigger than 1. Then the ring L has the Poincaré–Birkhoff–Witt property if and only if σiαβ = σαβ for some σαβ ∈ K. (3.20) The proof is in Section 4.9. Making the redefinitions of the generators, xiα Aαα′x iα′ and ∂̄iβ Bβ′ β ∂̄iβ′ with some A ∈ GL(N ′,K) and B ∈ GL(N,K) we can transform the matrix σαβ to the diagonal form, with the diagonal (1, . . . , 1, 0, . . . , 0). Therefore, the ring L is formed by several copies of the rings Diffh(n), V(n) and V∗(n). 4 Proofs of statements in Section 3.2 4.1 Poincaré–Birkhoff–Witt property. Proof of Proposition 3.1 The explicit form of the defining relations for the ring Diffh(σ1, . . . , σn) is xixj = h̃ij + 1 h̃ij xjxi, 1 ≤ i < j ≤ n, (4.1) Differential Calculus on h-Deformed Spaces 11 ∂̄i∂̄j = h̃ij − 1 h̃ij ∂̄j ∂̄i, 1 ≤ i < j ≤ n, (4.2) xi∂̄j =  ∂̄jx i, 1 ≤ i < j ≤ n, h̃ij ( h̃ij − 2 )( h̃ij − 1 )2 ∂̄jx i, n ≥ i > j ≥ 1, (4.3) xi∂̄i = ∑ j 1 1− h̃ij ∂̄jx j − σi, i = 1, . . . , n. (4.4) Proof of Proposition 3.1. We can consider (4.1), (4.2) and (4.3) as the set of ordering relations and use the diamond lemma [2, 3] for the investigation of the Poincaré–Birkhoff– Witt property. The relations (4.1), (4.2) and (4.3) are compatible with the h(n)-weights of the generators xi and ∂̄i, i = 1, . . . , n, so we have to check the possible ambiguities involving the generators xi and ∂̄i, i = 1, . . . , n, only. The properties (2.5) and (2.8) show that the ambiguities of the forms xxx and ∂̄∂̄∂̄ are resolvable. It remains to check the ambiguities xi∂̄j ∂̄k and xjxk∂̄i. (4.5) It follows from the properties (2.5) and (2.8) that the choice of the order for the generators with indices j and k in (4.5) is irrelevant. Besides, it can be verified directly that the ring Diffh(σ1, . . . , σn), with arbitrary σ1, . . . , σn ∈ Ū(n) admits an involutive anti-automorphism ε, defined by ε ( h̃i ) = h̃i, ε ( ∂̄i ) = ϕix i, ε ( xi ) = ∂̄iϕ −1 i , (4.6) where ϕi := ψi ψi[−εi] = ∏ k : k>i h̃ik h̃ik − 1 , i = 1, . . . , n. By using the anti-automorphism ε we reduce the check of the ambiguity xjxk∂̄i to the check of the ambiguity xi∂̄j ∂̄k. Since the associated graded algebra with respect to the filtration (3.6) has the Poincaré– Birkhoff–Witt property, we have, in the check of the ambiguity xi∂̄j ∂̄k, to track only those ordered terms whose degree is smaller than 3. We use the symbol u ∣∣ l.d.t. to denote the part of the ordered expression for u containing these lower degree terms. Check of the ambiguity xi∂̄j ∂̄k. We calculate, for i, j, k = 1, . . . , n, ( xi∂̄j ) ∂̄k ∣∣ l.d.t. = (∑ u,v Rui vj [εu]∂̄ux v − δijσi ) ∂̄k ∣∣ l.d.t. = − ∑ u Rui kj [εu]∂̄uσk − δijσi∂̄k, (4.7) and xi ( ∂̄j ∂̄k )∣∣ l.d.t. = xi ∑ a,b Rab kj ∂̄b∂̄a ∣∣ l.d.t. = ∑ a,b Rab kj [−εi] ∑ c,d Rci db[εc]∂̄cx d − δibσi  ∂̄a ∣∣ l.d.t. = − ∑ a,b,c Rab kj [−εi] Rci ab[εc]∂̄cσa − ∑ a Rai kj [−εi]σi∂̄a. (4.8) Comparing the resulting expressions in (4.7) and (4.8) and collecting coefficients in ∂̄u, we find the necessary and sufficient condition for the resolvability of the ambiguity xi∂̄j ∂̄k: Rui kj [εu]σk[εu] + δijδ u kσi = ∑ a,b Rab kj [−εi] Rui ab[εu]σa[εu] + Rui kj [−εi]σi, (4.9) i, k, j, u = 1, . . . , n. 12 B. Herlemont and O. Ogievetsky Shifting by −εu and using the property (1.3) together with the ice condition (1.4), we rewri- te (4.9) in the form Rui kj(σk − σi[−εu]) + δijδ u kσi[−εu] = ∑ a,b Rab kj Rui ab σa. (4.10) For j = k the system (4.10) contains no equations. For j 6= k we have two cases: • u = j and i = k. This part of the system (4.10) reads explicitly (see (1.2)) σk − σk[−εj ] = 1 h̃kj (σk − σj). This is the system (3.7). • u = k and i = j. This part of the system (4.10) reads explicitly 1 h̃kj (σk − σj [−εk]) + σj [−εk] = 1 h̃2 kj σk + h̃2 kj − 1 h̃2 kj σj , which reproduces the same system (3.7). 4.2 General solution of the system (3.8). Proofs of Theorem 3.3 and Lemma 3.5 We shall interpret elements of Ū(n) as rational functions on h∗ with possible poles on hyperplanes h̃ij + a = 0, a ∈ Z, i, j = 1, . . . , n, i 6= j. Let M be a subset of {1, . . . , n}. The symbol RMŪ(n) denotes the subring of Ū(n) consisting of functions with no poles on hyperplanes h̃ij + a = 0, a ∈ Z, i, j ∈ M, j 6= i. The symbol NMŪ(n) denotes the subring of Ū(n) consisting of functions which do not depend on variables h̃i, i ∈ M. We shall say that an element f ∈ Ū(n) is regular in h̃j if it has no poles on hyperplanes h̃jm + a = 0, a ∈ Z, m = 1, . . . , n, m 6= j. 1. Partial fraction decompositions. We will use partial fraction decompositions of an ele- ment f ∈ Ū(n) with respect to a variable h̃j for some given j. The partial fraction decomposition of f with respect to h̃j is the expression for f of the form f = Pj(f) + regj(f), where the elements Pj(f) and regj(f) have the following meaning. The “regular” part regj(f) is an element, regular in h̃j . The “principal” in h̃j part Pj(f) is Pj(f) = ∑ k : k 6=j Pj;k(f), where Pj;k(f) = ∑ a∈Z ∑ νa∈Z>0 ukaνa( h̃jk − a )νa , (4.11) with some elements ukaνa ∈ NjŪ(n); the sums are finite. The fact that the ring Ū(n) admits partial fraction decompositions (that is, that the ele- ments ukaνa and regj(f) belong to Ū(n)) is a consequence of the formula 1( h̃jk − a )( h̃jl − b ) = 1( h̃kl + a− b ) ( 1 h̃jk − a − 1 h̃jl − b ) . 2. Let D be a domain (a commutative algebra without zero divisors) over K. Let f be an element of D⊗K Ū(n). Set Yij(f) := ∆i∆j ( h̃ijf ) . (4.12) Differential Calculus on h-Deformed Spaces 13 Lemma 4.1. If Yij(f) = 0 for some i and j, i 6= j, then f can be written in the form f = A h̃ij +B, (4.13) with some A,B ∈ D⊗K Ri,jŪ(n). Proof. We write f in the form f = A( h̃ij − a1 )ν1(h̃ij − a2 )ν2 · · · (h̃ij − aM)νM +B, where a1 < a2 < · · · < aM , ν1, ν2, . . . , νM ∈ Z>0, A,B ∈ D ⊗K Ri,jŪ(n) and the element A is not divisible by any factor in the denominator. There is nothing to prove if A = 0. Assume that A 6= 0. Then 0 = Yij(f) = h̃ijA( h̃ij − a1 )ν1 · · · (h̃ij − aM)νM − ( h̃ij − 1 ) A[−εi]( h̃ij − a1 − 1 )ν1 · · · (h̃ij − aM − 1 )νM (4.14) − ( h̃ij + 1 ) A[−εj ]( h̃ij − a1 + 1 )ν1 · · · (h̃ij − aM + 1 )νM + h̃ijA[−εi − εj ]( h̃ij − a1 )ν1 · · · (h̃ij − aM)νM + Yij(B). The denominator ( h̃ij−aM−1 ) appears only in the second term in the right hand side of (4.14). It has therefore to be compensated by ( h̃ij − 1 ) in the numerator. Hence the only allowed value of aM is aM = 0 and moreover we have νM = 1. Similarly, the denominator ( h̃ij − a1 + 1 ) appears only in the third term in the right hand side of (4.14) and has to be compensated by (h̃ij + 1) in the numerator. Hence the only allowed value of a1 is a1 = 0 and we have ν1 = 1. The inequalities a1 < a2 < · · · < aM imply that M = 1 and we obtain the form (4.13) of f . � 3. Let f ∈ D⊗K Ū(n). We shall analyze the linear system of finite-difference equations Yij(f) = 0 for all i, j = 1, . . . , n, (4.15) where Yij are defined in (4.12). First we prove a preliminary result. We recall Definition 3.2 of the vector spaces Wi, i = 1, . . . , n. We select one of the variables h̃i, say, h̃1. Lemma 4.2. Assume that an element f ∈ D⊗K Ū(n) satisfies the system (4.15). Then f = n∑ j=2 Fj + ϑ, (4.16) where ϑ ∈ D⊗K U(n) and Fj = uj ( h̃j ) χj ∈ D⊗KWj (4.17) with some univariate polynomials uj ( h̃j ) , j = 2, . . . , n, with coefficients in D. Proof. Since Y1m(f) = 0, m = 2, . . . , n, Lemma 4.1 implies that the partial fraction decompo- sition of f with respect to h̃1 has the form f = n∑ m=2 βm h̃m1 + ϑ, (4.18) 14 B. Herlemont and O. Ogievetsky where βm ∈ D ⊗K N1Ū(n), m = 2, . . . , n, and ϑ ∈ D [ h̃1 ] ⊗K N1Ū(n). Substituting the expres- sion (4.18) for f into the equation Y1j(f) = 0, j = 2, . . . , n, we obtain 0 = Y1j(f) = ∆1∆j  ∑ m : m 6=1,j h̃1jβm h̃m1 − βj + h̃1jϑ  = ∆1∆j  ∑ m : m 6=1,j h̃1jβm h̃m1 + h̃1jϑ  (4.19) = ∑ m : m 6=1,j ( h̃1jβm h̃m1 − ( h̃1j + 1 ) βm[−εj ] h̃m1 − ( h̃1j − 1 ) βm h̃m1 + 1 + h̃1jβm[−εj ] h̃m1 + 1 ) + ∆1∆j ( h̃1jϑ ) . We used that βm ∈ D⊗KN1Ū(n) in the third and fourth equalities. For any m 6= 1, j, the terms containing the denominator h̃m1 in the expression (4.19) for Y1j(f) read 1 h̃m1 ( h̃1jβm − ( h̃1j + 1 ) βm[−εj ] ) . Therefore, h̃1jβm − ( h̃1j + 1 ) βm[−εj ] is divisible, as a polynomial in h̃1, by h̃m1, or, what is the same, the value of h̃1jβm − ( h̃1j + 1 ) βm[−εj ] at h̃1 = h̃m is zero. This means that 0 = h̃mjβm − ( h̃mj + 1 ) βm[−εj ] = ∆j ( h̃mjβm ) . Therefore, the element h̃mjβm does not depend on h̃j for any j > 1. We conclude that βm = um ( h̃m )∏ k : k 6=1,m h̃mk with some univariate polynomial um. We have proved that the element f has the form (4.16) where Fj , j = 2, . . . , n, are given by (4.17) and the element ϑ is regular in h̃1. A direct calculation shows that for any j = 2, . . . , n, the element Fj , given by (4.17), is a solution of the linear system (4.15). Therefore the regular in h̃1 part ϑ by itself satisfies the system Yij(ϑ) = 0. It is left to analyze the regular part ϑ. We use induction in n. For n = 2, the element ϑ is, by construction, a polynomial in h̃1 and h̃2. This is the induction base. We shall now prove that ϑ is a polynomial, with coefficients in D, in all n variables h̃1, . . . , h̃n. For arbitrary n > 2 we have ϑ ∈ D [ h̃1 ] ⊗K Ū ′ (n − 1) where we have denoted by Ū ′ (n − 1) the subring N1Ū(n) of Ū(n) consisting of functions not depending on h̃1. Since Yij(ϑ) = 0 for i, j = 2, . . . , n, we can use the induction hypothesis with n− 1 variables h̃2, . . . , h̃n over the ring D′ = D [ h̃1 ] . We now select the variable h̃2. It follows from the induction hypothesis that ϑ = ∑ m : m6=1,2 γ′m(h̃m)∏ l : l 6=1,m h̃ml + ϑ′, (4.20) where γ′m ( h̃m ) , m = 3, . . . , n, are univariate polynomials, with coefficients in D′, and the ele- ment ϑ′ is a polynomial, with coefficients in D′, in the variables h̃2, . . . , h̃n. We rewrite the Differential Calculus on h-Deformed Spaces 15 equality (4.20) in the form ϑ = ∑ m : m 6=1,2 γm ( h̃m, h̃1 )∏ l : l 6=1,m h̃ml + ϑ′, (4.21) with some polynomials γm, m = 3, . . . , n, in two variables, with coefficients in D; the element ϑ′ is a polynomial in all variables h̃1, . . . , h̃n with coefficients in D. The equation Y12(ϑ) = 0 for ϑ given by (4.21) reads 0 = ∑ m : m 6=1,2  h̃12γm h̃m2 ∏ l : l 6=1,2,m h̃ml − ( h̃12 − 1 ) γm[−ε1] h̃m2 ∏ l : l 6=1,2,m h̃ml − ( h̃12 + 1 ) γm( h̃m2 + 1 ) ∏ l : l 6=1,2,m h̃ml + h̃12γm[−ε1]( h̃m2 + 1 ) ∏ l : l 6=1,2,m h̃ml + Y12(ϑ′). (4.22) The terms containing the denominator h̃m2 in (4.22) read 1 h̃m2 ∏ l : l 6=1,2,m h̃ml ( h̃12γm − ( h̃12 − 1 ) γm[−ε1] ) . Therefore, the expression h̃12γm − ( h̃12 − 1 ) γm[−ε1] is divisible, as a polynomial in h̃2, by h̃2m = h̃2 − h̃m, so 0 = h̃1mγm − ( h̃1m − 1 ) γm[−ε1] = ∆1 ( h̃1mγm ) . Thus the product h̃1mγm, m = 3, . . . , n, does not depend on h̃1. Since γm, m = 3, . . . , n, is a polynomial, this implies that γm = 0. We conclude that ϑ = ϑ′ and is therefore a polynomial in all variables h̃1, . . . , h̃n. � 4. Now we refine the assertion of Lemma 4.2. We shall, at this stage, obtain the general solution of the system (4.15) in a form which does not exhibit the symmetry with respect to the permutations of the variables h̃1, . . . , h̃n. We recall Definition 3.4 of the vector space H. Lemma 4.3. (i) The general solution of the linear system (4.15) for an element f ∈ D ⊗K Ū(n) has the form f = n∑ j=2 Fj + ϑ, (4.23) where Fj ∈ D⊗KWj and ϑ ∈ D⊗K H. (4.24) (ii) The elements Fj, j = 2, . . . , n, and ϑ are uniquely defined. 16 B. Herlemont and O. Ogievetsky Proof. (i) In Lemma 4.2 we have established the decomposition (4.23) with ϑ ∈ D ⊗K U(n). We now prove the assertion (4.24). We first study the case n = 2. Let p ∈ D [ h̃1, h̃2 ] be a polynomial such that Y12(p) = 0. Since ∆1∆2 ( h̃12p ) = 0 we have ∆2(h̃12p) ∈ D [ h̃2 ] . It is a standard fact that the operator ∆2 is surjective on D [ h̃2 ] . This can be seen, for example, by noticing that the set h̃↑m2 := h̃2 ( h̃2 + 1 ) · · · ( h̃2 +m− 1 ) , m ∈ Z≥0, is a basis of D [ h̃2 ] over D, and ∆2 ( h̃↑m2 ) = mh̃↑m−1 2 . The surjectivity of ∆2 implies that ∆2 ( h̃12p ) = ∆2 ( w ( h̃2 )) for some polynomial w ( h̃2 ) ∈ D [ h̃2 ] . Then ∆2 ( h̃12p − w ( h̃2 )) = 0 so h̃12p − w ( h̃2 ) = v ( h̃1 ) for some polynomial v ( h̃1 ) ∈ D [ h̃1 ] . Therefore p = v ( h̃1 ) + w ( h̃2 ) h̃12 = v ( h̃1 ) − v ( h̃2 ) h̃12 + v ( h̃2 ) + w ( h̃2 ) h̃12 . Since p is a polynomial we must have w = −v. Thus p = v ( h̃1 ) − v ( h̃2 ) h̃12 , that is, p is a D-linear combination of complete symmetric polynomials in h̃1, h̃2. For arbitrary n, our polynomial ϑ is symmetric since, by the above argument, it is symmetric in every pair h̃i, h̃j of variables. Moreover, considered as a polynomial in a pair h̃i, h̃j , it is a D-linear combination of complete symmetric polynomials in h̃i, h̃j . It is then immediate that ϑ is a D-linear combination of complete symmetric polynomials in h̃1, . . . , h̃n. To finish the proof of the statement that the formula (4.23) gives the general solution of the system (4.15) it is left to check that the complete symmetric polynomials HL, L = 0, 1, . . . , in the variables h̃1, . . . , h̃n satisfy the system (4.15). Let s be an auxiliary variable and H(s) = ∞∑ L=0 HLs L = ∏ k 1 1− sh̃k (4.25) be the generating function of the elements HL, L = 0, 1, . . . It is sufficient to show that the formal power series (4.25) satisfies the system (4.15). Fix i, j ∈ {1, . . . , n}, i 6= j, and let ζij = 1 (1−h̃is)(1−h̃js) . The element ∆i ( h̃ij( 1− h̃is )( 1− h̃js )) = 1 1− h̃js ( h̃ij 1− h̃is − h̃ij − 1 1− ( h̃i − 1 ) s ) = 1( 1− h̃iτ )( 1− ( h̃i − 1 ) τ ) does not depend on h̃j so Yij(ζij) = 0. Therefore Yij(H(s)) = 0 since the factors other than ζij in the product in the right hand side of (4.25) do not depend on h̃i and h̃j . (ii) Finally, the summands in (4.23) are uniquely defined since (4.23) is a partial fraction decomposition of the element f in h̃1. � Differential Calculus on h-Deformed Spaces 17 5. Proof of Lemma 3.5(i). Let t be an auxiliary indeterminate. Multiplying by t−L−1 and taking sum in L, we rewrite (3.9) in the form n∑ j=1 1 t− h̃j 1 χj = 1 n∏ j=1 ( t− h̃j ) . The left hand side is nothing else but the partial fraction decomposition, with respect to t, of the product in the right hand side. 6. Proof of Theorem 3.3. The assertion of the Theorem follows immediately from the decomposition (4.23) in Lemma 4.3 and the identity (3.9). 7. Proof of Lemma 3.5(ii) and (iii). (ii) The formula (3.10) follows from the uniqueness of the decomposition (4.23) in Lemma 4.3. The element f of the form (4.23) is Sn-invariant if and only if f ∈ H and the assertion (3.11) follows. (iii) For j = 1 formula (3.12) is the uniqueness statement of Lemma 4.3. In the proof of Lemma 4.3 we could have selected any h̃j instead of h̃1. 4.3 System (3.7) We proceed to the study of the system (3.7), that is, the system of equations Zij = 0, i, j = 1, . . . , n, (4.26) where Zij = h̃ij∆jσi − σi + σj = −∆j (( h̃ji + 1 ) σi ) + σj . for the n-tuple σ1, . . . , σn ∈ Ū(n). 1. We use the equations Z1j , j = 2, . . . , n, to express the elements σj , j = 2, . . . , n, in terms of the element σ1: σj = ∆j (( h̃j1 + 1 ) σ1 ) = h̃j1∆j(σ1) + σ1. (4.27) Substituting the expressions (4.27) into the equations Zi1, i = 2, . . . , n, we find h̃i1 ( ∆1 ( h̃i1∆iσ1 + σ1 ) −∆iσ1 ) = 0. Simplifying by h̃i1 we obtain Wi = 0, i = 2, . . . , n, (4.28) where Wi = ∆1 ( h̃i1∆iσ1 + σ1 ) −∆iσ1 = ∆i ( ∆1 (( h̃i1 + 1 ) σ1 ) − σ1 ) = ∆i ( h̃i1σ1 − ( h̃i1 + 2 ) σ1[−ε1] ) . Substituting the expressions (4.27) into the equations Zij , i, j = 2, . . . , n, we find h̃i1 ( h̃ij∆i∆jσ1 + ∆jσ1 −∆iσ1 ) = 0. Simplifying by h̃i1, we obtain, with the notation (4.12), Yij(σ1) = 0, i, j = 2, . . . , n. (4.29) This is our first conclusion which we formulate in the following lemma. 18 B. Herlemont and O. Ogievetsky Lemma 4.4. If σ1, . . . , σn ∈ Ū(n) is a solution of the system (4.26) then the element σ1 sa- tisfies the equations (4.28) and (4.29). Conversely, if an element σ1 ∈ Ū(n) satisfies the equa- tions (4.28) and (4.29) then we reconstruct a solution of the system (4.26) with the help of the formulas (4.27). 2. We shall now analyze the consequences imposed by the equations (4.28) on the partial fraction decomposition of the element σ1 with respect to h̃1. The full form of the expression Wi reads Wi = h̃i1σ1 − ( h̃i1 + 2 ) σ1[−ε1]− ( h̃i1 − 1 ) σ1[−εi] + ( h̃i1 + 1 ) σ1[−ε1 − εi]. (4.30) We write the element σ1 in the form (keeping the notation of Section 4.2) σ1 = A( h̃i1 − a1 )ν1 · · · (h̃i1 − aL)νL , (4.31) where a1 < a2 < · · · < aM , ν1, ν2, . . . , νM ∈ Z≥0 and A ∈ R1,iŪ(n) is not divisible by any factor in the denominator. Substitute the expression (4.31) into the equation Wi = 0. The denominator ( h̃i1−aL−1 ) is present only in term ( h̃i1− 1 ) σ1[−εi] in (4.30). It has therefore to be compensated by ( h̃i1− 1 ) . Hence the only allowed value of aL is aL = 0 and we have νL ≤ 1. Similarly, the denominator( h̃ij − a1 + 1 ) appears only in the term ( h̃i1 + 2 ) σ1[−ε1] in (4.30). It has to be compensated by ( h̃i1 + 2 ) . Hence the only allowed value of a1 is a1 = 0 and we have ν1 ≤ 1. It follows that the partial fraction decomposition of the element σ1 with respect to h̃1 reads σ1 = n∑ k=2 ( Ak h̃k1 + A′k h̃k1 + 1 ) +B, (4.32) where Ak, A ′ k, k = 2, . . . , n, do not depend on h̃1 and B is regular in h̃1. 3. The equations (4.28) impose further restrictions on the constituents of the decomposi- tion (4.32) of the element σ1. Substitute the decomposition (4.32) into the equation Wi = 0. The terms which have denominators of the form h̃i1 +m, m ∈ Z, in (4.30) are h̃i1 ( Ai h̃i1 + A′i h̃i1 + 1 ) − ( h̃i1 + 2 )( Ai h̃i1 + 1 + A′i h̃i1 + 2 ) − ( h̃i1 − 1 )(Ai[−εi] h̃i1 − 1 + A′i[−εi] h̃i1 ) + ( h̃i1 + 1 )(Ai[−εi] h̃i1 + A′i[−εi] h̃i1 + 1 ) . (4.33) In the expression (4.33), the terms with the denominator h̃i1 + 1 read h̃i1A ′ i − ( h̃i1 + 2 ) Ai + ( h̃i1 + 1 ) A′i[−εi] h̃i1 + 1 = −A ′ i +Ai h̃i1 + 1 +A′i −Ai +A′i[−εi]. Therefore, Ai +A′i = 0, i = 2, . . . , n. With this condition, the expression (4.33) vanishes. We conclude that σ1 = n∑ k=2 ( Ak h̃k1 − Ak h̃k1 + 1 ) +B. (4.34) Differential Calculus on h-Deformed Spaces 19 4. Now we substitute the obtained expression (4.34) for σ1 into the equation Wj = 0 with j 6= i and follow the singularities of the form h̃i1 +m, m ∈ Z. The singular terms are h̃j1 ( Ai h̃i1 − Ai h̃i1 + 1 ) − ( h̃j1 + 2 )( Ai h̃i1 + 1 − Ai h̃i1 + 2 ) − ( h̃j1 − 1 )(Ai[−εj ] h̃i1 − Ai[−εj ] h̃i1 + 1 ) + (h̃j1 + 1) ( Ai[−εj ] h̃i1 + 1 − Ai[−εj ] h̃i1 + 2 ) . (4.35) In the expression (4.35), the terms with the denominator h̃i1 read h̃j1Ai − ( h̃j1 − 1 ) Ai[−εj ] h̃i1 . Therefore, the numerator, as a polynomial in h̃1, must be divisible by the denominator h̃i1. The polynomial remainder of this division equals h̃ijAi − ( h̃ij + 1 ) Ai[−εj ] = ∆j ( h̃ijAi ) . Therefore, for any j = 2, . . . , n, j 6= i, the combination h̃ijAi does not depend on h̃j . It follows that Ai = αi ( h̃i )∏ l : l 6=1,i h̃il , i = 2, . . . , n, where each αi is a univariate polynomial. For the moment, we have found that σ1 = σ (s) 1 +B, where the element B is regular in h̃1 and σ (s) 1 = n∑ i=2 ( 1 h̃i1 − 1 h̃i1 + 1 ) αi ( h̃i )∏ l : l 6=1,i h̃il . A direct calculation shows that the element σ (s) 1 satisfies the equations (4.28) and (4.29), so it is left to analyze the regular in h̃1 part B. 5. Since the element B satisfies the system of equations (4.29), we can use the results of Lemma 4.3 with D = K[h̃1]. According to Lemma 4.3, we can write (with an obvious shift in indices) the partial fraction decomposition of the element B with respect to h̃2 in the form B = n∑ j=3 uj ( h̃j , h̃1 )∏ l : l 6=1,j h̃jl + C, where uj ( h̃j , h̃1 ) , j = 3, . . . , n, is a polynomial in h̃j , h̃1 and C is a linear combination of complete symmetric polynomials in h̃2, . . . , h̃n with coefficients in K[h̃1]. The equation W2(B) = 0 implies that the expression h̃21B − ( h̃21 + 2 ) B[−ε1] 20 B. Herlemont and O. Ogievetsky does not depend on h̃2. In the notation of paragraph 1 in Section 4.2, the part P2;j , j = 3, . . . , n, of this expression is 1∏ l : l 6=12,j h̃jl ( h̃21uj ( h̃j , h̃1 ) − ( h̃21 + 2 ) uj ( h̃j , h̃1 − 1 ) h̃j2 ) . Therefore, h̃21uj ( h̃j , h̃1 ) − ( h̃21 + 2 ) uj ( h̃j , h̃1 − 1 ) is divisible, as polynomial in h̃2, by h̃2j . So the value of h̃21uj ( h̃j , h̃1 ) − ( h̃21 + 2 ) uj ( h̃j , h̃1 − 1 ) at h̃2 = h̃j is zero, h̃j1uj ( h̃j , h̃1 ) − ( h̃j1 + 2 ) uj ( h̃j , h̃1 − 1 ) = 0. (4.36) Set uj = βj h̃j1 ( h̃j1 + 1 ) . (4.37) Then equation (4.36) becomes βj h̃j1 + 1 + βj [−ε1] h̃j1 + 1 = 0, or ∆1(βj) = 0, so βj depends only on h̃j . But then if βj 6= 0, the formula (4.37) shows that uj cannot be a polynomial in h̃1. We conclude that the principal part of the element B with respect to h̃2 vanishes, and B = C is a polynomial in all its variables. 6. We claim that C is a K-linear combination of the elements ∆1(HL), L = 1, 2, . . . , where HL are the complete symmetric polynomials in h̃1, . . . , h̃n. Consider first the case n = 2. Set C = ξ h̃21 ( h̃21 + 1 ) , where ξ is some polynomial in h̃1 and h̃2. With this substitution the equation W2(C) = 0 becomes ∆2 ( 1 h̃21 + 1 ∆1(ξ) ) = 0, that is, 1 h̃21 + 1 ∆1(ξ) = µ, where µ does not depend on h̃2. Note that by construction, the polynomial ξ is divisible by h̃21 ( h̃21 + 1 ) , which implies that µ is a polynomial in h̃1. Since ∆1 is surjective on polynomials, we can write µ = ∆2 1 ( z ( h̃1 )) for some univariate polynomial z, that is ∆1(ξ) = ( h̃21 + 1 ) ∆2 1 ( z ( h̃1 )) . We have( h̃21 + 1 ) ∆2 1 ( z ( h̃1 )) = ∆1 ( h̃21∆1 ( z ( h̃1 )) + z ( h̃1 )) . Differential Calculus on h-Deformed Spaces 21 Therefore, ∆1 ( ξ − h̃21∆1 ( z ( h̃1 )) − z ( h̃1 )) = 0, or ξ = h̃21∆1 ( z ( h̃1 )) + z ( h̃1 ) + w ( h̃2 ) , where w ( h̃2 ) is a polynomial in h̃2. That is, C = ∆1 ( z ( h̃1 )) h̃21 + 1 + z ( h̃1 ) + w ( h̃2 ) h̃21 ( h̃21 + 1 ) . (4.38) Since the element C is a polynomial, the denominator h̃21 in the second term in the right hand side of (4.38) shows that w = −z. Therefore, C = h̃21∆1 ( z ( h̃1 )) + z ( h̃1 ) − z ( h̃2 ) h̃21 ( h̃21 + 1 ) = ∆1 ( z ( h̃1 ) − z ( h̃2 ) h̃21 ) , as claimed. The claim for arbitrary n follows since for any j > 2 the element C is a linear combination of ∆1 ( HL ( h̃1, h̃j )) , L = 1, 2, . . . 7. We summarize the results of this section in the following proposition. Proposition 4.5. The general solution of the system (4.28) and (4.29) has the form σ1 = n∑ i=2 ( 1 h̃i1 − 1 h̃i1 + 1 ) αi ( h̃i )∏ l : l 6=1,i h̃il + ∆1(ν), ν ∈ H (4.39) and α2, . . . , αn are univariate polynomials. The elements α2, . . . , αn and ν are uniquely defined. 4.4 Potential. Proof of Proposition 3.6 First proof. We rewrite the formula (4.39) in the form σ1 = ∆1(σ), where σ = n∑ i=2 αi ( h̃i ) χi + ν ∈ W. Then the expressions for the elements σj , j = 2, . . . , n, see (4.27), read σj = ∆j (( h̃j1 + 1 ) n∑ i=2 αi ( h̃i )( h̃i1 + 1 ) χi + ( h̃j1 + 1 ) ∆1(ν) ) . (4.40) Since, for ν ∈ H, ∆1∆j ( h̃j1ν ) = 0, we find that ∆j (( h̃j1 + 1 ) ∆1(ν) ) = ∆j(ν). The term with i = j in the sum in the right hand side of (4.40) is simply αj ( h̃j ) χj . 22 B. Herlemont and O. Ogievetsky Since h̃j1 + 1 h̃i1 + 1 = h̃i1 + 1 + h̃ji h̃i1 + 1 = 1 + h̃ji h̃i1 + 1 , we can rewrite the term with i 6= j in the right hand side of (4.40) in the form ∆j (( h̃j1 + 1 ) αi ( h̃i )( h̃i1 + 1 ) χi ) = ∆j αi(h̃i) χi − αi ( h̃i )( h̃i1 + 1 ) ∏ l 6=i,j h̃il  = ∆j ( αi ( h̃i ) χi ) . Therefore, σj = ∆j(σ) for all j = 1, . . . , n. The proof of Proposition 3.6 is completed. Second proof. Let p ∈ U(n) be a polynomial such that ∆1∆2(p) = 0. Thus, ∆2(p) does not depend on h̃1 so, by surjectivity of ∆2 on polynomials in h̃2, there exists a polynomial p1 which does not depend on h̃1 and ∆2(p) = ∆2(p1). The polynomial p2 := p−p1 does not depend on h̃2. The next lemma generalizes this decomposition p = p1 + p2, ∆1(p1) = 0, ∆2(p2) = 0, (4.41) to the ring Ū(n). Lemma 4.6. Let f ∈ Ū(n). If ∆1∆2(f) = 0 then there exist elements f1, f2 ∈ Ū(n) such that f1 does not depend on h̃1, f2 does not depend on h̃2, and f = f1 + f2. (4.42) Proof. Decompose f into partial fractions with respect to h̃1. We have P1;2(f) = 0. Indeed, write P1;2(f) in the form P1;2(f) = u( h̃12 − a1 )ν1 · · · (h̃12 − aL )νL , where a1 < a2 < · · · < aL, ν1, ν2, . . . , νL ∈ Z>0 and u ∈ R1,2Ū(n) is not divisible by any factor in the denominator. Assume that u 6= 0. Then ∆1∆2(P1;2(f)) = u+ u[−ε1 − ε2]( h̃12 − a1 )ν1 · · · (h̃12 − aL )νL − u[−ε1]( h̃12 − a1 − 1 )ν1 · · · (h̃12 − aL − 1 )νL − u[−ε2]( h̃12 − a1 + 1 )ν1 · · · (h̃12 − aL + 1 )νL . The factor ( h̃12− aL− 1 ) appears only in the denominator of the second term in the right hand side and cannot be compensated by the numerator. Thus P1;2(f) = 0 (the consideration of the factor ( h̃12 − a1 + 1 ) in the denominator of the third term proves the claim as well). Now we write the part P1;j(f), j > 2, in the form (4.11), P1;j(f) = ∑ a∈Z ∑ νa∈Z>0 ujaνa( h̃1j − a )νa , Differential Calculus on h-Deformed Spaces 23 where ujaνa ∈ N1Ū(n) and the sums are finite. Then ∆1∆2(P1;j(f)) = ∑ a∈Z ∑ νa∈Z>0 ∆2(ujaνa) ( 1( h̃1j − a )νa − 1( h̃1j − a− 1 )νa ) . (4.43) We prove that the elements ujaνa do not depend on h̃2. Indeed, if this is not true then there is a minimal a ∈ Z for which ∆2(ujaνa) 6= 0 for some νa. But then the denominator (h̃1j − a)νa in the right hand side in (4.43) cannot be compensated. We conclude that f = f2,0 +g where f2,0 = ∑ j>2 P1;j(f) does not depend on h̃2 and g is regular in h̃1. We decompose g with respect to h̃2. As above, the part P2;1(g) vanishes and the calculation, parallel to (4.43), shows that P2;j(g), j > 2, does not depend on h̃1. Now we have f = f2,0 + f1,0 + f+, where f1,0 = ∑ j>2 P2;j(g) does not depend on h̃1 and f+ is regular in h̃1 and h̃2. We use the decomposition (4.41) for the regular part f+ and write f+ = f+ 1 + f+ 2 , where f+ 1 does not depend on h̃1 and f+ 2 does not depend on h̃2. This leads to the required decomposi- tion (4.42) with f1 = f1,0 + f+ 1 and f2 = f2,0 + f+ 2 . � Lemma 4.7. Let σ1, . . . , σk, k ≤ n, be a k-tuple of elements in Ū(n) such that ∆a(σb) = ∆b(σa), a, b = 1, . . . , k. Assume that σa belongs to the image of ∆a for all a = 1, . . . , k, that is, there exist elements f1, . . . , fk ∈ Ū(n) for which σa = ∆a(fa), a = 1, . . . , k. Then there exists a potential f ∈ Ū(n) such that σa = ∆a(f) = 0, a = 1, . . . , k. Proof. For k = 1 there is nothing to prove. Let now k > 1. We use the induction in k. By the induction hypothesis, there exist elements F,G ∈ Ū(n) such that σa = ∆a(F ) for a = 1, 3, . . . , k and σb = ∆b(G) for b = 2, 3, . . . , k. Then ∆c(F ) = ∆c(G) for c = 3, . . . , k and ∆1∆2(G) = ∆2∆1(F ). The element F −G does not depend on h̃c, c = 3, . . . , k, and ∆1∆2(F −G) = 0. According to Lemma 4.6, there exist two elements u, v ∈ Ū(n) such that u does not depend on h̃2, v does not depend on h̃1, and F −G = u− v. Then f := F + v = G+ u is the desired potential. � Second proof of Proposition 3.6. The symmetric, in i and j, part of the equation (3.7) is ∆iσj = ∆jσi. (4.44) The system (4.44) by itself does not imply the existence of a potential. However, the equa- tion (3.7) can be written in the form σj = ∆j (( h̃ji + 1 ) σi ) . So for each j = 1, . . . , n the element σj belongs to the image of the operator ∆j . Then, according to Lemma 4.7, there exists σ ∈ Ū(n) such that σj = ∆j(σ). 24 B. Herlemont and O. Ogievetsky 4.5 Polynomial potentials. Proof of Lemma 3.7 The operator q̌i defined by (3.14) can be an automorphism of the ring Diffh,σ(n) only if q̌i(σj) = σsi(j) = ∆si(j)(σ), i, j = 1, . . . , n. (4.45) On the other hand, q̌i(σj) = q̌i(∆j(σ)) = ∆si(j)(q̌i(σ)), i, j = 1, . . . , n. (4.46) Comparing (4.45) and (4.46) we obtain ∆j(σ − q̌i(σ)) = 0, i, j = 1, . . . , n, which implies that σ is Sn-invariant. The assertion now follows from Lemma 3.5(ii). 4.6 Central elements. Proof of Proposition 3.8 (i) To analyze the relation xjc(t) − c(t)xj = 0, we shall write the expression xjc(t) − c(t)xj in the ordered form, in the order ∂̄xx. The element c0(t) = ∑ i e(t) 1 + h̃it Γi is central in the homogeneous ring Diffh,0(n), see the calculation in [16, Proposition 3]. Hence we have to track only those ordered terms whose filtration degree, see (3.6), is smaller than 3. As before, we use the symbol u ∣∣ l.d.t. to denote these lower degree terms in an expression u. We have ( xjc(t)− c(t)xj )∣∣ l.d.t. = ( − e(t) 1 + h̃jt σj − ρ(t)[−εj ] + ρ(t) ) xj . Thus the element c(t) commutes with the generators xj , j = 1, . . . , n, if and only if the poly- nomial ρ(t) satisfies the system (3.16). The use of the anti-automorphism (4.6) shows that the element c(t) then commutes with the generators ∂̄j , j = 1, . . . , n, as well. (ii) We check the case j = 1. The calculation for σ ∈ Wj is similar. Since the combination e(t) 1+h̃1t does not depend on h̃1, we have, for ρ(t) = e(t) 1+h̃1t σ, ∆1ρ(t) = e(t) 1 + h̃1t ∆1σ = e(t) 1 + h̃1t σ1. For j > 1 we have σ = ( h̃1j + 1 ) ∆jσ, j = 2, . . . , n, (4.47) and we calculate ∆jρ(t) = e(t)( 1 + h̃1t )( 1 + h̃jt )∆j (( 1 + h̃jt ) σ ) = e(t)( 1 + h̃1t )( 1 + h̃jt )(tσ + ( 1 + ( h̃j − 1 ) t ) ∆jσ ) = e(t) 1 + h̃jt ∆jσ, according to the formula (4.47). (iii) The proof is the same as for the ring Diffh(n), see [16, Lemma 8]. Differential Calculus on h-Deformed Spaces 25 4.7 Rings of fractions. Proof of Lemma 3.9 (i) The set BD := { h̃i, x ′◦i, ci }n i=1 , where x′◦i := xiψ′i, i = 1, . . . , n, generates the localized ring S−1 x Diffh,σ(n). Moreover, the complete set of the defining relations for the generators from the set BD does not remember about the potential σ. It reads h̃ih̃j = h̃j h̃i, h̃ix ′◦j = x′◦j ( h̃i + δji ) , x′◦ix′◦j = x′◦jx′◦i, i, j = 1, . . . , n, ci are central, i = 1, . . . , n. The proof is the same as for the ring Diffh(n), see [16]. The isomorphism is now clear. (ii) Assume that ι : Diffh,σ(n) → Diffh,σ′(n) is an isomorphism of filtered rings over Ū(n). To distinguish the generators, we denote the generators of the ring Diffh,σ′(n) by x′i and ∂̄′i. The εi-weight subspace Ei of the ring Diffh,σ(n) consists of elements of the form θxi where θ is a polynomial in the elements Γj , j = 1, . . . , n, with coefficients in Ū(n). Since the space of the elements of Ei of filtration degree ≤ 1 is Ū(n)xi, we must have ι : xi 7→ µix ′i, ∂̄i 7→ ∂̄′iνi (4.48) with some invertible elements µi, νi ∈ Ū(n), i = 1, . . . , n. Let γi := µiνi, i = 1, . . . , n. The defining relation (4.4) and the corresponding relation for the ring Diffh,σ′(n) shows that the formulas (4.48) may define an isomorphism only if γi = γj [εj ], i, j = 1, . . . , n, (4.49) and γiσ ′ i = σi, i = 1, . . . , n. (4.50) The condition (4.49) implies that γi = γ for some γ ∈ K. The condition (4.50) then becomes γσ′i = σi and the assertion follows. 4.8 Lowest weight representations. Proof of Proposition 3.10 We need the following identity (see [16, Lemma 5]):∑ j 1 h̃j + t−1 Q+ j = 1− e(t)[−ε] e(t) (4.51) and its several consequences. At t = ( 1− h̃m )−1 , m = 1, . . . , n, the equality (4.51) becomes∑ j 1 h̃jm + 1 Q+ j = 1. (4.52) Then,∑ i 1 1 + th̃i 1 h̃ik + 1 Q+ i = 1 1 + t ( h̃k − 1 )∑ i ( 1 h̃ik + 1 − t 1 + th̃i ) Q+ i = 1 1 + t ( h̃k − 1 ) e(t)[−ε] e(t) . (4.53) We used (4.51) and (4.52) in the last equality. The substitution h̃i −h̃i + 1, i = 1, . . . , n, and t −t into (4.53) gives∑ i 1 1 + t ( h̃i − 1 ) 1 h̃ki + 1 Q−i = 1 1 + th̃k e(t) e(t)[−ε] . (4.54) 26 B. Herlemont and O. Ogievetsky Proof of Proposition 3.10. Since the element c(t) is central, it is sufficient to calculate its value on the vector \| 〉. Denote c(t)\| 〉 = ω(t)\| 〉. We have ∂̄jx i = ∑ k,l Ψik jlx l∂̄k + ∑ k Ψik jkσk, (4.55) where Ψ is the skew inverse of the operator R, see (1.6) (we refer, e.g., to [15, Section 4.1.2] for details on skew inverses). Since the generators ∂̄i, i = 1, . . . , n, annihilate the vector \| 〉, see (3.18), we find, in view of (4.55), that ω(t) = ∑ i,k e(t) 1 + th̃i Ψik ikσk − ρ(t) = ∑ i,k e(t) 1 + th̃i 1 h̃ik + 1 Q+ i Q−k σk − ρ(t) = e(t)[−ε] ∑ k 1 1 + t(h̃k − 1) Q−k σk − ρ(t). (4.56) We used (1.7) in the second equality and (4.53) in the third equality. We shall verify (3.19) for every representative of the space W. As in the proof of Proposi- tion 3.8(ii), it is sufficient to establish (3.19) for σ = A ( h̃1 ) χ1 , where A is a univariate polynomial. Then σj = 1 h̃1j + 1 σ, j = 2, . . . , n, (4.57) and, according to Proposition 3.8(ii), ρ(t) = e(t) 1 + th̃1 σ. (4.58) Denote the underlined sum in (4.56) by ξ. Taking into account (4.57) we calculate ξ = 1 1 + t ( h̃1 − 1 )(σ − σ[−ε1]) Q−1 +σ n∑ j=2 1 1 + t ( h̃j − 1 ) 1 h̃1j + 1 Q−j = − σ[−ε1] Q−1 1 + t ( h̃1 − 1 ) + σ n∑ j=1 1 1 + t ( h̃j − 1 ) 1 h̃1j + 1 Q−j = − σ[−ε1] Q−1 1 + t ( h̃1 − 1 ) + 1 1 + th̃1 σe(t) e(t)[−ε] . We have used (4.54) in the last equality. Note that σ[−ε1] Q−1 = A ( h̃1 − 1 ) χ1[−ε1] χ1[−ε1] χ1 = A ( h̃1 − 1 ) χ1 = σ[−ε], Differential Calculus on h-Deformed Spaces 27 so ξ = − σ[−ε] 1 + t ( h̃1 − 1 ) + 1 1 + th̃1 σe(t) e(t)[−ε] . Substituting the obtained expression for ξ into (4.56) and taking into account (4.58) we conclude that ω(t) = e(t)[−ε] ( − σ[−ε] 1 + t ( h̃1 − 1 ) + 1 1 + th̃1 σe(t) e(t)[−ε] ) − e(t) 1 + th̃1 σ = − e(t)[−ε] 1 + t ( h̃1 − 1 )σ[−ε] = −ρ(t)[−ε], as stated. 4.9 Several copies. Proof of Lemma 3.11 Assume that, say, N > 1. Repeating the calculations (4.7) and (4.8) for one copy in Section 4.1, we find, for i, j, k = 1, . . . , n, ( xiα∂̄jβ ) ∂̄kγ ∣∣ l.d.t. = (∑ u,v Rui vj [εu]∂̄uβx vα − δijσiαβ ) ∂̄kγ ∣∣ l.d.t. = − ∑ u Rui kj [εu]∂̄uβσkαγ − δijσiαβ ∂̄kγ , (4.59) xiα ( ∂̄jβ ∂̄kγ )∣∣ l.d.t. = xiα ∑ a,b Rab kj ∂̄bγ ∂̄aβ ∣∣ l.d.t. = ∑ a,b Rab kj [−εi] ∑ c,d Rci db[εc]∂̄cγx dα − δibσiαγ  ∂̄aβ ∣∣ l.d.t. = − ∑ a,b,c Rab kj [−εi] Rci ab[εc]∂̄cγσaαβ − ∑ a Rai kj [−εi]σiαγ ∂̄aβ. (4.60) Take β 6= γ. Equating the coefficients in ∂̄uβ, u = 1, . . . , n, in (4.59) and (4.60), we find Rui kj [εu]σkαγ [εu] = Rui kj [−εi]σiαγ , i, k, j, u = 1, . . . , n. (4.61) Equating the coefficients in ∂̄uγ , u = 1, . . . , n, in (4.59) and (4.60), we find δijδ u kσiαβ = ∑ a,b Rab kj [−εi] Rui ab[εu]σaαβ[εu], i, k, j, u = 1, . . . , n. (4.62) Shifting by −εu and using the property (1.3) we rewrite the equality (4.61) in the form Rui kj (σkαγ − σiαγ [−εu]) = 0. 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[22] Zhelobenko D.P., Representations of reductive Lie algebras, Nauka, Moscow, 1994. https://doi.org/10.1007/BF02101512 https://doi.org/10.1016/0001-8708(78)90010-5 https://doi.org/10.1007/BF01877233 https://doi.org/10.1063/1.531780 https://arxiv.org/abs/q-alg/9508022 https://doi.org/10.1088/0305-4470/36/20/310 https://arxiv.org/abs/hep-th/0003210 https://doi.org/10.1063/1.532779 https://arxiv.org/abs/q-alg/9712026 https://doi.org/10.1090/S0002-9947-2011-05367-5 https://arxiv.org/abs/0912.1101 https://doi.org/10.1016/j.jalgebra.2007.04.020 https://arxiv.org/abs/math.QA/0606259 https://doi.org/10.1007/s10688-010-0023-0 https://arxiv.org/abs/0912.4055 https://doi.org/10.3842/SIGMA.2011.064 https://arxiv.org/abs/1101.2647 https://doi.org/10.1007/s10468-012-9397-4 https://doi.org/10.1007/s11856-017-1571-2 https://arxiv.org/abs/1510.05258 https://doi.org/10.1016/0034-4877(73)90006-2 https://doi.org/10.1090/conm/294/04973 https://doi.org/10.1134/S0040577917080104 https://arxiv.org/abs/1612.08001 https://doi.org/10.1090/conm/391/07342 https://arxiv.org/abs/math-ph/0412087 http://hdl.handle.net/2066/147527 https://doi.org/10.1016/0920-5632(91)90143-3 https://doi.org/10.1016/0920-5632(91)90143-3 https://doi.org/10.1070/RM1962v017n01ABEH001123 https://doi.org/10.1070/RM1962v017n01ABEH001123 https://doi.org/10.1007/BF02577133 1 Introduction 2 Coordinate rings of h-deformed vector spaces 3 Generalized rings of h-deformed differential operators 3.1 Two examples 3.2 Main question and results 3.2.1 Main question 3.2.2 Poincaré–Birkhoff–Witt property 3.2.3 -system 3.2.4 Potential 3.2.5 A characterization of polynomial potentials 3.2.6 Center 3.2.7 Rings of fractions 3.2.8 Lowest weight representations 3.2.9 Several copies 4 Proofs of statements in Section 3.2 4.1 Poincaré–Birkhoff–Witt property. Proof of Proposition 3.1 4.2 General solution of the system (3.8). Proofs of Theorem 3.3 and Lemma 3.5 4.3 System (3.7) 4.4 Potential. Proof of Proposition 3.6 4.5 Polynomial potentials. Proof of Lemma 3.7 4.6 Central elements. Proof of Proposition 3.8 4.7 Rings of fractions. Proof of Lemma 3.9 4.8 Lowest weight representations. Proof of Proposition 3.10 4.9 Several copies. Proof of Lemma 3.11 References

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