Differential Calculus on h-Deformed Spaces

We construct the rings of generalized differential operators on the ${\bf h}$-deformed vector space of ${\bf gl}$-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 ${\bf h}$-deformed differential operators $\operatorname{Diff}_{{\bf h},\sigma}(n)$ is labeled by a rational function $\sigma$ 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 $\operatorname{Diff}_{{\bf h},\sigma}(n)$.


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 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 finitedifference 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 (t j , . . . , t j ) for j = 0, 1, . . . , n − 2. Thus for each σ ∈ W we have a ring Diff h,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 Diff h,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 Diff h (n) which is the reduction algebra, with respect to gl n , of the classical ring of polynomial differential operators. The second ring is related to the reduction algebra, with respect to gl n , of the algebra U(gl n+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 Diff h,σ (n) and construct an isomorphism between a certain ring of fractions of the ring Diff h,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 S n the symmetric group on n letters. The symbol s i stands for the transposition (i, i + 1).
Let h(n) be the abelian Lie algebra with generatorsh i , i = 1, . . . , n, and U(n) its universal enveloping algebra. Seth ij =h i −h j ∈ h(n). We defineŪ(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 h ij + a −1 , a ∈ Z, i, j = 1, . . . , n, i = j. Let Let ε j , j = 1, . . . , n, be the elementary translations of the generators of U(n), ε j :h i →h i + δ j i . For an element p ∈Ū(n) we denote ε j (p) by p[ε j ]. We shall use the finite-difference operators ∆ j defined by We denote by e L , L = 0, . . . , n, the elementary symmetric polynomials in the variables h 1 , . . . ,h n , and by e(t) the generating function of the polynomials e L , We denote by R ∈ EndŪ (n) Ū (n) n ⊗Ū (n)Ū (n) n the standard solution of the dynamical Yang-Baxter equation of type A. The nonzero components of the operator R are We shall need the following properties of R: 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 The nonzero components of the operator Ψ are, see [13], where
The ring V(n, N ) is the reduction algebra, with respect to gl n , of the semi-direct product of gl n and the abelian Lie algebra V ⊕V ⊕· · ·⊕V (N times) where V is the (tautological) n-dimensional gl n -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 x iα , i = 1, . . . , n, α = 1, . . . , N , the set of all ordered monomials in x iα is a basis of the leftŪ(n)-module V(n, N ).
is an arbitrary array of functions inh i , 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.
The coordinate ring V * (n, N ) of N copies of the "dual" h-deformed vector space is the factor-ring ofF * (n, N ) by the relations Again, the ring V * (n, N ) is the reduction algebra, with respect to gl n , of the semi-direct product of gl n and the abelian Lie algebra V * ⊕ V * ⊕ · · · ⊕ V * (N times) where V * is the gl n -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 leftŪ(n)-module V * (n, N ). (2.8) Again, the Poincaré-Birkhoff-Witt property of the algebra defined by the relations (2.7), together 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 dif ferential operators

Two examples
Before presenting the main question we consider two examples. 1. We denote by W n the algebra of polynomial differential operators in n variables. It is the algebra with the generators X j , D j , j = 1, . . . , n, and the defining relations The map, defined on the set {e ij } n i,j=1 of the standard generators of gl n by extends to a homomorphism U(gl n ) → W n . The reduction algebra of W n ⊗ U(gl n ) with respect to the diagonal embedding of U(gl n ) was denoted by Diff h (n) in [13]. It is generated, overŪ(n), by the images x i and ∂ i , i = 1, . . . , n, of the generators X i and D i . Let where the elements ψ i are defined in (1.1). Then The h(n)-weights of the generators are given by (2.2) and (2.6). Moreover, the set of the defining relations, overŪ(n), for the generators x i and∂ i , i = 1, . . . , n, consists of (2.4), (2.7) (with N = 1) and (3.1) (see [13,Proposition 3.3]).
The algebra Diff h (n, N ), formed by N copies of the algebra Diff h (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 gl n (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 gl n into gl n+1 . The resulting reduction algebra R U(gl n+1 ) gl n , or simply R gl n+1 gl n , was denoted by AZ n in [22]. It is generated, overŪ(n), by the elements x i , y i , i = 1, . . . , n, andh n+1 = z − (n + 1), where x i and y i are the images of the standard generators e i,n+1 and e n+1,i of U(gl n+1 ) and z is the image of the standard generator e n+1,n+1 . Let where the elements ψ i are defined in (1.1) (they depend onh 1 , . . . ,h n only). The h(n)-weights of the generators are given by (2.2) and (2.6) whilẽ The set of the remaining defining relations consists of (2.4), (2.7) (with N = 1) and The algebra AZ n 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 AZ n has a central element In the factor-algebra AZ n of AZ n by the ideal, generated by the element (3.3), the relation (3.2) is replaced by k + 1, i = 1, . . . , n.

Main question
Both rings, Diff h (n) and AZ n 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 crosscommutation 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 investigate possible generalizations of the rings Diff h (n) and AZ n . More precisely, given n elements σ 1 , . . . , σ n ofŪ(n), we let Diff h (σ 1 , . . . , σ n ) be the ring, overŪ(n), with the generators x i and∂ i , i = 1, . . . , n, subject to the defining relations (2.4), (2.7) (with N = 1) and the oscillator-like relations 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 δ i j in the right hand side of (3.5)) is dictated by the h(n)-weight considerations.
The assignment defines the structure of a filtered algebra on Diff h (σ 1 , . . . , σ n ). The associated graded algebra is the homogeneous algebra Diff h (0, . . . , 0). This homogeneous algebra has the desired Poincaré-Birkhoff-Witt property because it is the reduction algebra, with respect to gl n , of the semi-direct product of gl n and the abelian Lie algebra V ⊕ V * . The standard argument shows that the ring Diff h (σ 1 , . . . , σ n ) can be viewed as a deformation of the homogeneous ring Diff h (0, . . . , 0): for the generating set x i ,∂ i , where x i = x i , all defining relations are the same except (3.5) in which σ i gets replaced by σ i ; one can consider as the deformation parameter. Thus our aim is to study the conditions under which this deformation is flat.

Poincaré-Birkhof f-Witt property
It turns out that the Poincaré-Birkhoff-Witt property is equivalent to the system of finitedifference equations for the elements σ 1 , . . . , σ n ∈Ū(n).
We postpone the proof to Section 4.1.

∆-system
The system (3.7) is closely related to the following linear system of finite-difference equations for one element σ ∈Ū(n): We shall call it the "∆-system". The ∆-system can be written in the form We describe the most general solution of the system (3.8).
Definition 3.2. Let W j , j = 1, . . . , n, be the vector space of the elements ofŪ(n) of the form and χ j is defined in (1.1). Let W be the sum of the vector spaces W j , j = 1, . . . , n. The proof is in Section 4.2.
The sum W j is not direct.
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, The symmetric group S n acts on the ringŪ(n) and on the space W by permutations of the variablesh 1 , . . . ,h n . We have where W Sn denotes the subspace of S n -invariants in W.
(iii) Select j ∈ {1, . . . , n}. Then we have a direct sum decomposition 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 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 (t j , . . . , t j ) 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.

Potential
We shall give a general solution of the system (3.7).
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 H L , L = 1, 2, . . . , and let 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 .

A characterization of polynomial potentials
The polynomial potentials σ ∈ W can be characterized in different terms. The rings Diff h (n) and AZ n admit the action of Zhelobenko automorphismsq 1 , . . . ,q n−1 [9,21]. Their action on the generators x i and∂ i , i = 1, . . . , n, is given by (see [13]) Lemma 3.7. The ring Diff h,σ (n) admits the action of Zhelobenko automorphisms if and only if σ is a polynomial, The proof is in Section 4.5.
In the examples discussed in Section 3.1, the ring Diff h (n) corresponds to σ = H 1 and the ring AZ n corresponds to The question of constructing an associative algebra which contains U(gl n ) and whose reduction with respect to gl n is Diff h,σ (n) for σ = H k , k > 2, will be discussed elsewhere.

Center
In [16] we have described the center of the ring Diff h (n). The center of the ring Diff h,σ (n), σ ∈ W, admits a similar description. Let Let where t is an auxiliary variable and ρ(t) a polynomial of degree n − 1 in t with coefficients inŪ(n).
(i) Let σ ∈ W and σ j = ∆ j σ, j = 1, . . . , n. The elements c 1 , . . . , c n are central in the ring Diff h,σ (n) if and only if the polynomial ρ satisfies the system of finite-difference equations 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 ∈ W k for each k = 1, . . . , n, that is, for The solution of the system (3.16) for the element σ of the form (3.17) is, up to an additive constant from K, (iii) The center of the ring Diff h,σ (n) is isomorphic to the polynomial ring K[t 1 , . . . , t n ]; the isomorphism is given by t j → c j , j = 1, . . . , n.
The proof is in Section 4.6.

Rings of fractions
In [16] we have established an isomorphism between certain rings of fractions of the ring Diff h (n) and the Weyl algebra W n . It turns out that when we pass to the analogous ring of fractions of the ring Diff h,σ (n), we loose the information about the potential σ. Thus we obtain the isomorphism with the same, as for the ring Diff h (n), ring of fractions of the Weyl algebra W n . We denote, as for the ring Diff h (n), by S −1 x Diff h,σ (n) the localization of the ring Diff h,σ (n) with respect to the multiplicative set S x generated by x j , j = 1, . . . , n. (i) The rings S −1 x Diff h,σ (n) and S −1 x Diff h,σ (n) are isomorphic. (ii) However, the rings Diff h,σ (n) and Diff h,σ (n) are isomorphic, as filtered rings overŪ(n) (where the filtration is defined by (3.6)), if and only if The proof is in Section 4.7.

Lowest weight representations
The ring Diff h,σ (n) has an n-parametric family of lowest weight representations, similar to the lowest weight representations of the ring Diff h (n), see [16]. We recall the definition. Let D n be anŪ(n)-subring of Diff h,σ (n) generated by {∂ i } n i=1 . Let λ := {λ 1 , . . . , λ n } be a sequence, of length n, of complex numbers such that λ i − λ j / ∈ Z for all i, j = 1, . . . , n, i = j. Denote by M λ the one-dimensional K-vector space with the basis vector | . The formulas The proof is in Section 4.8.
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 The proof is in Section 4.9.
Making the redefinitions of the generators, K) and B ∈ GL(N, K) we can transform the matrix σ αβ to the diagonal form, with the diagonal (1, . . . , 1, 0, . . . , 0). Therefore, the ring L is formed by several copies of the rings Diff h (n), V(n) and V * (n).
4 Proofs of statements in Section 3. The explicit form of the defining relations for the ring Diff h (σ 1 , . . . , σ n ) is x j x k∂ i . where By using the anti-automorphism we reduce the check of the ambiguity x j x k∂ i to the check of the ambiguity x i∂ 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 x i∂ 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 x i∂ j∂k . We calculate, for i, j, k = 1, . . . , n, and 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 x i∂ j∂k : i, k, j, u = 1, . . . , n.
Shifting by −ε u and using the property (1.3) together with the ice condition (1.4), we rewrite (4.9) in the form (4.10) For j = k the system (4.10) contains no equations. For j = k we have two cases: • u = j and i = k. This part of the system (4.10) reads explicitly (see (1.2)) This is the system (3.7).
• u = k and i = j. This part of the system (4.10) reads explicitly which reproduces the same system (3.7).

General solution of the system (3.8). Proofs of Theorem 3.3 and Lemma 3.5
We shall interpret elements ofŪ(n) as rational functions on h * with possible poles on hyperplanes h ij + a = 0, a ∈ Z, i, j = 1, . . . , n, i = j. Let M be a subset of {1, . . . , n}. The symbol R MŪ (n) denotes the subring ofŪ(n) consisting of functions with no poles on hyperplanesh ij + a = 0, a ∈ Z, i, j ∈ M, j = i. The symbol N MŪ (n) denotes the subring ofŪ(n) consisting of functions which do not depend on variablesh i , i ∈ M. We shall say that an element f ∈Ū(n) is regular inh j if it has no poles on hyperplanesh jm + a = 0, a ∈ Z, m = 1, . . . , n, m = j. 1. Partial fraction decompositions. We will use partial fraction decompositions of an element f ∈Ū(n) with respect to a variableh j for some given j. The partial fraction decomposition of f with respect toh j is the expression for f of the form where the elements P j (f ) and reg j (f ) have the following meaning. The "regular" part reg j (f ) is an element, regular inh j . The "principal" inh j part P j (f ) is with some elements u kaνa ∈ N jŪ (n); the sums are finite. The fact that the ringŪ(n) admits partial fraction decompositions (that is, that the elements u kaνa and reg j (f ) belong toŪ(n)) is a consequence of the formula 2. Let D be a domain (a commutative algebra without zero divisors) over K. Let f be an element of D ⊗ KŪ (n). Set If Y ij (f ) = 0 for some i and j, i = j, then f can be written in the form Proof . We write f in the form where a 1 < a 2 < · · · < a M , ν 1 , ν 2 , . . . , ν M ∈ Z >0 , A, B ∈ D ⊗ K R i,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 = 0. Then The denominator h ij − a M − 1 appears only in the second term in the right hand side of (4.14). It has therefore to be compensated by h ij − 1 in the numerator. Hence the only allowed value of a M is a M = 0 and moreover we have ν M = 1. Similarly, the denominator h ij − a 1 + 1 appears only in the third term in the right hand side of (4.14) and has to be compensated by (h ij + 1) in the numerator. Hence the only allowed value of a 1 is a 1 = 0 and we have ν 1 = 1. The inequalities a 1 < a 2 < · · · < a M imply that M = 1 and we obtain the form (4.13) of f .
3. Let f ∈ D ⊗ KŪ (n). We shall analyze the linear system of finite-difference equations Y ij (f ) = 0 for all i, j = 1, . . . , n, (4.15) where Y ij are defined in (4.12). First we prove a preliminary result. We recall Definition 3.2 of the vector spaces W i , i = 1, . . . , n. We select one of the variablesh i , say,h 1 .

Lemma 4.2.
Assume that an element f ∈ D ⊗ KŪ (n) satisfies the system (4.15). Then where ϑ ∈ D ⊗ K U(n) and with some univariate polynomials u j h j , j = 2, . . . , n, with coefficients in D.
Proof . Since Y 1m (f ) = 0, m = 2, . . . , n, Lemma 4.1 implies that the partial fraction decomposition of f with respect toh 1 has the form where β m ∈ D ⊗ K N 1Ū (n), m = 2, . . . , n, and ϑ ∈ D h 1 ⊗ K N 1Ū (n). Substituting the expression (4.18) for f into the equation Y 1j (f ) = 0, j = 2, . . . , n, we obtain We used that β m ∈ D ⊗ K N 1Ū (n) in the third and fourth equalities. For any m = 1, j, the terms containing the denominatorh m1 in the expression (4.19) for Y 1j (f ) read Therefore, the elementh mj β m does not depend onh j for any j > 1. We conclude that with some univariate polynomial u m . We have proved that the element f has the form (4.16) where F j , j = 2, . . . , n, are given by (4.17) and the element ϑ is regular inh 1 .
A direct calculation shows that for any j = 2, . . . , n, the element F j , given by (4.17), is a solution of the linear system (4.15). Therefore the regular inh 1 part ϑ by itself satisfies the system Y ij (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 inh 1 andh 2 . This is the induction base. We shall now prove that ϑ is a polynomial, with coefficients in D, in all n variablesh 1 , . . . ,h n .
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 variablesh 1 , . . . ,h n .
We recall Definition 3.4 of the vector space H. where F j ∈ D ⊗ K W j and ϑ ∈ D ⊗ K H. 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 h 1 ,h 2 be a polynomial such that Y 12 (p) = 0. Since ∆ 1 ∆ 2 h 12 p = 0 we have ∆ 2 (h 12 p) ∈ D h 2 . It is a standard fact that the operator ∆ 2 is surjective on D h 2 . This can be seen, for example, by noticing that the set is a basis of D h 2 over D, and The surjectivity of ∆ 2 implies that ∆ 2 h 12 p = ∆ 2 w h 2 for some polynomial w h 2 ∈ D h 2 .
Since p is a polynomial we must have w = −v. Thus that is, p is a D-linear combination of complete symmetric polynomials inh 1 ,h 2 . For arbitrary n, our polynomial ϑ is symmetric since, by the above argument, it is symmetric in every pairh i ,h j of variables. Moreover, considered as a polynomial in a pairh i ,h j , it is a D-linear combination of complete symmetric polynomials inh i ,h j . It is then immediate that ϑ is a D-linear combination of complete symmetric polynomials inh 1 , . . . ,h n .
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 H L , L = 0, 1, . . . , in the variablesh 1 , . . . ,h n satisfy the system (4.15). Let s be an auxiliary variable and be the generating function of the elements H L , 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 = j, and let does not depend onh j so Y ij (ζ ij ) = 0. Therefore Y ij (H(s)) = 0 since the factors other than ζ ij in the product in the right hand side of (4.25) do not depend onh i andh j .
(ii) Finally, the summands in (4.23) are uniquely defined since (4.23) is a partial fraction decomposition of the element f inh 1 . 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

Proof of
The left hand side is nothing else but the partial fraction decomposition, with respect to t, of the product in the right hand side.

2.
We shall now analyze the consequences imposed by the equations (4.28) on the partial fraction decomposition of the element σ 1 with respect toh 1 . The full form of the expression W i reads (4.30) We write the element σ 1 in the form (keeping the notation of Section 4.2) where a 1 < a 2 < · · · < a M , ν 1 , ν 2 , . . . , ν M ∈ Z ≥0 and A ∈ R 1,iŪ (n) is not divisible by any factor in the denominator. Substitute the expression (4.31) into the equation (4.30). It has therefore to be compensated by h i1 − 1 . Hence the only allowed value of a L is a L = 0 and we have ν L ≤ 1. Similarly, the denominator h ij − a 1 + 1 appears only in the term h i1 + 2 σ 1 [−ε 1 ] in (4.30). It has to be compensated by h i1 + 2 . Hence the only allowed value of a 1 is a 1 = 0 and we have ν 1 ≤ 1.
It follows that the partial fraction decomposition of the element σ 1 with respect toh 1 reads where A k , A k , k = 2, . . . , n, do not depend onh 1 and B is regular inh 1 .
3. The equations (4.28) impose further restrictions on the constituents of the decomposition (4.32) of the element σ 1 . Substitute the decomposition (4.32) into the equation W i = 0. The terms which have denominators of the formh i1 + m, m ∈ Z, in (4.30) arẽ In the expression (4.33), the terms with the denominatorh i1 + 1 read Therefore, With this condition, the expression (4.33) vanishes. We conclude that . Now we substitute the obtained expression (4.34) for σ 1 into the equation W j = 0 with j = i and follow the singularities of the formh i1 + m, m ∈ Z. The singular terms arẽ In the expression (4.35), the terms with the denominatorh i1 read Therefore, the numerator, as a polynomial inh 1 , must be divisible by the denominatorh i1 . The polynomial remainder of this division equals Therefore, for any j = 2, . . . , n, j = i, the combinationh ij A i does not depend onh j . It follows that where each α i is a univariate polynomial. For the moment, we have found that where the element B is regular inh 1 and σ (s) A direct calculation shows that the element σ where u j h j ,h 1 , j = 3, . . . , n, is a polynomial inh j ,h 1 and C is a linear combination of complete symmetric polynomials inh 2 , . . . ,h n with coefficients in K[h 1 ].
The equation W 2 (B) = 0 implies that the expressioñ does not depend onh 2 . In the notation of paragraph 1 in Section 4.2, the part P 2;j , j = 3, . . . , n, of this expression is Then equation (4.36) becomes or ∆ 1 (β j ) = 0, so β j depends only onh j . But then if β j = 0, the formula (4.37) shows that u j cannot be a polynomial inh 1 . We conclude that the principal part of the element B with respect toh 2 vanishes, and B = C is a polynomial in all its variables.
Consider first the case n = 2. Set where ξ is some polynomial inh 1 andh 2 . With this substitution the equation W 2 (C) = 0 becomes that is, where µ does not depend onh 2 . Note that by construction, the polynomial ξ is divisible bỹ h 21 h 21 + 1 , which implies that µ is a polynomial inh 1 . Since ∆ 1 is surjective on polynomials, we can write µ = ∆ 2 1 z h 1 for some univariate polynomial z, that is We have Therefore, where w h 2 is a polynomial inh 2 . That is, Since the element C is a polynomial, the denominatorh 21 in the second term in the right hand side of (4.38) shows that w = −z. Therefore, as claimed.
The claim for arbitrary n follows since for any j > 2 the element C is a linear combination of ∆ 1 H L h 1 ,h j , L = 1, 2, . . .

7.
We summarize the results of this section in the following proposition.

Potential. Proof of Proposition 3.6
First proof. We rewrite the formula (4.39) in the form Then the expressions for the elements σ j , j = 2, . . . , n, see (4.27), read Since, for ν ∈ H, we find that The term with i = j in the sum in the right hand side of (4.40) is simply we can rewrite the term with i = j in the right hand side of (4.40) in the form Therefore, 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 onh 1 so, by surjectivity of ∆ 2 on polynomials inh 2 , there exists a polynomial p 1 which does not depend onh 1 and ∆ 2 (p) = ∆ 2 (p 1 ). The polynomial p 2 := p − p 1 does not depend onh 2 . The next lemma generalizes this decomposition to the ringŪ(n).  Proof . Decompose f into partial fractions with respect toh 1 . We have P 1;2 (f ) = 0. Indeed, write P 1;2 (f ) in the form where a 1 < a 2 < · · · < a L , ν 1 , ν 2 , . . . , ν L ∈ Z >0 and u ∈ R 1,2Ū (n) is not divisible by any factor in the denominator. Assume that u = 0. Then The factor h 12 − a L − 1 appears only in the denominator of the second term in the right hand side and cannot be compensated by the numerator. Thus P 1;2 (f ) = 0 (the consideration of the factor h 12 − a 1 + 1 in the denominator of the third term proves the claim as well). Now we write the part P 1;j (f ), j > 2, in the form (4.11), where u jaνa ∈ N 1Ū (n) and the sums are finite. Then (4.43) We prove that the elements u jaνa do not depend onh 2 . Indeed, if this is not true then there is a minimal a ∈ Z for which ∆ 2 (u jaνa ) = 0 for some ν a . But then the denominator (h 1j − a) νa in the right hand side in (4.43) cannot be compensated.
We conclude that f = f 2,0 + g where f 2,0 = j>2 P 1;j (f ) does not depend onh 2 and g is regular We decompose g with respect toh 2 . As above, the part P 2;1 (g) vanishes and the calculation, parallel to (4.43), shows that P 2;j (g), j > 2, does not depend onh 1 . Now we have where f 1,0 = j>2 P 2;j (g) does not depend onh 1 and f + is regular inh 1 andh 2 .
We use the decomposition (4.41) for the regular part f + and write f does not depend onh 1 and f + 2 does not depend onh 2 . This leads to the required decomposition (4.42) with f 1 = f 1,0 + f + 1 and f 2 = f 2,0 + f + 2 .
The element F − G does not depend onh c , 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 onh 2 , v does not depend onh 1 , and F − G = u − v. Then is the desired potential.
Second proof of Proposition 3.6. The symmetric, in i and j, part of the equation (3.7) is (4.44) The system (4.44) by itself does not imply the existence of a potential. However, the equation (3.7) can be written in the form σ j = ∆ j h ji + 1 σ i . So for each j = 1, . . . , n the element σ j belongs to the image of the operator ∆ j . Then, according to Lemma 4.7, there exists σ ∈Ū(n) such that σ j = ∆ j (σ).

Polynomial potentials. Proof of Lemma 3.7
The operatorq i defined by (3.14) can be an automorphism of the ring Diff h,σ (n) only if On the other hand, is central in the homogeneous ring Diff h,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 Thus the element c(t) commutes with the generators x j , j = 1, . . . , n, if and only if the polynomial 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 ∈ W j is similar. Since the combination e(t) 1+h 1 t does not depend onh 1 , we have, for ρ(t) = e(t) 1+h 1 t σ, For j > 1 we have and we calculate ∆ j ρ(t) = e(t) 1 +h 1 t 1 +h j t ∆ j 1 +h j t σ = e(t) 1 +h 1 t 1 +h j t tσ 4.7 Rings of fractions. Proof of Lemma 3.9 , where x •i := x i ψ i , i = 1, . . . , n, generates the localized ring S −1 x Diff h,σ (n). Moreover, the complete set of the defining relations for the generators from the set B D does not remember about the potential σ. It reads The proof is the same as for the ring Diff h (n), see [16]. The isomorphism is now clear.
(ii) Assume that ι : Diff h,σ (n) → Diff h,σ (n) is an isomorphism of filtered rings overŪ(n). To distinguish the generators, we denote the generators of the ring Diff h,σ (n) by x i and∂ i .
The ε i -weight subspace E i of the ring Diff h,σ (n) consists of elements of the form θx i where θ is a polynomial in the elements Γ j , j = 1, . . . , n, with coefficients inŪ(n). Since the space of the elements of E i of filtration degree ≤ 1 isŪ(n)x i , we must have The condition (4.49) implies that γ i = γ for some γ ∈ K. The condition (4.50) then becomes γσ i = σ i and the assertion follows.

Lowest weight representations. Proof of Proposition 3.10
We need the following identity (see [16,Lemma 5]): Then, e(t) . 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 havē
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 (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 Proposition 3.8(ii), it is sufficient to establish (3.19) for where A is a univariate polynomial.