Structure constants of diagonal reduction algebras of gl type

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


Introduction
This paper completes the work [7]: it contains a derivation of basic relations for the diagonal reduction algebras of gl type, their low dimensional examples and properties.
Let g be a Lie algebra, k ⊂ g its reductive Lie subalgebra and V an irreducible finitedimensional g-module, which decomposes, as an k-module, into a direct sum of irreducible kmodules V i with certain multiplicities m i , (1.1) Here W i = Hom k (V i , V ) are the spaces of multiplicities, m i = dim W i . While the multiplicities m i present certain combinatorial data, the spaces W i of multiplicities itself may exhibit a 'hidden structure' of modules over certain special algebras [4]. The well-known example is the Olshanski centralizer construction [9], where g = gl n+m , k = gl m and the spaces W i carry the structure of irreducible Yangian Y (gl n )-modules. In general, the multiplicity spaces W i are irreducible modules over the centralizer U(g) k of k in the universal enveloping algebra U(g) [8]. However, these centralizers have a rather complicated algebraic structure and are hardly convenient for applications. Besides, under the above assumptions, the direct sum W = ⊕ i W i becomes a module over the reduction (or Mickelsson) algebra. The reduction algebra is defined as follows. Suppose k is given with a triangular decomposition k = n − + h + n.
(1. 2) Denote by I + the left ideal of A := U(g), generated by elements of n, I + := An . Then the reduction algebra S n (A), related to the pair (g, k), is defined as the quotient Norm(I + )/I + of the normalizer of the ideal I + over I + . It is equipped with a natural structure of the associative algebra. By definition, for any g-module V the space V n of vectors, annihilated by n, is a module over S n (A). Since V is finite-dimensional, V n is isomorphic to ⊕ i W i , so the latter space can be viewed as an S n (A)-module as well; the zero-weight component of S n (A), which contains a quotient of the centralizer U(g) k , preserves each multiplicity space W i . The representation theory of the reduction algebra S n (A) is closely related to the theory of branching rules g ↓ k for the restrictions of representations of g to k.
The reduction algebra simplifies after the localization over the multiplicative set generated by elements h γ + k, where γ ranges through the set of roots of k, k ∈ Z, and h γ is the coroot corresponding to γ. Let U(h) be the localization of the universal enveloping algebra U(h) of the Cartan subalgebra h of k over the above multiplicative set. The localized reduction algebra Z n (A) is an algebra over the commutative ring U(h); the principal part of the defining relations is quadratic but the relations may contain linear or degree 0 terms, see [10,6].
Besides, the reduction algebra admits another description as a (localized) double coset space n − A\A/An endowed with the multiplication map defined by means of the insertion of the extremal projector [6] of Asherova-Smirnov-Tolstoy [3]. The centralizer A k is a subalgebra of Z n (A).
It was shown in [7] that the general reduction algebra Z n (A) admits a presentation over U(h) such that the defining relations are ordering relations for the generators, in an arbitrary order, compatible with the natural partial order on h * . The set of ordering relations for the general reduction algebra Z n (A) was shown in [7] to be "algorithmically efficient" in the sense that any expression in the algebra can be ordered with the help of this set.
The structure constants of the reduction algebra are in principle determined with the help of the extremal projector P or the tensor J studied by Arnaudon, Buffenoir, Ragoucy and Roche [1]. However the explicit description of the algebra is hardly achievable directly.
In the present paper, we are interested in the special restriction problem, when g is the direct sum of two copies of a reductive Lie algebra a and k is the diagonally embedded a. The resulting reduction algebra for the symmetric pair (a ⊕ a, a) we call diagonal reduction algebra DR(a) of a. The theory of branching rules for a ⊕ a ↓ a is the theory of decompositions of the tensor products of a-modules into a direct sum of irreducible a-modules.
We restrict ourselves here to the Lie algebra a = gl n of the general linear group. In this situation finite-dimensional irreducible modules over g are tensor products of two irreducible gl nmodules, the decomposition (1.1) is the decomposition of the tensor product into the direct sum of irreducible gl n -modules, and the multiplicities m i are the Littlewood-Richardson coefficients.
The reduction algebra DR(gl n ) for brevity will be denoted further by Z n . In [7] we suggested a set R of relations for the algebra Z n . We demonstrated that the set R is equivalent, over U(h), to the set of the defining ordering relations provided that all relations from the set R are valid.
The main goal of the present paper is the verification of all relations from the system R. There are two principal tools in our derivation. First, we use the braid group action by the Zhelobenko automorphisms of reduction algebras [10,6]. Second, we employ the stabilization phenomenon, discovered in [7], for the multiplication rule and for the defining relations with respect to the standard embeddings gl n ֒→ gl n+1 ; stabilization provides a natural way of extending relations for Z n to relations for Z n+1 (Z n is not a subalgebra of Z n+1 ). We perform necessary calculations for low n (at most n = 4); the braid group action and the stabilization law allow to extend the results for general n.
As an illustration, we write down the complete lists of defining relations in the form of ordering relations for the reduction algebras DR(sl 3 ) and DR(sl 2 ). Although for a finite n the task of deriving the set of defining (ordering) relations for DR(sl n ) is achievable in a finite time, it is useful to have the list of relations for small n in front of the eyes.
We return to the stabilization and cut phenomena and make more precise statements concerning now the embedding of the Lie algebra gl n ⊕gl 1 into the Lie algebra gl n+1 (more generally, of gl n ⊕ gl m into gl n+m ). As a consequence we find that cutting preserves the centrality: the cut of a central element of the algebra Z n+m is central in the algebra Z n ⊗ Z m . We also show that, similarly to the Harish-Chandra map, the restriction of the cutting to the center is a homomorphism. As an example, we derive the Casimir operators for the algebra DR(sl 2 ) by cutting the Casimir operators for the algebra DR(sl 3 ).
The relations in the diagonal reduction algebra have a quadratic and a degree zero part. The algebra, defined by the homogeneous quadratic part of the relations, tends, in a quite simple regime, to a commutative algebra (the homogeneous algebra can be thus considered as a "dynamical" deformation of a commutative algebra; "dynamical" here means that the left and right multiplications by elements of the ring U(h) differ). This observation about the limit is used in the proof in [7] of the completeness of the set of derived relations over the field of fractions of U(h). We prove the completeness by establishing the equivalence between the set of derived relations and the set of ordering relations.
The stabilization law enables one to give a definition of the reduction "algebra" Z ∞ related to the diagonal embedding of the inductive limit gl ∞ of gl n into gl ∞ ⊕ gl ∞ (strictly speaking, Z ∞ is not an algebra, some relations have an infinite number of terms).
We also discuss the diagonal reduction algebra for the special linear Lie algebra sl n ; it is a direct tensor factor in Z n .
Such a precise description, as the one we give for Z n , is known for a few examples of the reduction algebras: the most known is related to the embedding of gl n to gl n+1 [10]. Its representation theory was used for the derivation of precise formulas for the action of the generators of gl n on the Gelfand-Zetlin basic vectors [2]. The reduction algebra for the pair (gl n , gl n+1 ) is based on the root embedding gl n ⊂ gl n+1 of Lie algebras. In contrast to this example, the diagonal reduction algebra DR(a) is based on the diagonal embedding of a into a ⊕ a, which is not a root embedding of reductive Lie algebras.

Notation
Let E ij , i, j = 1, . . . , n, be the standard generators of the Lie algebra gl n , with the commutation relations where δ jk is the Kronecker symbol. We shall also use the root notation H α , E α , E −α , . . . for elements of gl n .
Let E (1) ij and E (2) ij , i, j = 1, . . . , n, be the standard generators of the two copies of the Lie algebra gl n in g := gl n ⊕ gl n , ij .
The elements e ij span the diagonally embedded Lie algebra k ≃ gl n , while E ij form an adjoint k-module p. The Lie algebra k and the space p constitute a symmetric pair, that is, In the sequel, h a means the element e aa of the Cartan subalgebra h of the subalgebra k ∈ gl n ⊕gl n and h ab the element e aa − e bb . Let {ε a } be the basis of h * dual to the basis {h a } of h, ε a (h b ) = δ ab . We shall use as well the root notation h α , e α , e −α for elements of k, and H α , E α , E −α for elements of p.
The Lie subalgebra n in the triangular decomposition (1.2) is spanned by the root vectors e ij with i < j and the Lie subalgebra n − by the root vectors e ij with i > j. Let b + and b − be the corresponding Borel subalgebras, b + = h ⊕ n and b − = h ⊕ n − . Denote by ∆ + and ∆ − the sets of positive and negative roots in the root system ∆ = ∆ + ∪ ∆ − of k: ∆ + consists of roots ε i − ε j with i < j and ∆ − consists of roots ε i − ε j with i > j. Let Q be the root lattice, Q := {γ ∈ h * | γ = α∈∆ + ,nα∈Z n α α}. It contains the positive cone Q + , For λ, µ ∈ h * , the notation λ > µ (2.1) means that the difference λ − µ belongs to Q + , λ − µ ∈ Q + . This is a partial order in h * . We fix the following action of the cover of the symmetric group S n (the Weyl group of the diagonal k) on the Lie algebra gl n ⊕ gl n by automorphismś Here σ i = (i, i + 1) is an elementary transposition in the symmetric group. We extend naturally the above action of the cover of S n to the action by automorphisms on the associative algebra A ≡ A n := U(gl n ) ⊗ U(gl n ). The restriction of this action to h coincides with the natural action σ(h k ) = h σ(k) , σ ∈ S n , of the Weyl group on the Cartan subalgebra.
Besides, we use the shifted action of S n on the polynomial algebra U(h) (and its localizations) by automorphisms; the shifted action is defined by It becomes the usual action for the variables by (2.2) for any σ ∈ S n we have It will be sometimes convenient to denote the commutator [a, b] of two elements a and b of an associative algebra bŷ (2.4)

Reduction algebra Z n
In this section we recall the definition of the reduction algebras, in particular the diagonal reduction algebras of the gl type. We introduce the order for which the ordering relations for the algebra Z n will be discussed. The formulas for the Zhelobenko automorphisms for the algebra Z n are given; some basic facts about the standard involution, anti-involution and central elements for the algebra Z n are presented at the end of the section. 1. Let U(h) andĀ be the rings of fractions of the algebras U(h) and A with respect to the multiplicative set, generated by elements Define Z n to be the double coset space ofĀ by its left ideal I + :=Ān, generated by elements of n, and the right ideal I − := n −Ā , generated by elements of n − , Z n :=Ā/(I + + I − ).
The space Z n is an associative algebra with respect to the multiplication map Here P is the extremal projector [3] for the diagonal gl n . It is an element of a certain extension of the algebra U(gl n ) satisfying the relations e ij P = P e ji = 0 for all i and j such that 1 ≤ i < j ≤ n. The algebra Z n is a particular example of a reduction algebra; in our context, Z n is defined by the coproduct (the diagonal inclusion) U(gl n ) → A.
2. The main structure theorems for the reduction algebras are given in [7,Section 2].
In the sequel we choose a weight linear basis {p K } of p (p is the k-invariant complement to k in g, g = k + p) and equip it with a total order ≺. The total order ≺ will be compatible with the partial order < on h * , see (2.1), in the sense that µ K < µ L ⇒ p K ≺ p L . We shall sometimes write I ≺ J instead of p I ≺ p J . For an arbitrary element a ∈Ā let a be its image in the reduction algebra; in particular, p K is the image in the reduction algebra of the basic vector p K ∈ p.
3. In our situation we choose the set of vectors E ij , i, j = 1, . . . , n, as a basis of the space p. The weight of E ij is ε i − ε j . The compatibility of a total order ≺ with the partial order < on h * means the condition The order in each subset {E ij |i − j = a} with a fixed a can be chosen arbitrarily. For instance, we can set Denote the images of the elements E ij in Z n by z ij . We use also the notation t i for the elements z ii and t ij := t i − t j for the elements z ii − z jj . The order (3.2) induces as well the order on the generators z ij of the algebra Z n : The statement (d) in the paper [7, Section 2] implies an existence of structure constants B (ab),(cd),(ij),(kl) ∈ U(h) and D (ab),(cd) ∈ U(h) such that for any a, b, c, d = 1, . . . , n we have In particular, the algebra Z n (in general, the reduction algebra related to a symmetric pair (k, p), g := k + p) is Z 2 -graded; the degree of z ab is 1 and the degree of any element from U(h) is 0. The relations (3.3) together with the weight conditions are the defining relations for the algebra Z n . Note that the denominators of the structure constants B (ab),(cd),(ij),(kl) and D (ab),(cd) are products of linear factors of the formh ij + ℓ, i < j, where ℓ ≥ −1 is an integer, see [7].
4. The algebra Z n can be equipped with the action of Zhelobenko automorphisms [6]. Denote byq i the Zhelobenko automorphismq i : Z n → Z n corresponding to the transposition σ i ∈ S n . It is defined as follows [6]. First we define a mapq i : A →Ā/I + by Hereê i,i+1 stands for the adjoint action of the element e i,i+1 , see (2.4). The operatorq i has the property for any x ∈ A and h ∈ h; σ • h is defined in (2.2). With the help of (3.5), the mapq i can be extended to the map (denoted by the same symbol)q i :Ā →Ā/I − by settingq i (a(h)x) = (σ i • a(h))q i (x) for any x ∈ A and a(h) ∈ U(h). One can further prove thatq i (I + ) = 0 anď q i (I − ) ⊂ (I − + I + )/I + , so thatq i can be viewed as a linear operatorq i : Z n → Z n . Due to [6], this is an algebra automorphism, satisfying (3.5).
The operatorsq i satisfy the braid group relations [10]: and the inversion relation [6]: Besides, the outer automorphism of the Dynkin diagram of gl n induces the involutive automorphism ω of Z n , where i ′ = n + 1 − i. The operations ǫ and ω commute, ǫω = ωǫ. Central elements of the subalgebra U(gl n ) ⊗ 1 ⊂ A, generated by n Casimir operators of degrees 1, . . . , n, as well as central elements of the subalgebra 1 ⊗ U(gl n ) ⊂ A project to central elements of the algebra Z n . In particular, central elements of degree 1 project to central elements and I (n,t) := t 1 + · · · + t n (3.10) of the algebra Z n . The difference of central elements of degree two projects to the central element of the algebra Z n . The images of other Casimir operators are more complicated.

Main results
This section contains the principal results of the paper. We first give preliminary information on the new basis in which the defining relations for the algebra Z n can be written down in an economical fashion. The braid group action on the new generators is then explicitly given in Subsection 4.2. The complete set of the defining relations for the algebra Z n is written down in Subsection 4.3. The regime for which both the set of the derived defining relations and the set of the defining ordering relation have a controllable "limiting behavior" is introduced in Subsection 4.4. Subsection 4.5 deals with the diagonal reduction algebra for sl n ; the quadratic Casimir operator for DR(sl n ) as well as for the diagonal reduction algebra for an arbitrary semi-simple Lie algebra k is given there. Subsection 4.6 is devoted to the stabilization and cut phenomena with respect to the embedding of the Lie algebra gl n ⊕gl m into the Lie algebra gl n+m ; the theorem about the behavior of the centers of the diagonal reduction algebra under the cutting is proved there.

New variables
We shall use the following elements of U(h): which can be used to convert the definition (4.1) into a linear over the ring U(h) change of variables: For example, In terms of the new variablest's, the linear in t central element (3.10) reads

Braid group action
Sinceq 2 i (x) = x for any element x of zero weight, the braid group acts as its symmetric group quotient on the space of weight 0 elements. It follows from (4.1) andq i (t 1 ) = t 1 for all i > 1 thatq for any σ ∈ S n .
The action of the Zhelobenko automorphisms, see Section 3, on the generators z kl looks as follows: Denote i ′ = n + 1 − i, as before. The braid group action (4.4) is compatible with the antiinvolution ǫ and the involution ω (note that ω(h ij ) =h j ′ i ′ ), see (3.7) and (3.8), in the following sense: Let w 0 be the longest element of the Weyl group of gl n , the symmetric group S n . Similarly to the squares of the transformations corresponding to the simple roots, see (3.6), the action ofq 2 w 0 is the conjugation by a certain element of U(h).
Lemma 1. We havě The proof shows that the formula (4.7) works for an arbitrary reductive Lie algebra, with S = α∈∆ +h α .

Proposition 2. The action ofq w 0 on generators readš
The proofs of Lemma 1 and Proposition 2 are in Section 5.

Defining relations
To save space we omit in this section the symbol ⋄ for the multiplication in the algebra Z n . It should not lead to any confusion since no other multiplication is used in this section. Each relation which we will derive will be of a certain weight, equal to a sum of two roots. From general considerations the upper estimate for the number of terms in a quadratic relation of weight λ = α + β is the number |λ| of quadratic combinations z α ′ z β ′ with α ′ + β ′ = λ. There are several types, excluding the trivial one, λ = 2(ε i − ε j ), |λ| = 1: where i, j and k are pairwise distinct. Then |λ| = 2.
Below we write down relations for each type (and subtype) separately. The relations of the types 1 and 2 have a simple form in terms of the original generators z ij . To write the relations of the types 3 and 4, it is convenient to renormalize the generators z ij with i = j. Namely, we setz (4.11) In terms of the generatorsz ij , the formulas (4.4) for the action of the automorphismsq i translate as follows: 1. The relations of the type 1 are: Then, for any four pairwise different indices i, j, k and l, we have the following relations of the type 2: With this notation the first group of the relations of the type 3 is: The relations (4.14) can be written in a more compact way with the help of both systems, z ij andz ij , of generators. Let now Moreover, after an extra redefinition:z kl =z kl B lk for k > l, the left hand side of the second line in (4.15) becomes, up to a common factor, the same as the left hand side of the first line, The second group of relations of the type 3 reads: 4a. The relations of the weight zero (the type 4) are also divided into 2 groups. This is the first group of the relations: As follows from the proof, the relations (4.17) hold for the diagonal reduction algebra for an arbitrary reductive Lie algebra: the images of the generators, corresponding to the Cartan subalgebra, commute. 4b. Finally, the second group of the relations of the type 4 is where i = j. Main statement. Denote by R the system (4.12), (4.13), (4.14), (4.16), (4.17) and (4.18) of the relations. The derivation of the system R of the relations is given in Section 5. The validity in Z n of relations from the set R, together with the results from [7], completes the proof of Theorem 3 (Section 5.4).

Limit
Let R ≺ be the set of ordering relations (3.3). Denote by R 0 the homogeneous (quadratic) part of the system R and by R ≺ 0 the homogeneous part of the system R ≺ . 1. Placing coefficients from U(h) in all relations from R 0 to the same side (to the right, for example) from the monomials p L ⋄ p M , one can give arbitrary numerical values to the variables h α (α's are roots of k).
The structure of the extremal projector P or the recurrence relation (5.4) implies that the system R 0 admits, for an arbitrary reductive Lie algebra, the limit at h α i = c i h, h → ∞ (α i ranges through the set of simple positive roots of k and c i are generic positive constants). Moreover, this homogeneous algebra becomes the usual commutative (polynomial) algebra in this limit; so this limiting behavior of the system R 0 , used in the proof, generalizes to a wider class of reduction algebras, related to a pair (g, k) as in the introduction.
2. The limiting procedure from paragraph 1 establishes the bijection between the set of relations and the set of unordered pairs (L, M ), where L, M are indices of basic vectors of p. The proof in [7] shows that over D(h) the system R can be rewritten in the form of ordering relations for an arbitrary order on the set { p L } of generators. Here D(h) is the field of fractions of the ring U(h).
By definition, the relations from R ≺ are labeled by pairs (L, M ) with L > M . The above bijection induces therefore a bijection between the sets R and R ≺ . 4.5 sl n 1. Denote the subalgebra of Z n , generated by two central elements (3.9) and (3.10), by Y n ; the algebra Y n is isomorphic to Z 1 .
Since the extremal projector for sl n is the same as for gl n , the diagonal reduction algebra DR(sl n ) for sl n is naturally a subalgebra of Z n . The subalgebra DR(sl n ) is complementary to Y n in the sense that Z n = Y n ⊗ DR(sl n ).
The algebra DR(sl n ) is generated by z ij , i, j = 1, . . . , n, i = j, and t i,i+1 := t i − t i+1 , i = 1, . . . , n − 1 (and the Cartan subalgebra h, generated by h i,i+1 , of the diagonally embedded sl n ). The elements t i,i+1 form a basis in the space of "traceless" combinations c m t m (traceless means that c m = 0), c m ∈ U(h). 2. The action of the braid group restricts onto the traceless subspace: The traceless subspace with respect to the generators t i and the traceless subspace with respect to the generatorst i (that is, the space of linear combinations c mtm , c m ∈ U(h), with c m = 0) coincide. Indeed, in the expression of t l as a linear combination oft k 's (the second line in (4.2)), we find, calculating residues and the value at infinity, that the sum of the coefficients is 1, Therefore, in the decomposition of the difference t i − t j as a linear combination oft k 's, the sum of the coefficients vanishes, so it is traceless with respect tot k 's; t l,l+1 is a linear combination oft 12 ,t 23 , . . . ,t l,l+1 (and vice versa). It should be however noted that in contrast to (4.2), the coefficients in these combinations do not factorize into a product of linear monomials, the lowest example ist 34 : 3. One can directly see that the commutations between z ij and the differences t k − t l close. The renormalization (4.11) is compatible with the sl-condition and, as we have seen, the set {t i,i+1 } of generators can be replaced by the set {t i,i+1 }. Therefore, one can work with the generatorsz ij , i, j = 1, . . . , n, i = j, andt i,i+1 := t i − t i+1 , i = 1, . . . , n − 1. A direct look at the relations (4.12), (4.13), (4.14), (4.16), (4.17) and (4.18) shows that the only non-trivial verification concerns the relations (4.16); one has to check here the following assertion: when z moves throught i,i+1 , only traceless combinations oft l 's appear in the right hand side. Write a relation from the list (4.16) in the formz ijtl = m χ (i,j,l,m) mtmzij + · · · , χ (i,j,l,m) m ∈ U(h), where dots stand for terms withzz. The assertion follows from the direct observation that for all i, j and l the sum of the coefficients χ It clearly depends only on the differences h i − h j and belongs therefore to the center of the subalgebra DR(sl n ). One can write this central element in the form where C uv is the inverse Cartan matrix of sl n . In general, let k be a semi-simple Lie algebra of rank r with the Cartan matrix a ij . Let b ij be the symmetrized Cartan matrix and ( , ) the scalar product on h * induced by the invariant non-degenerate bilinear form on k, so that For each i = 1, . . . , r let α ∨ i be the coroot vector corresponding to the simple root α i , so that Let ρ ∈ h * be the half-sum of all positive roots. Write where n i are nonnegative integers. Let t α i be the images of (2) be the coroot vectors of the diagonally embedded Lie algebra k. The generalization of the central element (4.19) to the reduction algebra DR(k) reads

Stabilization and cutting
In [7] we discovered the stabilization and cut phenomena which are heavily used in our derivation of the set of defining relations for the diagonal reduction algebras of gl-type. The consideration in [7] uses the standard (by the first coordinates) embedding of gl n into gl n+1 . In this subsection we shall make several more precise statements about the stabilization and cut considering now the embedding of gl n ⊕ gl 1 into gl n+1 (more generally, gl n ⊕ gl m into gl n+m ). These precisions are needed to establish the behavior of the center of the diagonal reduction algebra: namely we shall see that cutting preserves the centrality. Notation: h in this subsection denotes the Cartan subalgebra of gl n+m . Consider an embedding of gl n ⊕ gl m into gl n+m , given by an assignment e ij → e ij , i, j = 1, . . . , n, and e ab → e n+a,n+b , a, b = 1, . . . , m, where e kl in the source are the generators of gl n ⊕ gl m and target e kl are in gl n+m . This rule together with the similar rule E ij → E ij and E ab → E n+a,n+b defines an embedding of the Lie algebra (gl n ⊕ gl m ) ⊕ (gl n ⊕ gl m ) into the Lie algebra gl n+m ⊕gl n+m and of the enveloping algebras A n ⊗A m = U(gl n ⊕gl n )⊗U(gl m ⊕gl m ) into A n+m = U(gl n+m ⊕ gl n+m ). This embedding clearly maps nilpotent subalgebras of gl n ⊕ gl m to the corresponding nilpotent subalgebras of gl n+m and thus defines an embedding ι n,m : Z n ⊗ Z m → Z n+m of the corresponding double coset spaces. However, the map ι n,m is not a homomorphism of algebras. This is because the multiplication maps are defined with the help of projectors, which are different for gl n ⊕ gl m and gl n+m .
However, as we will explain now we can control certain differences between the two multiplication maps. Let V n,m be the left ideal of the algebra Z n+m generated by elements z ia with i = 1, . . . , n and a = n + 1, . . . , n + m; let V ′ n,m be the right ideal of the algebra Z n+m generated by elements z ai with i = 1, . . . , n and a = n + 1, . . . , n + m.
Write any element λ ∈ Q + (the positive cone of the root lattice of gl n+m ) in the form λ = n+m k=1 λ k ε k . The element λ can be presented as a sum where λ ′ is an element of the root lattice of gl n ⊕ gl m , and λ ′′ is proportional to the simple root ε n − ε n+1 : λ ′ = n+m k=1 λ ′ k ε k with n k=1 λ ′ k = n+m k=n+1 λ ′ k = 0 and λ ′′ = c(ε n − ε n+1 ).

Lemma 4.
The left ideal V n,m ⊂ Z n+m consists of images in Z n+m of sums ia X ia E ia with X ia ∈Ā n+m , i = 1, . . . , n and a = n + 1, . . . , n + m. The right ideal V n,m ⊂ Z n+m consists of images in Z n+m of sums ai E ai Y ai with Y ai ∈Ā n+m , i = 1, . . . , n and a = n + 1, . . . , n + m.
For any X ∈Ā n+m , i = 1, . . . , n and a = n + 1, . . . , n + m consider the product X ⋄ λ z ia . The product X ⋄ λ z in is zero if λ ′′ = 0 (the component λ ′′ is defined by (4.20)). Indeed, in this case in each summand of P λ one of e γ ′ k ′ is equal to some e jb , j = 1, . . . , n and b = n+1, . . . , n+m. Choose an ordered basis of n + which ends by all such e jb (ordered arbitrarily); any element of U(n + ) can be written as a sum of ordered monomials, that is, monomials in which all such e jb stand on the right. Since [e jb , E ia ] = 0 for any i, j = 1, . . . , n and a, b = n + 1, . . . , n + m, the product e γ ′ k ′ E ia belongs to the left ideal I + and thus X ⋄ λ z ia = 0 in Z n+m .
If λ ′′ = 0 then generators of n + in monomials entering the decomposition of P λ are among the elements e ij , 1 ≤ i < j ≤ n, and e ab , n + 1 ≤ a < b ≤ n + m and thus their adjoint action leaves the space, spanned by all E ia , i = 1, . . . , n, a = n + 1, . . . , n + m invariant, so X ⋄ λ z ia can be presented as an image of the sum jb X jb E jb with X jb ∈Ā n+m , j = 1, . . . , n, b = n + 1, . . . , n + m. Thus, the left ideal, generated by all z ia is contained in the vector space of images in Z n+m of sums jb X jb E jb .
Moreover, for any X ∈Ā n+m the element X ⋄z ia is the image of XE ia + j,b: j<i, b>a X (jb) E jb for some X (jb) and the double induction on i and a proves the inverse inclusion.
The second part of lemma is proved similarly.
Corollary 5. We have the following decomposition of the free left (and right) U(h)-modules: where I n,m := V n,m + V ′ n,m .
Proof . The double coset space Z n+m is a free left and right U(h)-module with a basis consisting of images of ordered monomials on elements E ij , i, j = 1, . . . , n + m; recall that we always use orders compatible with the partial order < on h * , see (c) in Section 3, paragraph 2. We can choose an order for which all ordered monomials are of the form XY Z, where X is a monomial on E ai with i = 1, . . . , n and a = n + 1, . . . , n + m, Z is a monomial on E ia with i = 1, . . . , n and a = n + 1, . . . , n + m while Y is a monomial on E ij with i, j = 1, . . . , n or i, j = n + 1, . . . , n + m. Then we apply the lemma above.
For a moment denote for each k > 0 the multiplication map in Z k by ⋄ (k) : Z k ⊗ Z k → Z k (instead of the default notation ⋄, see (3.1)); denote also for each k, l > 0 by ⋄ (k,l) the multiplication map ⋄ (k) ⊗ ⋄ (l) in Z k ⊗ Z l . Let h n and h m be the Cartan subalgebras of gl n and gl m , respectively. Denote the space Z n ⊗ U(hn) U(h) ⊗ U(hm) Z m by U(h) · (Z n ⊗ Z m ). The composition law ⋄ (n,m) naturally extends to the space U(h) · (Z n ⊗ Z m ) equipping it with an associative algebra structure (we keep the same symbol ⋄ (n,m) for the extended composition law in U(h) · (Z n ⊗ Z m )). Also, the map ι n,m admits a natural extension to a map ι n,m : U(h) · (Z n ⊗ Z m ) → Z n+m denoted by the same symbol and defined by the rule ι n,m (ϕx) := ϕ ι n,m (x) for any ϕ ∈ U(h) and x ∈ Z n ⊗ Z m . The statement of Proposition 6 remains valid for this extension as well, that is, one can take x, y ∈ U(h) · (Z n ⊗ Z m ) in the formulation.
Proof of Proposition 6. Denote by P n,m := P n ⊗P m the projector for the Lie algebra gl n ⊕gl m .
It is sufficient to prove the following statement. Suppose X and Y are (non-commutative) polynomials in E ij with i, j = 1, . . . , n Then the product of x and y in Z n+m coincides with the image in Z n+m of X P n,m Y modulo the left ideal V n,m and modulo the right ideal V ′ n,m ). Due to the structure of the projector the condition λ ′′ = 0, see (4.20), implies that the product X ⋄ λ Y related to gl n ⊕ gl m coincides with product X ⋄ λ Y related to gl n+m .
Let now λ ′′ = 0. Then each monomial e γ ′ 1 · · · e γ ′ t ′ in the decomposition of P λ , see (4.21), contains generators e ia with i ∈ {1, . . . , n} and a ∈ {n+1, . . . , n+m}; these e ia can be assumed to be right factors of the corresponding monomial (like in the proof of Lemma 4). The commutator of any such generator e ia with every factor in Y is a linear combination of the elements E jb with j ∈ {1, . . . , n} and b ∈ {n + 1, . . . , n + m}. Moving the resulting E jb to the right we see that the product X ⋄ λ Y is the image in Z n+m of an element of the form s X s Y s where each Y s belongs to the left ideal ofĀ n+m generated by E jb with j ∈ {1, . . . , n} and b ∈ {n + 1, . . . , n + m} (one can say more: each Y s can be written in a form j,b Y (jb) s E jb where each Y (jb) s ∈Ā n+m does not involve generators E ck with k ∈ {1, . . . , n} and c ∈ {n + 1, . . . , n + m}; we don't need this stronger form). Thus, due to Lemma 4, X ⋄ λ Y ∈ V n,m .
Similarly, each X s participating in the sum s X s Y s , see above, belongs to the right ideal ofĀ n+m generated by the elements E bj with j ∈ {1, . . . , n} and b ∈ {n + 1, . . . , n + m}. So, again by Lemma 4, X ⋄ λ Y ∈ V ′ n,m .
Suppose that we have a relation whereā k = ι n,m (a k ),b k = ι n,m (b k ) and z ∈ J n,m = V n,m ∩ V ′ n,m . On the other hand, suppose we have the following relation in Z n+m : where all a k and b k are elements of Z n ⊗ Z m ,ā k = ι n,m (a k ),b k = ι n,m (b k ), and u ∈ I n,m = V n,m + V ′ n,m . Then the elements a k and b k satisfy the relation (4.23) and u ∈ J n,m . Indeed, suppose that the relation (4.25) is satisfied and k a k ⋄ (n,m) b k = v for some v ∈ Z n ⊗ Z m . It follows from Proposition 6 that kā k ⋄ (m+n)bk −v belongs to J n,m ; herev = ι n,m (v). Then (4.25) implies thatv ∈ I n,m and thusv = 0 due to Corollary 5. Thus v = 0, since the map ι n,m is an inclusion, and u ∈ J n,m .
We refer to the implication (4.23) ⇒ (4.24) as stabilization. Call cutting the (almost inverse) implication (4.25) ⇒ (4.23) which can be understood as a procedure of getting relations in Z n ⊗ Z m from relations in Z n+m ; we say that (4.23) is the cut of (4.25). Clearly all relations in Z n ⊗ Z m can be obtained by cutting appropriate relations in Z n+m .
Let π n,m : Z n+m → U(h) · (Z n ⊗ Z m ) be the composition of the projectionπ n,m of Z n+m onto ι n,m (U(h) · Z n ⊗ Z m ) = U(h) · ι n,m (Z n ⊗ Z m ) along I n,m , see (4.22), and of the inverse to the inclusion ι n,m : π n,m = ι −1 n,m •π n,m .
We have the following consequence of Proposition 6 and Corollary 5. Proof . Denote X = π n,m (x). Then, by definition, x = ι n,m (X) + z, where z ∈ I n,m . Since x is central, it is of zero weight; so X and z are of zero weight as well. Thus each monomial entering the decomposition of z contains both types of generators, E ai and E ia , where i ∈ {1, . . . , n} and a ∈ {n + 1, . . . , n + m}, which implies that z ∈ J n,m = V n,m ∩ V ′ n,m . Take any Y ∈ Z n ⊗ Z m . We now prove that X ⋄ (n,m) Y − Y ⋄ (n,m) X = 0. Denote y = ι n,m (Y ). Due to Proposition 6, where z ′ ∈ J n,m = V n,m ∩ V ′ n,m . Since x is central in Z n+m , the right hand side of (4.26) is equal to which is an element of I n,m = V n,m ⊕ V ′ n,m since z, z ′ ∈ J n,m . On the other hand, the left hand side of (4.26) belongs to U(h) · ι n,m (Z n ⊗ Z m ). Thus, by Corollary 5, both sides of (4.26) are equal to zero and X ⋄ (n,m) Y − Y ⋄ (n,m) X = 0 since the map ι n,m is injective.
The map π n,m obeys properties similar to those of the Harish-Chandra map U(g) h → U(h) (U(g) h is the space of elements of zero weight). For instance, its restriction to the center of Z n+m is a homomorphism. More precisely, if x is a central element of Z n+m , then π n,m (x ⋄ (m+n) y) = π n,m (x) ⋄ (n,m) π n,m (y) and π n,m (y ⋄ (m+n) x) = π n,m (y) ⋄ (n,m) π n,m (x) (4.27) for any y ∈ Z n+m . Indeed, let X = π n,m (x), Y = π n,m (y). Then where u ∈ I n,m while, as it was noted in the proof of Proposition 7, z ∈ J n,m . Moreover, it is clear that z can be written in the form z = a z ′ a z a , where z a ∈ V n,m and z ′ a ∈ V ′ n,m (for instance, use the order as in the proof of Corollary 5). Then (dropping for brevity the multiplication symbol ⋄ (m+n) ) we have ι n,m (X)ι n,m (Y ) = (x + z)(y + u) = x + a z ′ a z a (y +z ′ +z) = xy + a z ′ a z a (y +z ′ +z) + xz +z ′ x ≡ xy mod I n,m . (4.28) Herez ∈ V n,m andz ′ ∈ V ′ n,m . In the last equality we used the centrality of x. Due to Proposition 6, (4.28) is precisely equivalent to the fist part of (4.27). The second part of (4.27) is proved similarly.

Tensor J
The multiplication map ⋄ in Z n (we return to the original notation) is given by the prescription (3.1), as in any reduction algebra. It can be formally expanded into a series over the root lattice of certain bilinear maps as follows. Set All these are associative algebras. Besides, both algebras U(b ± ) are U(h)-bimodules. The algebra U 12 (b) admits three commuting actions of U(h). Two of them are given by the assignments for any X ∈ U(h), Y ∈ U(b − ) and Z ∈ U(b + ). The third action associates to any X ∈ U(h), Present the projector P in an ordered form: the summation is over γ ∈ Q + and i ∈ Z ≥0 ; everyF γ,i is an element of U(n − ) of the weight −γ, everyÈ γ,i is an element of U(n + ) of the weight γ andH γ,i ∈ U(h). Let J be the following element of U 12 (b): Due to the PBW theorem in U(gl n ) the tensor J is uniquely defined by the projector P ; it is of total weight zero: hJ = Jh for any h ∈ h. We have the weight decomposition of J with respect to the adjoint action of h in the second tensor factor of U 12 (b): where J λ consists of all the terms, corresponding toF λ,iÈλ,iHλ,i in (5.1) (contributing to λ ∈ Q + in the summation), By definition of J, the multiplication ⋄ in the double coset space Z n can be described by the relation where m( i c i ⊗ d i ) is the image in Z n of the element i c i d i . Moreover, in (5.2) we can replace all productsÈ γ,i b in the second tensor factor by the adjoint action ofÈ γ,i on b (in fact, for E γ,i = e γm · · · e γ 1 , we can replaceÈ γ,i b by [È γ,i , b] or byê γm · · ·ê γ 1 (b), see (2.4)) and likewise all products aF γ,i in the first tensor factor by the opposite adjoint action ofF γ,i on a. We have a decomposition of the product ⋄ into a sum over Q + : If a and b are weight elements of Z n of weights ν(a) and ν(b), then the product a ⋄ λ b is the image in Z n of the sum i a i b i , where the weight of each b i is ν(b) + λ, and the weight of each a i is ν(a) − λ.
The tensor J satisfies the Arnaudon-Buffenoir-Ragoucy-Roche (ABRR) difference equation [1], see also [5] for the translation of the results of [1] to the language of reduction algebras. To describe the equation, let ϑ = 1 2 n k=1h 2 k ∈ U(h); for any positive root γ ∈ ∆ + , denote by T γ the following linear operator on the vector space U 12 (b): The ABRR equation means the relation [1, 5]: This relation is equivalent to the following system of recurrence relations for the weight components J λ : whereh λ := k λ khk for λ = k λ k ε k . The recurrence relations (5.4) together with the initial condition J 0 = 1 ⊗ 1 uniquely determine all weight components J λ . It should be noted that the recurrence relations (5.4) provides less information about the structure of the denominators (from U(h)) of the summands of the extremal projector P than the information implied by the product formula (see [3]) for the extremal projector.
Using (5.4) we get in particular:

Braid group action
The proof of the relations (4.1) and (4.4) consists of the following arguments, valid for any reduction algebra. Let α be any simple root of gl n , α = ε i − ε i+1 and g α the corresponding sl 2 subalgebra of gl n . It is spanned by the elements e α = e i,i+1 , e −α = e i+1,i and h α = h i − h i+1 . Letσ α =σ i be the corresponding automorphism of the algebra A andq α =q i the Zhelobenko automorphism of Z n . Assume that Y ∈ A belongs, with respect to the adjoint action of g α , to an irreducible finite-dimensional g α -module of dimension 2j + 1, j ∈ {0, 1/2, 1, . . . }. Assume further that Y is homogeneous, of weight 2m, [h α , Y ] = 2mY . Identify Y with its image in Z n . Thenq α (Y ) coincides with the image in Z n of the element This can be checked directly using [6,Proposition 6.5].
In the realization of irreducible sl 2 -modules as the spaces of homogeneous polynomials in two variables u and v, , u), or, in the basis |j, k := x j+k y j−k (j labels the representation; k = 0, 1, . . . , 2j), Proof of Lemma 1, Subsection 4.2. To see this, write a reduced expression forq w 0 ,q w 0 = q α i 1 · · ·q α i M with α i 1 , . . . , α i M simple roots. Thenq w 0 =q α i M · · ·q α i 1 as well. Writing, forq 2 w 0 , the second expression after the first one, we get squares ofq α is 's (which are conjugations bẙ h −1 α is 's; they thus commute) one after another. Moving these conjugations to the left through the remainingq's, we produce, exactly like in the construction of a system of all positive roots from a reduced expression for the longest element of the Weyl group of a reductive Lie group, the conjugation by the product (4.8) over all positive roots.

Derivation of relations
The set of defining relations in Z n divides into several different types, see Section 4.3. We prove the necessary amount of relations of each type and get the rest by applying the transformations from the braid group as well as the anti-involution ǫ, see (3.7).
We never use the automorphism ω, defined in (3.8), in the derivation of relations. However, the involution ω is compatible with our set of relations in the sense explained in Section 5.4.
In the following we denote by the symbol ≡ the equalities of elements fromĀ modulo the sum (I − + I + ) of two ideals I − and I + defined in the beginning of Section 3. Moreover, for any two elements X and Y of the algebraĀ we may regard the expressions X ⋄ Y and X ⋄ λ Y as the sums of elements fromĀ defined in (5.2) and (5.3). The sum X ⋄ λ Y is finite. By the construction, all but a finite number of terms in the product X ⋄ Y belong to (I − + I + ). Unlike to the system of notation adopted in Section 3, our proof of each relation in Z n will use equalities inĀ taken modulo (I − + I + ). Elements z 12 and z 13 are images in Z n of E 12 and E 13 . Consider the product E 12 ⋄ λ E 13 . Since the adjoint action of gl n preserves the space p, see Section 2, this product is the sum of such monomials E ij E kl , with coefficients in U(h), that (i): the weight ε k − ε l of E kl is equal to the weight ε 1 − ε 3 of E 13 plus λ ∈ Q + , while (ii): the weight ε i − ε j of E ij is equal to the weight ε 1 − ε 2 of E 12 minus λ. Assume that E 12 ⋄ λ E 13 = 0. By (i), λ = −ε 1 + ε 3 + ε k − ε l and it can be positive only if k = 1 and l ≥ 3. So, the condition (i) implies that either λ = 0 or λ = ε 3 − ε l with l > 3. The possibility λ = ε 3 − ε l , l > 3, is excluded by the condition (ii). Therefore, λ = 0 and Similarly, for λ ∈ Q + , which can non-trivially contribute to the product E 13 ⋄ E 12 , the analogue of the condition (i) on the weight λ gives the restriction λ = 0 or λ = ε 2 −ε k , k > 2; the analogue of the condition (ii) further restricts λ: λ = 0 or λ = ε 2 − ε 3 , so we have since J ε 2 −ε 3 = −e 32 ⊗ e 23 (h 23 + 1) −1 as it follows from the ABRR equation, see (5.5), or from the precise explicit expression for the projector P , see [3]. Thus, since E 12 and E 13 commute in the universal enveloping algebra Comparing (5.10) and (5.11) we find (5.9). Applying to (5.9) the anti-involution ǫ, see (3.7), we get the relation The rest of the relations (4.12) are obtained from (5.9) and (5.12) by applying different transformationsq w from the Weyl group.
2. Now we prove in Z n the relation We begin by the proof of this relation in Z 4 . We proceed in the same manner as for the derivation of the relation (5.9), Combining the three latter equalities we obtain (5.13) in Z 4 . The difference of the left and right hand sides of (5.13) in Z n is a linear combination of monomials in z ij of the total weight ε 1 + ε 2 − ε 3 − ε 4 . The weight is non-trivial, so the monomials can be only quadratic. Due to the stabilization phenomenon, each monomial should contain z ij with i > 4 or j > 4, but, by the weight arguments, there is no such non-zero possibility, which completes the proof of the relation (5.13) in Z n .
The rest of relations (4.13) is then obtained by applications of the transformations from the braid group.