Symmetric quantum Weyl algebras

We study the symmetric powers of four algebras: q -oscillator algebra, q -Weyl algebra, h -Weyl algebra and U ( sl 2 ) . We provide explicit formulae as well as combinatorial interpretation for the normal coordinates of products of arbitrary elements in the above algebras


Introduction
This paper takes part in the time-honored tradition of studying an algebra by first choosing a "normal" or "standard" basis for it B, and second, writing down explicit formulae and, if possible, combinatorial interpretation for the representation of the product of a finite number of elements in B, as linear combination of elements in B.
This method has been successfully applied to many algebras, most prominently in the theory of symmetric functions (see [7]).We shall deal with algebras given explicitly as the quotient of a free algebra, generated by a set of letters L, by a number of relations.We choose normal basis for our algebras by fixing an ordering of the set of the letters L, and defining B to be the set of normally ordered monomials, i.e., monomials in which the letters appearing in it respect the order of L.
We consider algebras of the form Sym n (A), i.e, symmetric powers of certain algebras.Let us recall that for each n ∈ N, there is a functor Sym n : C-alg −→ C-alg from the category of associative C-algebras into itself defined on objects as follows: if A is a C-algebra, then Sym n (A) denotes the algebra whose underlying vector space is the n-th symmetric power of A: where S n denotes the group of permutations on n letters.The product of m elements in Sym n (A) is given by the rule for all (a ij ) ∈ A [ [1,m]]×[ [1,n]] .Notice that if A is an algebra, then A ⊗n is also an algebra.S n acts on A ⊗n by algebra automorphisms, and thus we have a well defined invariant subalgebra (A ⊗n ) Sn ⊆ A ⊗n .The following result is proven in [8].The main goal of this paper is the study of the symmetric powers of certain algebras that may be regarded as quantum analogues of the Weyl algebra.Let us recall the well-known This algebra admits a natural representation as indicated in the The symmetric powers of the Weyl algebra have been studied from several point of view in papers such as [1], [3], [6], [10].Our interest in the subject arose from the construction of non-commutative solitonic states in string theory, based on the combinatorics of the annihilation ∂ ∂x and creation x operators given in [3].In [8] we gave explicit formula, as well as combinatorial interpretation for the normal coordinates of monomials ∂ a 1 x b 1 . . .∂ an x bn .This formulae allow us to find explicit formulae for the product of a finite number of elements in the symmetric powers of the Weyl algebra.Looking at Proposition 1 ones notices that the definition of Weyl algebra relies on the notion of the derivative operator ∂ ∂x from classical infinitesimal calculus.The classical derivative ∂ ∂x admits two well-known discrete deformations, the so called q-derivative ∂ q and the h-derivative ∂ h .The main topic of this paper is to introduce the corresponding q and h-analogues for the Weyl algebra, and generalize the results established in [8] to these new contexts.
For the q-calculus, we will actually introduce two q-analogues of the Weyl algebra: the q-oscillator algebra (section 2) and the q-Weyl algebra (section 3).Needless to say, the q-oscillator algebra also known as the q-boson algebra [11], and q-Weyl algebra are deeply related.The main difference between them is that while the q-oscillator is the algebra generated by ∂ q and x, a third operator, the q-shift s q is also present in the q-Weyl algebra.We believe that s q is as fundamental as ∂ q and x.The reason it has passed unnoticed in the classical case is that for q = 1, s q is just the identity operator.For both q-analogues of the Weyl algebra we are able to find explicit formulae and combinatorial interpretation for the product rule in their symmetric powers algebras.
For the h-calculus, also known as the calculus of finite differences, we introduce the h-Weyl algebra in section 4.Besides the annihilator ∂ h and the creator x operators, also includes an h-shift operator s h , which again reduces to the identity for h = 0. We give explicit formulae and combinatorial interpretation for the product rule in the symmetric powers of the h-Weyl algebra.
In section 5 we deal with an algebra of a different sort.Since our method has proven successful for dealing with the Weyl algebra (and it q and hdeformations); and it is known that the Weyl algebra is isomorphic to the universal enveloping algebra of the Heisenberg Lie algebra, it is an interest problem to apply our constructions for other Lie algebras.We consider here only the simplest case, that of sl 2 .We give explicit formulae for the product rule in the symmetric powers of U (sl 2 ).
Although some of the formulae in this paper are rather cumbersome, all of them are just the algebraic embodiment of fairly elementary combinatorial facts.The combinatorial statements will be further analyzed in [9].

Notations and conventions
• N denotes the set of natural numbers.For x ∈ N n and i ∈ N, we denote by x <i the vector (x 1 , . . ., x i−1 ) ∈ N i−1 , by x ≤i the vector (x 1 , . . ., x i ) ∈ N i , by x >i the vector (x i+1 , . . ., x n ) ∈ N n−i and by x ≥i the vector (x i , . . ., x n ) ∈ N n−i+1 .
• For a set X, ♯(X) := cardinality of X, and C X := free associative algebra generated by X.
• Let S be a set and , for all i = 1, . . ., m.
• The q-analogue n integer is [n] := 1 − q n 1 − q .For k ∈ N, we will use 2 Symmetric q-oscillator algebra In this section we define the q-oscillator algebra and study its symmetric powers.Let us introduce several fundamental operators in q-calculus (see [13] for a nice introduction to q-calculus).
Definition 2. The operators ∂ q , s q , x, q, ĥ : C[x, q, h] −→ C[x, q, h] are given as follows We call ∂ q the q-derivative and s q the q-shift.
Definition 3. The algebra C x, y [q, h]/I qo , where I qo is the ideal generated by the relation yx = qxy + h is called the q-oscillator algebra.
We have the following analogue of Proposition 1.
Notice that if we let q → 1, y becomes central and we recover the Weyl algebra.We order the letters of the q-oscillator algebra as follows: q < x < y < h.

Assume we are given
Using this notation we have where min = min(|a|, |b|).For k > min, we set N qo (A, k) to be equal to 0.
Let us introduce some notation needed to formulate Theorem 2 below which provides explicit formula for the normal coordinates Given k ∈ N, we let P k (U, V ) be the set of all maps p : Figure 1 shows an example of such a map.We only show the finite part of p, all other points in V being mapped to ∞.

PSfrag replacements
Definition 5.The value of the crossing number map c : ] and any k ∈ N, we have that Proof.The proof is by induction.The only non-trivial case is the following where min = min(|a|, |b|).Normalizing the left-hand side of (3) we get a recursive relation The other recursion needed being q c(p) satisfies both recursions.
Consider the identity (2) in the representation of the q-oscillator algebra defined in Proposition 2. Apply both sides of the identity (2) to x t for t ∈ N, and use Theorem 2 to get the fundamental Corollary 1.Given (t, a, b) ∈ N×N n ×N n the following identity holds Our next theorem gives a fairly simple formula for the product of m elements in Sym n (C x, y [q, h]/I qo ).Fix a matrix Proof.Using the product rule given in (1), the identity (2) and the distributive property we obtain where min j = min(|a σ j |, |b σ j |).
3 Symmetric q-Weyl algebra In this section we study the symmetric powers of the q-Weyl algebra.
Definition 6.The q-Weyl algebra is given by C x, y, z [q]/I q , where I q is the ideal generated by the following relations: We have the following q-analogue of Proposition 1.
Notice that if we let q → 1, we recover the Weyl algebra.We order the letters of the q-Weyl algebra as follows: q < x < y < z.Given a ∈ N and I ⊂ [[1, a]], we define the crossing number of I to be χ(I) := ♯({(i, j) : i > j, i ∈ I, j ∈ I c }). 2. z a y b = q ab y b z a .
3. y a x b = q ab x b y a .Proof. 2. and 3. are obvious, let us to prove 1.It should be clear that where f I (j) = z, if j / ∈ I and f I (j) = y, if j ∈ I.The normal form of a j=1 f I (j) is q χ(I) y ♯(I) z a−♯(I) .Thus, where min = min(a, b), I ⊂ [[1, a]] and ♯(I) = k.

Assume we are given
Using this notation, we have the where k runs over all vectors k = (k 1 , . . ., ).We set N q (A, k) = 0 for k ∈ N n−1 not satisfying the previous conditions.
Our next theorem follows from Theorem 4 by induction.It gives an explicit formula for the normal coordinates N q (A, k) in the q-Weyl algebra of n i=1 Theorem 5. Let A, k be as in the previous definition, we have Applying both sides of the identity (4) in the representation of the q-Weyl algebra given in Proposition 3 to x t and using Theorem 5, we obtain the remarkable Corollary 2. For any given (t, a, b, c) ∈ N × N n × N n × N n , the following identity holds Our next theorem provides an explicit formula for the product of m elements in the algebra Sym n (C x, y, z [q]/I q ).Fix and similarly for |b σ j | and |c σ j |.We have the following: , where min j = min(|a σ j |, |b σ j |, |c σ j |).

Symmetric h-Weyl algebra
In this section we introduce the h-analogue of the Weyl algebra in the hcalculus, and study its symmetric powers.A basic introduction to h-calculus may be found in [13].
Definition 9.The h-Weyl algebra is the algebra C x, y, z [h]/I h , where I h is the ideal generated by the following relations: given by ρ(x) = x, ρ(y) = ∂ h , ρ(z) = s h and ρ(h) = ĥ defines a representation of the h-Weyl algebra.
Notice that if we let h → 0, z becomes a central element and we recover the Weyl algebra.We order the letters on the h-Weyl algebra as follows x < y < z < h.Also, for a ∈ N and k . With this notation, we have the Proof. 2. is obvious and 3. is similar to 1.We prove 1. by induction.It is easy to check that z a x = xz a + az a h.Furthermore,
The condition for p and q in the definition above might seem unmotivated.They appear naturally in the course of the proof of Theorem 8 below, which is proved using induction and Theorem 7.
Theorem 8. Let A, p and q be as in the previous definition, we have Figure 2 illustrates the combinatorial interpretation of Theorem 8.As we try to moves the z's or the y's above the 'x', a subset of the x may get killed.The z's do not die in this process but the y's do turning themselves into z' s.Our next result provides an explicit formula for the product of m elements in the algebra Sym A σ j denotes the vector (A 1σ −1 1 (j) , . . ., A mσ −1 m (j) ) ∈ (N 3 ) m and set and similarly for |b σ j | and |c σ j |.

Theorem 9. For any
holds in Sym n (C x, y, z /I h ), where σ ∈ {id} × S m−1 n , p j , q j are that (A σ j , p j , q j ) satisfy the condition of the definition above.r(A σ j , p j , q Theorem 9 is proven similarly to Theorem 6. 5 Deformation quantization of (sl * 2 ) n /S n We denote by sl 2 the Lie algebra of all 2 × 2 complex matrices of trace zero.
sl * 2 is the dual vector space.It carries a natural structure of Poisson manifold.We consider a deformation quantization of the Poisson orbifold (sl * 2 ) n /S n .It is proven in [4] that the quantized algebra of the Poisson manifold sl * 2 is isomorphic to U (sl 2 ) the universal enveloping algebra of sl 2 , after setting the formal parameter appearing in [4] to be 1.Thus we regard (U (sl 2 ) ⊗n ) Sn ∼ = Sym n (U (sl 2 )) as the quantized algebra associated to the Poisson orbifold (sl * 2 ) n /S n .It is well-known that U (sl 2 ) can be identified with the algebra C x, y, z /I sl 2 where I sl 2 is the ideal generated by the following relations: The next result can be found in [2], [5].Given s, n ∈ N with 0 ≤ s ≤ n, the s-th elementary symmetric function  Proof.Formula 1. is proved by induction.It is equivalent to another formula for the normalization of z a x b given in [12].2. and 3. are similar to Theorem 7, part 1. Formula 2. express the fact that as we try to move the z's above the y's some of the y's may get killed.This argument justify the b k factor.The a k factor arises from the fact that each 'y' may be killed for any of the z's.The (−2) k factor follows from the fact that each killing of a 'y' is weighted by a −2.N sl 2 (A, k, s, p, q)X r(A,k,s,p,q) (6)

Theorem 1 .a σ − 1
The map s : Sym n (A) −→ (A ⊗n ) Sn given by s (i) , for all a i ∈ A defines an algebra isomorphism.

For
k ∈ N, we let χ k : N −→ N the map given by χ k (a) = ♯(I)=k I⊂[[1,a]] q χ(I) , for all a ∈ N. We have the following Theorem 4. Given a, b ∈ N, the following identities hold in C x, y, z [q]/I q 1. z a x b = min k=0 χ k (a)[b] k x b−k y k z a−k , where min = min(a, b).

Figure 2 :
Figure 2: Combinatorial interpretation of N h .

1≤i 1 <
•••<is≤n x i 1 . . .x is on variables x 1 , . . ., x n is denoted by e n s (x 1 , . . ., x n ).For b ∈ N, the notation e n s (b) := e n s (b, b − 1, . . ., b − n + 1) we will used.Given a, n ∈ N such that a ≤ n, we set (a) n = a(a − 1) . . .(a − n + 1).Theorem 10.Given a, b ∈ N, the following identities hold in C x, y, z /I sl 21. z a x b = s,k (a) k (b) k k! e k k−s (−a − b + 2k)x b−k y s z a−k ,where the sum runs over all k, s ∈ N such that 0 ≤ s ≤ k ≤ min(a, b).

2 .
z a y b = b k=0 b k (−2a) k y b−k z a .3. y a x b = ) k x b y a−k .
A = (A 1 , . . ., A n ) ∈ (N 3 ) n and A i = (a i , b i , c i ), for i ∈ [[1, n]].Set X A i = x a i y b i z c i for i ∈ [[1, n]], furthermore set a = (a 1 , . . ., a n ) ∈ N n , b = (b 1 , . . ., b n ) ∈ N n , c = (c 1 , . . ., c n ) ∈ Nn and |A| = (|a|, |b|, |c|) ∈ N 3 .Using this notation, we have the Definition 11.The normal coordinates N sl 2 (A, k, s, p, q) of n i=1 X A i in the algebra C x, y, z /I sl 2 are given via the identity n i=1 X A i = k,s,p,q