Symmetric quantum Weyl algebras

ANNALES MATHÉMATIQUES

BLAISE PASCAL

Rafael Díaz, Eddy Pariguan

Symmetric quantum Weyl algebras

Volume 11, n^o2 (2004), p. 187-203.

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Symmetric quantum Weyl algebras

Rafael Díaz

Eddy Pariguan

Abstract

We study the symmetric powers of four algebras: q-oscillator alge- bra, q-Weyl algebra, h-Weyl algebra andU(sl₂). We provide explicit formulae as well as combinatorial interpretation for the normal coor- dinates of products of arbitrary elements in the above algebras.

1 Introduction

This paper takes part in the time-honored tradition of studying an algebra by first choosing a “normal” or “standard” basis for itB, and second, writing down explicit formulae and, if possible, combinatorial interpretation for the representation of the product of a finite number of elements inB, as linear combination of elements inB.

This method has been successfully applied to many algebras, most promi- nently 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 lettersL, and definingBto be the set of normally ordered monomials, i.e., monomials in which the letters appearing in it respect the order ofL.

We consider algebras of the form Symⁿ(A), i.e, symmetric powers of certain algebras. Let us recall that for each n ∈ N, there is a functor Symⁿ:C-alg −→C-alg from the category of associativeC-algebras into it- self defined on objects as follows: ifAis aC-algebra, thenSymⁿ(A)denotes

1The first author is partially supported by UCV.

2The second author is partially supported by FONACIT.

the algebra whose underlying vector space is then-th symmetric power ofA:

Symⁿ(A) = (A^⊗n)/ha₁⊗. . .⊗a_n−a_σ−1(1)⊗. . .⊗a_σ−1(n) :a_i ∈A, σ ∈Sni, whereSn denotes the group of permutations onn letters. The product ofm elements inSymⁿ(A)is given by the rule

(n!)^m−1

m

Y

i=1 n

O

j=1

a_ij

!

= X

σ∈{id}×S^m−1n n

O

j=1 m

Y

i=1

a_iσ−1 i (j)

!

(1.1)

for all (a_ij) ∈A[[1,m]]×[[1,n]]. Notice that if A is an algebra, thenA^⊗n is also an algebra. Sn 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].

Theorem 1.1: The map s: Symⁿ(A)−→(A^⊗n)^Sn given by s

n

O

i=1

a_i

!

= 1 n!

X

σ∈Sn n

O

i=1

a_σ−1(i), for all a_i∈A defines an algebra isomorphism.

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

Definition 1.2: The algebra Chx, yi[h]/hyx−xy−hi is called the Weyl algebra.

This algebra admits a natural representation as indicated in the

Proposition 1.3: The map ρ : Chx, yi[h]/hyx−xy−hi −→ End(C[x, h]) given byρ(x)(f) =xf,ρ(y)(f) =h ∂f /∂x,ρ(h)(f) =hf for anyf ∈C[x, h]

defines a representation of the Weyl algebra.

The symmetric powers of the Weyl algebra have been studied from sev- eral 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 ˆx operators given in [3]. In [8] we gave explicit formula, as well as combinato- rial interpretation for the normal coordinates of monomials∂^a1x^b1. . . ∂^anx^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 1.3 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 calledq-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 theq-calculus, we will actually introduce twoq-analogues of the Weyl algebra: theq-oscillator algebra (section 2) and theq-Weyl algebra (section 3). Needless to say, theq-oscillator algebra also known as theq-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, aˆ third operator, theq-shifts_qis also present in theq-Weyl algebra. We believe thats_q is as fundamental as∂_q andx. The reason it has passed unnoticed inˆ the classical case is that forq= 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 xˆ operators, also includes an h-shift operator s_h, which again reduces to the identity forh= 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 h- deformations); 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 ofsl₂. We give explicit formulae for the product rule in the symmetric powers ofU(sl₂).

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

• Ndenotes the set of natural numbers. Forx∈Nⁿandi∈N, we denote byx_<ithe vector(x₁, . . . , x_i−1)∈Nⁱ⁻¹, byx_≤ithe vector(x₁, . . . , x_i)∈

Nⁱ, by x_>i the vector (x_i+1, . . . , x_n) ∈ Nⁿ⁻ⁱ and by x≥i the vector (x_i, . . . , x_n)∈Nⁿ⁻ⁱ⁺¹.

• Given (U, <)an ordered set ands∈U, we setU_>s :={u∈U:u > s}.

• | |:Nⁿ−→Ndenotes the map such that|x|:=Pn

i=1x_i, for allx∈Nⁿ.

• For a set X, ](X) := cardinality of X, and ChXi:= free associative algebra generated by X.

• Given natural numbers a₁, . . . , a_n ∈ N, min(a₁, . . . , a_n) denotes the smallest number in the set {a₁, . . . , a_n}.

• Given n∈N, we set [[1, n]] ={1, . . . , n}.

• Let Sbe a set and A: [[1, m]]×[[1, n]]−→S anS-valued matrix. For σ ∈ (Sn)^m and j ∈ [[1, n]], A^σ_j : [[1, m]] −→ S denotes the map such that A^σ_j(i) =A_iσ⁻¹

i (j), for all i= 1, . . . , m.

• The q-analogue n integer is [n] := 1−qⁿ

1−q . For k ∈ N, we will use [n]_k:= [n][n−1]. . .[n−k+ 1].

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 toq-calculus).

Definition 2.1: The operators ∂_q, s_q,x,ˆ q,ˆˆh : C[x, q, h] −→ C[x, q, h] are given as follows

∂_qf(x) = ^f(qx)−f_(q−1)x^(x) s_q(f)(x) =f(qx) q(fˆ ) =qf ˆ

x(f) =xf ˆh(f) =hf

for allf ∈C[x, q, h]. We call∂_q theq-derivative ands_q theq-shift.

Definition 2.2: The algebra Chx, yi[q, h]/I_qo, whereI_qo is the ideal gener- ated by the relationyx=qxy+his called theq-oscillator algebra.

We have the following analogue of Proposition 1.3.

Proposition 2.3: The mapρ:Chx, yi[q, h]/I_qo−→End(C[x, q, h])given by ρ(x) = ˆx, ρ(y) =h∂_q,ρ(q) = ˆq and ρ(h) = ˆhdefines a representation of the q-oscillator algebra.

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 A= (A₁, . . . , A_n) ∈(N²)ⁿ, and A_i = (a_i, b_i) ∈N², fori∈[[1, n]]. SetX^Ai=x^aiy^bi, fori∈[[1, n]]. We seta= (a₁, . . . , a_n)∈Nⁿ, b= (b₁, . . . , b_n) ∈Nⁿ, c= (c₁, . . . , c_n)∈Nⁿ and |A|= (|a|,|b|)∈N². Using this notation we have

Definition 2.4: Thenormal coordinatesN_qo(A, k) of

n

Y

i=1

X^Ai∈Chx, yi[h, q]/I_qo

are given by the identity

n

Y

i=1

X^Ai =

min

X

k=0

N_qo(A, k)X^|A|−(k,k)h^k (2.1) wheremin = min(|a|,|b|). Fork >min, we setN_qo(A, k) to be equal to0.

Let us introduce some notation needed to formulate Theorem 2.6 be- low which provides explicit formula for the normal coordinates N_qo(A, k) of

n

Y

i=1

X^Ai. Given (A₁, . . . , A_n) ∈ (N²)ⁿ choose disjoint totally ordered sets (U_i, <_i), (V_i, <_i) such that ](U_i) = a_i and ](V_i) = b_i, for i ∈ [[1, n]]. De- fine a total order set(U∪V ∪ {∞}, <), whereU =∪ⁿ_i=1U_i, V =∪ⁿ_i=1V_i and

∞ 6∈U∪V, as follows: Givenu, v∈U∪V ∪ {∞}we say thatu≤vif and only if a least one of the following conditions hold

u∈V_i, v∈V_j and i≤j; u∈V_i, v∈U_j and i≤j;

u∈U_i, v∈V_j and i≤j; u, v∈U_i and u≤_iv;

u, v∈V_i and u≤_iv; v=∞.

Givenk ∈N, we letP_k(U, V)be the set of all maps p:V −→U∪ {∞}such that

• p restricted top⁻¹(U)is injective, and](p⁻¹(U)) =k.

• If(v, p(v))∈V_i×U_j theni < j.

Figure1 shows an example of such a map. We only show the finite part of p, all other points inV being mapped to∞.

R. D´ıaz and E. Pariguan

Givenk∈N, we letPk(U, V) be the set of all mapsp:V −→U∪ {∞}such that

• prestricted top⁻¹(U) is injective, and(p⁻¹(U)) =k.

• If (v, p(v))∈Vi×Uj theni < j.

Figure 1 shows an example of such a map. We only show the finite part of p, all other points inV being mapped to∞.

U1 V1 U2 V2 U3 V3 ∞

Figure 1: Combinatorial interpretation ofNqo.

Definition 2.5: The value of the crossing number mapc :Pk(U, V) −→ N when evaluated onp∈Pk(U, V) is given by

c(p) =({(s, t)∈V ×U|s < t < p(s), t∈p(U>s)^c}).

Theorem 2.6: For anyA= (A1, . . . , An)∈(N²)ⁿwithAi = (ai, bi)∈N² for i∈[[1, n]] and anyk∈N, we have that

Nqo(A, k) =

p∈Pk(U,V)

q^c(p).

Proof: The proof is by induction. The only non-trivial case is the following y

n i=1

X^Ai = min k=0

q^|a|−kNqo(A, k)X|A|−(k,k+1)h^k

+ min k=0

Nqo(A, k) a−k

i=1

qⁱ⁻¹X|A|−(k+1,k)h^k (2.2) where min = min(|a|,|b|). Normalizing the left-hand side of (2.2) we get a recursive relation

Nqo(((0,1)A), k) =Nqo(A, k) + a−k

i=1

qⁱ⁻¹Nqo(A, k−1).

192

Figure 1: Combinatorial interpretation of N_qo.

Definition 2.5: The value of the crossing number map c:P_k(U, V) −→N when evaluated onp∈P_k(U, V) is given by

c(p) =]({(s, t)∈V ×U|s < t < p(s), t∈p(U_>s)^c}).

Theorem 2.6: For any A = (A₁, . . . , A_n) ∈(N²)ⁿ with A_i = (a_i, b_i) ∈N² fori∈[[1, n]] and any k∈N, we have that

N_qo(A, k) = X

p∈Pk(U,V)

q^c(p).

Proof: The proof is by induction. The only non-trivial case is the following y

n

Y

i=1

X^Ai =

min

X

k=0

q^|a|−kN_qo(A, k)X|A|−(k,k+1)

h^k

+

min

X

k=0

N_qo(A, k)

a−k

X

i=1

qⁱ⁻¹X|A|−(k+1,k)h^k (2.2) where min = min(|a|,|b|). Normalizing the left-hand side of (2.2) we get a recursive relation

N_qo(((0,1)A), k) =N_qo(A, k) +

a−k

X

i=1

qⁱ⁻¹N_qo(A, k−1).

The other recursion needed beingN_qo(((1,0)A), k) =N_qo(A, k). It is straight- forward to check thatN_qo(A, k) = X

p∈P_k(U,V)

q^c(p) satisfies both recursions.

Consider the identity(2.1)in the representation of theq-oscillator algebra defined in Proposition 2.3. Apply both sides of the identity(2.1) to x^t for t∈N, and use Theorem2.6to get the fundamental

Corollary 2.7: Given (t, a, b)∈N×Nⁿ×Nⁿ the following identity holds

n

Y

i=1

[t+|a_>i| − |b_>i|]_bi = X

p∈Pk(U,V)

q^c(p)[t]|b|−k.

Our next theorem gives a fairly simple formula for the product of m elements inSymⁿ(Chx, yi[q, h]/I_qo). Fix a matrixA: [[1, m]]×[[1, n]]−→N², (A_ij) = ((a_ij),(b_ij)). Recall that givenσ∈(Sn)^m andj∈[[1, n]],A^σ_j denotes the vector(A_1σ−1

1 (j), . . . , A_mσ−1

m(j))∈(N²)^m and X_j^Aij =x^a_j^ijy^b_j^ij. Set

|A^σ_j|= (|a^σ_j|,|b^σ_j|)where |a^σ_j|=

m

X

i=1

a_iσ−1

i (j) and |b^σ_j|=

m

X

i=1

b_iσ−1 i (j). Theorem 2.8: For any A: [[1, m]]×[[1, n]]−→N², the identity

(n!)^m−1

m

Y

i=1 n

Y

j=1

X_j^Aij =X

σ,k n

Y

j=1

N_qo(A^σ_j, k_j)

! _n Y

j=1

X^|A

σ j|−(kj,kj)

j h^|k|

whereσ∈ {id} ×S^m−1n andk ∈Nⁿ, holds inSymⁿ(Chx, yi[q, h]/I_qo).

Proof: Using the product rule given in (1.1), the identity (2.1) and the distributive property we obtain

(n!)^m−1

m

Y

i=1 n

Y

j=1

X_j^Aij = X

σ∈{id}×S^m−1n n

Y

j=1 m

Y

i=1

X

A_iσ−1 i (j) j

= X

σ∈{id}×S^m−1n n

Y

j=1 minj

X

k=0

N_qo(A^σ_j, k)x^|a

σ j|−k j y^|b

σ j|−k

j h^k

!

= X

σ,k n

Y

j=1

N_qo(A^σ_j, k_j)

! _n Y

j=1

X^|A

σ j|−(kj,kj)

j h^|k|

where min_j= min(|a^σ_j|,|b^σ_j|).

3 Symmetric q-Weyl algebra

In this section we study the symmetric powers of theq-Weyl algebra.

Definition 3.1: Theq-Weyl algebra is given byChx, y, zi[q]/I_q, where I_q is the ideal generated by the following relations:

zx=xz+y yx=qxy zy=qyz We have the followingq-analogue of Proposition1.3.

Proposition 3.2: The map ρ : Chx, y, zi[q]/I_q −→ End(C[x, q]) given by ρ(x) = ˆx, ρ(y) =s_q, ρ(z) =∂_q and ρ(q) = ˆq defines a representation of the q-Weyl algebra.

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 thecrossing number of I to be

χ(I) :=]({(i, j) :i > j, i∈I, j ∈I^c}).

For k∈N, we letχ_k :N−→Nthe map given byχ_k(a) = X

](I)=k I⊂[[1,a]]

q^χ(I), for

alla∈N. We have the following

Theorem 3.3: Givena, b∈N, the following identities hold inChx, y, zi[q]/I_q 1. z^ax^b=

min

X

k=0

χ_k(a)[b]_kx^b−ky^kz^a−k, where min = min(a, b).

2. z^ay^b=q^aby^bz^a. 3. y^ax^b =q^abx^by^a.

Proof: 2. and 3. are obvious, let us to prove1. It should be clear that z^ax^b= X

I⊂[[1,a]]

[b]_](I)x^b−](I)

a

Y

j=1

f_I(j),

wheref_I(j) =z, ifj /∈Iandf_I(j) =y, ifj∈I. The normal form of

a

Y

j=1

f_I(j)

isq^χ(I)y^](I)z^a−](I). Thus, z^ax^b=

min

X

k=0

X

I

q^χ(I)

!

[b]_kx^b−ky^kz^a−k=

min

X

k=0

χ_k(a)[b]_kx^b−ky^kz^a−k,

wheremin = min(a, b),I ⊂[[1, a]]and ](I) =k.

Assume we are givenA= (A₁, . . . , A_n)∈(N³)ⁿ, whereA_i= (a_i, b_i, c_i), for i∈[[1, n]], alsoX^Ai=x^aiy^biz^ci fori∈[[1, n]]. We seta= (a₁, . . . , a_n)∈Nⁿ, b = (b₁, . . . , b_n) ∈ Nⁿ, c = (c₁, . . . , c_n) ∈ Nⁿ and |A| = (|a|,|b|,|c|) ∈ N³. Using this notation, we have the

Definition 3.4: Thenormal coordinatesN_q(A, k)of

n

Y

i=1

X^Ai ∈Chx, y, zi[q]/I_q

are given via the identity

n

Y

i=1

X^Ai= X

k∈Nⁿ⁻¹

N_q(A, k)X|A|+(−|k|,|k|,−|k|)

(3.1) where k runs over all vectors k = (k₁, . . . , kn−1)∈Nⁿ⁻¹ such that 0≤k_i ≤ min(|c_≤i| − |k_<i|, a_i+1). We setN_q(A, k) = 0 fork ∈Nⁿ⁻¹not satisfying the previous conditions.

Our next theorem follows from Theorem 3.3 by induction. It gives an explicit formula for the normal coordinates N_q(A, k) in the q-Weyl algebra of

n

Y

i=1

X^Ai.

Theorem 3.5: Let A, k be as in the previous definition, we have N_q(A, k) =q^{Pn−1i=1 λ(i)}

n−1

Y

j=1

χ_kj(|c_≤j| − |k_<j|)[a_j+1]_kj,

where λ(i) =b_i+1(|c≤i| − |k≤i|) + (a_i+1−k_i)(|b≤i|+|k_<i|).

Applying both sides of the identity(3.1) in the representation of the q- Weyl algebra given in Proposition3.2tox^tand using Theorem3.5, we obtain the remarkable

Corollary 3.6: For any given (t, a, b, c) ∈ N×Nⁿ×Nⁿ×Nⁿ, the following identity holds

n

Y

i=1

q^γ(i)[t+|a_>i|+|c_>i|]_ci=X

k

q^β(k)

n−1

Y

j=1

χ_kj(|c_≤j| − |k_<j|)[a_j+1]_kj

! [t]_|c|−|k|

where k∈Nⁿ⁻¹ such that0≤k_i≤min(|c≤i| − |k_<i|, a_i+1), γ(i) =b_i(t+|a_>i|−|c>i−1|), andβ(k) =

n−1

X

i=1

λ(i)

!

+ (|b|+|k|)(t− |c|+|k|).

Our next theorem provides an explicit formula for the product of mele- ments in the algebraSymⁿ(Chx, y, zi[q]/I_q). FixA: [[1, m]]×[[1, n]]−→N³, with (A_ij) = ((a_ij),(b_ij),(c_ij)). Recall that given σ ∈ S^mn and j ∈ [[1, n]], A^σ_j denotes the vector (A_1σ−1

1 (j), . . . , A_mσ−1

m(j)) ∈ (N³)^m and set X_j^Aij = x^a_j^ijy_j^bijz^c_j^ij, forj∈[[1, n]]. SetA^σ_j = (|a^σ_j|,|b^σ_j|,|c^σ_j|), where|a^σ_j|=

m

X

i=1

a_iσ−1 i (j)

and similarly for|b^σ_j| and |c^σ_j|. We have the following:

Theorem 3.7: For any A: [[1, m]]×[[1, n]]−→N³, the identity (n!)^m−1

m

Y

i=1 n

Y

j=1

X_j^Aij =X

σ,k n

Y

j=1

N_q(A^σ_j, k^j)

! _n Y

j=1

X^|A

σ

j|+(−|k^j|,|k^j|,−|k^j|) j

where σ∈ {id} ×S^m−1n and k= (k¹, . . . , kⁿ)∈(N^m−1)ⁿ, holds in Symⁿ(Chx, y, zi[q]/I_q).

Proof:

(n!)^m−1

m

Y

i=1 n

Y

j=1

X_j^Aij = X

σ∈{id}×S^m−1n n

Y

j=1 m

Y

i=1

X

Aiσ−1 i (j) j

= X

σ n

Y

j=1 minj

X

k=0

N_q(A^σ_j, k)x^|a

σ j|−|k|

j y^|b

σ j|+|k|

j z^|c

σ j|−|k|

j

!

= X

σ,k n

Y

j=1

N_q(A^σ_j, k^j)

! _n Y

j=1

X^|A

σ

j|+(−|k^j|,|k^j|,−|k^j|)

j ,

wheremin_j= min(|a^σ_j|,|b^σ_j|,|c^σ_j|).

4 Symmetric h-Weyl algebra

In this section we introduce the h-analogue of the Weyl algebra in the h- calculus, and study its symmetric powers. A basic introduction toh-calculus may be found in[13].

Definition 4.1: The operators∂_h, s_h,x,ˆ ˆh:C[x, h]−→C[x, h]are given by

∂_hf(x) = ^f(x+h)−f_h ^(x) s_h(f)(x) =f(x+h) x(f) =ˆ xf ˆh(f) =hf forf ∈C[x, h]. We call∂_htheh-derivative and s_h theh-shift.

Definition 4.2: Theh-Weyl algebrais the algebraChx, y, zi[h]/I_h, whereI_h is the ideal generated by the following relations:

yx=xy+z zx=xz+zh yz=zy

Proposition 4.3: The map ρ : Chx, y, zi[h]/I_h −→ End(C[x, h]) given by ρ(x) = ˆx, ρ(y) =∂_h,ρ(z) =s_h and ρ(h) = ˆhdefines a representation of the h-Weyl algebra.

Notice that if we leth→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 = (k₁, . . . , k_n) ∈ Nⁿ, we set a

k

:= a!

Qk_i!(a− |k|)!. With this notation, we have the

Theorem 4.4: Givena, b∈N, the following identities hold inChx, y, zi[h]/I_h

1. z^ax^b=

b

X

k=0

b k

a^kx^b−kz^ah^k. 2. z^ay^b=y^bz^a.

3. y^ax^b = X

k∈N^a

b k

x^b−|k|y^a−s(k)z^s(k)h^|k|−s(k), where k∈N^a is such that 0≤ |k| ≤band s(k) =]({i:k_i6= 0}).

Proof: 2. is obvious and3.is similar to 1.We prove 1. by induction. It is easy to check thatz^ax=xz^a+az^ah. Furthermore,

z^ax^b+1 =

b

X

k=0

b k

a^kx^b−k(z^ax)h^k

=

b

X

k=0

b k

a^kx^b+1−kz^ah^k+

b+1

X

k=1

b k−1

a^kx^b+1−kz^ah^k

=

b+1

X

k=0

b+ 1 k

a^kx^b+1−kz^ah^k.

Assume we are given A= (A₁, . . . , A_n) ∈(N³)ⁿ and A_i = (a_i, b_i, c_i), for i∈[[1, n]]. SetX^Ai =x^aiy^biz^ci fori ∈[[1, n]]. We set a= (a₁, . . . , a_n)∈Nⁿ, b = (b₁, . . . , b_n) ∈ Nⁿ, c = (c₁, . . . , c_n) ∈ Nⁿ and |A| = (|a|,|b|,|c|) ∈ N³. Using this notation, we have the

Definition 4.5: Thenormal coordinatesN_h(A, p, q)of

n

Y

i=1

X^Ai ∈Chx, y, zi[h]/I_h

are given via the identity

n

Y

i=1

X^Ai=X

p,q

N_h(A, p, q)X^r(A,p,q)h|q|+|p|−s(p)

(4.1) where the sum runs over all vectorsq= (q₁, . . . , q_n−1)∈Nⁿ⁻¹ such that 0≤q_j ≤a_j+1 and p= (p₁, . . . , p_n−1)withp_j ∈N^{|b≤j|−|s(p<j)|}.

Also,r(A, p, q) = (|a|−|p|−|q|,|b|−s(p),|c|+s(p)), wheres(p) =Pn−1 j=1s(p_j).

We setN_h(A, p, q) = 0 forp, q not satisfying the previous conditions.

The condition forpandqin the definition above might seem unmotivated.

They appear naturally in the course of the proof of Theorem 4.6 below, which is proved using induction and Theorem 4.4.

Theorem 4.6: LetA, p and q be as in the previous definition, we have N_h(A, p, q) =

n−1

Y

i=1

a_i+1 q_i

a_i+1−q_i p_i

(|c_≤i|+|s(p_<i)|)^qi.

Figure 2illustrates the combinatorial interpretation of Theorem 4.6. 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 intoz’ s.

y z x y z x

Figure 2: Combinatorial interpretation of N_h.

Our next result provides an explicit formula for the product ofmelements in the algebraSymⁿ(Chx, y, zi[h]/I_h). Fix A: [[1, m]]×[[1, n]]−→N³, with (A_ij) = ((a_ij),(b_ij),(c_ij)). Recall that given σ ∈S^mn and j ∈[[1, n]], A^σ_j de- notes the vector(A_1σ−1

1 (j), . . . , A_mσ−1

m(j))∈(N³)^m and set X_j^Aij =x^a_j^ijy_j^bijz^c_j^ij, forj ∈ [[1, n]]. Set A^σ_j = (|a^σ_j|,|b^σ_j|,|c^σ_j|), where |a^σ_j|=

m

X

i=1

a_iσ−1

i (j) and simi- larly for|b^σ_j| and |c^σ_j|.

Theorem 4.7: For any A: [[1, m]]×[[1, n]]−→N³, the identity (n!)^m−1

m

Y

i=1 n

Y

j=1

X_j^Aij =X

σ,p,q n

Y

j=1

N_h(A^σ_j, p^j, q^j)

! _n Y

j=1

X^r(A

σ j,p^j,q^j) j h^kj(p,q)

holds in Symⁿ(Chx, y, zi/I_h), where σ ∈ {id} ×S^m−1n , p^j, q^j are such that (A^σ_j, p^j, q^j)satisfy the condition of the definition above. r(A^σ_j, p^j, q^j) = (|a^σ_j|−

|p^j| − |q^j|,|b^σ_j| −s(p^j),|c^σ_j|+s(p^j)) and k_j(p, q) =|q^j| − |p^j| −s(p^j).

Theorem 4.7 is proven similarly to Theorem3.7.

5 Deformation quantization of (sl

)

/

We denote bysl₂the Lie algebra of all2×2complex matrices of trace zero.

sl^∗₂is the dual vector space. It carries a natural structure of Poisson manifold.

We consider a deformation quantization of the Poisson orbifold(sl^∗₂)ⁿ/Sn. It is proven in [4] that the quantized algebra of the Poisson manifold sl^∗₂ is isomorphic toU(sl₂)the universal enveloping algebra ofsl₂, after setting the formal parameter~appearing in [4] to be1. Thus we regard(U(sl₂)^⊗n)^Sn ∼= Symⁿ(U(sl₂)) as the quantized algebra associated to the Poisson orbifold (sl^∗₂)ⁿ/Sn. It is well-known that U(sl₂) can be identified with the algebra Chx, y, zi/I_sl2 where I_sl2 is the ideal generated by the following relations:

zx=xz+y yx=xy−2x zy=yz−2z The next result can be found in[2],[5].

Proposition 5.1: The map ρ : Chx, y, zi/I_sl2 −→ End(C[x₁, x₂]) given by ρ(x) =x₂ ∂

∂x₁,ρ(y) =x₁ ∂

∂x₁−x₂ ∂

∂x₂ and ρ(z) =x₁ ∂

∂x₂ defines a represen- tation of the algebraChx, y, zi/I_sl2.

Given s, n∈Nwith0≤s≤n, thes-th elementary symmetric function X

1≤i1<...<is≤n

x_i1. . . x_ison variablesx₁, . . . , x_nis denoted byeⁿ_s(x₁, . . . , x_n). For b∈N, the notation eⁿ_s(b) := eⁿ_s(b, b−1, . . . , b−n+ 1) we will used. Given a, n∈Nsuch thata≤n, we set(a)_n=a(a−1). . .(a−n+ 1).

Theorem 5.2: Given a, b∈N, the following identities hold inChx, y, zi/I_sl2 1. z^ax^b=X

s,k

(a)_k(b)_k

k! e^k_k−s(−a−b+ 2k)x^b−ky^sz^a−k,

where the sum runs over all k, s∈N such that0≤s≤k ≤min(a, b).

2. z^ay^b=

b

X

k=0

b k

(−2a)^ky^b−kz^a.

3. y^ax^b=

a

X

k=0

a k

(−2b)^kx^by^a−k.

Proof: Formula 1. is proved by induction. It is equivalent to another formula for the normalization of z^ax^b given in [12]. 2. and 3. are similar to Theorem 4.4, part1. Formula2.express the fact that as we try to move the z’s above they’s some of they’s may get killed. This argument justify the

b k

factor. Thea^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.

Assume we are given A = (A₁, . . . , A_n) ∈ (N³)ⁿ and A_i = (a_i, b_i, c_i), for i ∈ [[1, n]]. Set X^Ai = x^aiy^biz^ci for i ∈ [[1, n]], furthermore set a = (a₁, . . . , a_n) ∈ Nⁿ, b = (b₁, . . . , b_n) ∈ Nⁿ, c = (c₁, . . . , c_n) ∈ Nⁿ and |A| = (|a|,|b|,|c|)∈N³. Using this notation, we have the

Definition 5.3: The normal coordinates N_sl2(A, k, s, p, q) of

n

Y

i=1

X^Ai in the algebraChx, y, zi/I_sl2 are given via the identity

n

Y

i=1

X^Ai = X

k,s,p,q

N_sl2(A, k, s, p, q)Xr(A,k,s,p,q) (5.1) wherek, s, p, q ∈Nⁿ⁻¹are such that0≤s_i≤k_i≤min(|c≤i| − |k_<i|, a_i+1), 0 ≤ p_i ≤ b_i+1, 0 ≤ q_i ≤ |b≤i|+ |s_<i| − |p_<i| − |q_<i|, for i ∈ [[1, n−1]].

Moreover, r(A, k, s, p, q) = (|a| − |k|,|b|+|s| − |p| − |q|,|c| − |k|). We set N_sl2(A, k, s, p, q) = 0fork, s, p, q not satisfying the previous conditions.

Theorem 5.4 provides an explicit formula for the normal coordinates N_sl2(A, k, s, t)of

n

Y

i=1

X^Ai. Its proof goes by induction using Theorem 5.2.

Theorem 5.4: With the notation of the definition above, we have N_sl2(A, k, s, p, q) = (−2)^|p|+|q|

n−1

Y

i=1

α_iβ_iγ_i b_i+1

p_i

(|c_≤i| − |k_≤i|)^pi(a_i+1−k_i)^qi,

whereα_i= (|c_≤i| − |k_<i|)_ki(a_i+1)_ki

k_i! ,β_i=e^k_kⁱ

i−si(−a_i+1− |c_≤i|+|k_<i|+ 2k_i), andγ_i=

|b_≤i|+|s_<i| − |p_<i| − |q_<i| q_i

, for alli∈[[1, n−1]].

Our final result provides an explicit formula for the product ofmelements in the algebra Symⁿ(Chx, y, zi/I_sl2). Fix A : [[1, m]]×[[1, n]] −→ N³, with (A_ij) = ((a_ij),(b_ij),(c_ij)). Recall that given σ ∈S^mn and j ∈[[1, n]], A^σ_j de- notes the vector(A_1σ⁻¹

1 (j), . . . , A_mσm⁻¹_(j))∈(N³)^m. SetX_j^Aij =x^a_j^ijy_j^bijz_j^cij, for j∈[[1, n]]and |A^σ_j|= (|a^σ_j|,|b^σ_j|,|c^σ_j|), where|a^σ_j|=

m

X

i=1

a_iσ−1

i (j) and similarly for|b^σ_j|, |c^σ_j|, and k, s, p, q∈(N^m−1)ⁿ.

Theorem 5.5: For any A: [[1, m]]×[[1, n]]−→N³, the identity (n!)^m−1

m

Y

i=1 n

Y

j=1

X_j^Aij = X

σ,k,s,p,q n

Y

j=1

N_sl2(A^σ_j, k^j, s^j, p^j, q^j)

! _n Y

j=1

X^rj(A

σ

j,k^j,s^j,p^j,q^j) j

holds inSymⁿ(Chx, y, zi/I_sl2), wherek = (k¹, . . . , kⁿ)∈(N^m−1)ⁿ, and similar fors, p, q,r_j(A^σ_j, k^j, s^j, p^j, q^j) = (|a^σ_j| − |k^j|,|b^σ_j|+|s^j| − |p^j| − |q^j|,|c^σ_j| − |k^j|, and σ∈ {id} ×S^m−1n .

Theorem 5.5 is proven similarly to Theorem 3.7.

Acknowledgment

We thank Nicolas Andruskiewitsch, Sylvie Paycha and Carolina Teruel. We also thank the organizing committee of Geometric and Topological Methods for Quantum Field Theory, Summer School 2003, Villa de Leyva, Colombia.

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Rafael Díaz Eddy Pariguan

Instituto Venezolano de Inves- tigaciones Científicas

Universidad Central de Venezue- la

Departamento de Matemáticas Departamento de Matemáticas

Altos de Pipe. Los Chaguaramos

Caracas 21827 Caracas 1020

Venezuela Venezuela

radiaz@ivic.ve eddyp@euler.ciens.ucv.ve