Symmetry, Integrability and Geometry: Methods and Applications From slq(2) to a Parabosonic Hopf Algebra

A Hopf algebra with four generators among which an involution (reflection) operator, is introduced. The defining relations involve commutators and anticommutators. The discrete series representations are developed. Designated by sl 1(2), this algebra en- compasses the Lie superalgebra osp(1j2). It is obtained as a q = 1 limit of the slq(2) algebra and seen to be equivalent to the parabosonic oscillator algebra in irreducible repre- sentations. It possesses a noncocommutative coproduct. The Clebsch{Gordan coefficients (CGC) of sl 1(2) are obtained and expressed in terms of the dual 1 Hahn polynomials. A generating function for the CGC is derived using a Bargmann realization.


Introduction
On the one hand, algebraic structures are natural descriptors of symmetries. On the other, the exact solutions of the dynamical equations of physical systems, when they exist, are typically presented in terms of special functions and orthogonal polynomials. Not surprisingly hence, the relations between solvable models, special functions, symmetries and their algebraic translations is of considerable interest.
The presence of reflection operators has been seen to arise in many contexts, physical and mathematical, related in particular, to the first two of the above areas. To give some examples, recall that in integrable many-body problems of the Calogero type, operators with reflections play a key role in expressing the constants of motion that are in involution [10,2]. There is currently much activity also in the study of Dunkl harmonic analysis [17].
Recently, we have examined univariate polynomials that are eigenfunctions of operators of Dunkl type, that is of operators that are first order in the derivative and involve reflections. We have thus discovered certain families of "classical" orthogonal polynomials that had hitherto escaped notice [23,24].
It has been found that these polynomials can be identified as a q → −1 limits of some q-orthogonal polynomials, the simplest among them being the little −1 Jacobi polynomials introduced in [22].
In [18] and [19] this approach was generalized to Dunkl shift operators. This provided a theoretical framework for the Bannai-Ito and the dual −1 Hahn polynomials.
With this perspective, it is thus natural to examine algebraic structures involving reflection operators and it is the purpose of this paper to contribute to such a study. For related investigations see, e.g. [1,8,6,7].
2 Def inition of the sl −1 (2) algebra and its relation with the osp(1|2) Lie superalgebra We define sl −1 (2) as the algebra which is generated by the four elements J 0 , J ± and R subject to the relations The operator R is an involution operator, i.e. it satisfies the property The Casimir operator Q, which by definition commutes with all the generators (R, J 0 , J ± ), is where µ is a constant and ρ n are the positive matrix elements of the representation. Moreover, demand that ρ 0 = 0 in order to obtain the standard discrete series bounded from below and with n = 0, 1, 2, . . . . The operator R commutes with J 0 and hence can be diagonalized in the basis e n . A simple analysis based on the properties of R, leads to the conclusion that Re n = ǫ(−1) n e n , n = 0, 1, 2, . . . , (2.3) where ǫ = ±1 is a fixed parameter in a given representation. Expressing the commutation relations in the basis e n gives the following equation for ρ n ρ 2 n + ρ 2 n+1 = 2(n + µ + 1/2) with general solution where κ is an arbitrary constant. The condition ρ 0 = 0 means that κ = −µ and we thus have The Casimir operator (2.2), as should be, is a multiple of the identity operator Qe n = −ǫµe n on the module with the basis {e n }.
The matrix elements can be presented in the form in terms of the "mu-numbers" We define also the "mu-factorials" by If we assume that µ > −1/2 then ρ 2 n > 0 for n = 1, 2, 3, . . . , and we thus obtain a unitary infinite-dimensional representation of the algebra sl −1 (2). The value of the Casimir operator is Q = −ǫµ in this representation.
Thus, the discrete series representation is fixed by two parameters ǫ = ±1 and µ > −1/2. Let us now indicate the connection that sl −1 (2) has with the simplest Lie superalgebra osp(1|2). Consider the elements K ± = J 2 ± . It is easy to verify that J 0 , K + and K − satisfy together the commutation relations of the sl(2) algebra Hence, J 0 , J ± , K ± form a basis for the Lie superalgebra osp(1|2) [4]. The operators J 0 , K ± belong to the even part of this algebra, while the operators J ± belong to the odd part. The Casimir operator (2.2) of the sl −1 (2) algebra contains the involution operator R which commutes with the operators J 0 and J + J − . Hence the square Q 2 of the Casimir operator will commute with all the generators of the sl −1 (2) algebra. However its expression will contain only the operators J 0 , J ± and not R: This operator coincides with the Casimir operator of the Lie superalgebra osp(1|2) [4]. We see that the Casimir operator Q of the algebra sl −1 (2) can be considered as a "square root" of the Casimir operator for the algebra osp(1|2).
In the next section we show that the algebra sl −1 (2) can be obtained as a q → −1 limit of the algebra sl q (2). This justifies the name of the algebra.
3 The sl −1 (2) algebra as a limit of the sl q (2) algebra Consider the algebra generated by three operators J 0 , J ± , with commutation relations [3] [J 0 , J ± ] = ±J ± , where q is a real parameter. The Casimir operator Q, commuting with J 0 and J ± is In what follows we restrict ourselves to discrete series representations of the algebra (3.1). This means representations that have bases e n , n = 0, 1, . . . such that J 0 e n = (n + ν)e n , J − e n = r n e n−1 , J + e n+1 = r n+1 e n+1 .
As usual, the condition r 0 = 0 is assumed. It is easily verified that The parameter ν is related to the value of the Casimir operator in these representations. The Fock-Bargmann realization of the algebra (3.1) can be defined on the space of polynomials in the variable z by the formulas: In this realization the basis vectors e n (z) are the monomials e n (z) = γ n z n , where γ n = 1 √ r 1 r 2 · · · r n is the normalization coefficient. When 0 < q < 1, the algebra defined by (3.1) is equivalent to the quantum sl q (2) algebra defined by the relations Indeed, under the identifications the commutation relations (3.1) are transformed into the commutation relations (3.3). When q → 1 the algebra sl q (2) with the defining relations (3.1) becomes the sl(2) algebra: There is also a nontrivial limit when q → −1. It is obvious that the commutation relations (3.1) become the commutation relations (2.1) when q = −1. The limit process for the matrix coefficients r n is more subtle however.
Assume first that n = 0, 2, 4, . . . is even. Then Hence lim q→−1 When n is odd, we have Thus, for integer values of the parameter ν the limit q → −1 of the matrix elements r n gives the expected matrix elements ρ n of the discrete series of the sl −1 (2) algebra.
When ν is not an integer, the limit of r n is not well defined. In this case we can assume that the limiting matrix element ρ 2 n is obtained by a linear interpolation from the integer ν case. If ν = j is integer, the involution operator R can also be obtained in the limit q → −1 Indeed, we have q J 0 e n = q n+j e n .

So, in the limit
This uniquely characterizes the involution operator with the property R 2 = I.
We thus see that the generators J 0 , J ±1 and R of sl −1 (2) can be obtained from the algebra (3.1) when the representation parameter is a positive integer ν = µ = 1, 2, 3, . . . . If ν is a real positive parameter, then the limiting process is not well defined and we postulate that in the limit q → −1 the matrix elements ρ n correspond to the matrix elements r n with ν real and positive.
Note that the q → −1 limit considered here is different from the well known special case of sl q (2) for q a root of unity [15]. In the latter case the operators J ± are nilpotent J N ± = 0, where N is the order of the root of unity and hence all irreducible representations are restricted to be of dimension N . In our case we have infinite-dimensional representations.

Relation with the parabosonic oscillator and the Fock-Bargmann realization
Consider the commutator [J − , J + ]. We have Remembering the expression (2.2) for the Casimir operator, we find that For representations with a fixed value ǫ = ±1, we have Q = −ǫµ and hence This relation, (4.1), defines the parabosonic oscillator algebra [20,12,11,16] with operators J − , J + , R satisfying the commutation relations (4.1) and {R, J ± } = 0 together with the condition R 2 = I. Conversely, assume that the operators J − , J + , R form a representation of the parabosonic oscillator algebra. We can define the operator J 0 as J 0 = {J + , J − }/2. Then it is easily verified that the operators J 0 , J + , J − , R satisfy the relations (2.1) defining the sl −1 (2) algebra.
Thus, if one restricts to irreducible representations with a fixed value of the Casimir operator Q = −ǫµ, the algebra sl −1 (2) is equivalent to the parabosonic oscillator algebra.
For definiteness, in what follows we will use representations for which ǫ = 1.
We can construct the Fock-Bargmann representation of the sl −1 (2) algebra in terms of first order differential-difference operators. Indeed, one can use the well known realization of the parabosonic operators [12,16] where R x is the reflection (parity) operator defined by Rf (x) = f (−x) for every function f (x). The operator J − coincides in this realization with the standard Dunkl operator [2]. Note that when ν is integer, the realization (4.2) can be obtained as a limit q → −1 from the realization (3.2).
The basis e n (x) is here realized by the monomials e n (x) = γ n x n , with some constants γ n . If we take

Coproduct and the Clebsch-Gordan coef f icients
The most important property of the sl −1 (2) algebra is that it admits an "addition rule", or a coproduct which can be inferred from the well known coproduct of the quantum algebra sl q (2). Assume that we have two independent representations of the algebra (3.1) on the linear spaces S 1 and S 2 . Let S 1 ⊗ S 2 be the direct product of these spaces. We will denote by A ⊗ B, the direct products of operators acting on the spaces S 1 and S 2 , A ∈ End(S 1 ), B ∈ End(S 2 ). It is readily verified that the elements again satisfy the commutation relations (3.1) of the sl q (2) algebra [3]. (Here I stands for the identity operator).
Assuming that the representation parameter ν is a positive integer, we have a well-defined q → −1 limit from sl q (2) to sl −1 (2). The operator q J 0 in this limit becomes ǫR with ǫ = ±1. It is thus natural to expect that for arbitrary representation parameter µ > −1/2, the sl −1 (2) algebra admits a coproduct rule.
It can be defined as follows. For two independent representations of the sl −1 (2) algebra with the Casimir parameters µ 1 , µ 2 , let us introduce the following operatorsJ 0 ,J ± ,R that act on the direct product of the spaces S 1 , S 2 : Then the operatorsJ 0 ,J ± ,R satisfy the commutation relations (2.1), i.e. they are again generators of the algebra sl −1 (2). The verification of this statement is elementary. Note that a similar coproduct was proposed in [1,8] for the parabosonic oscillator algebra, in the identification of its Hopf algebra structure.
In what follows we restrict ourselves to representations with ǫ 1 = ǫ 2 = 1 and µ 1 > −1/2, In the Fock-Bargmann realization, S 1 and S 2 are spaces of polynomials in the arguments, say, x and y. We define representations with the parameters µ 1 and µ 2 on these spaces by the formulas and J (y) 0 = y∂ y + µ 2 + 1/2, J (y) The Casimir operators take the constant values Q 1 = −µ 1 , Q 2 = −µ 2 on these representations. Following (5.1), the operators of the coproduct are given as J 0 = x∂ x + y∂ y + µ 1 + µ 2 + 1,J + = xR y + y, The corresponding Casimir operator Q =J +J−R − J 0 − 1/2 R commutes with the "local" Casimir operators Q 1 and Q 2 and with the operatorsJ 0 ,J ± but not with the operators J In view of (5.1), the operatorJ 0 can take the eigenvalues µ 1 +µ 2 +N +1, where N = 0, 1, 2, . . . . We denote by Φ N,q , the eigenstate with fixed eigenvalues of the total Casimir operator and ofJ 0 : This state can be decomposed as a linear combination of direct product of states: with coefficients W s;N,k that can be called the Clebsch-Gordan coefficients of the sl −1 (2) algebra.
The state with the maximal absolute value |q N | = µ 1 + µ 2 + N + 1/2 of the Casimir operatorQ, corresponds to the stateẽ 0 satisfying the conditions: In order to determine the sign of the eigenvalue q N , we notice that This means on the one hand that On the other hand, by (2.3)Rẽ 0 =ǫẽ 0 and henceǫ = (−1) N , whereǫ stands for the eigenvalue of the parity operatorR on the stateẽ 0 . We thus have Taking into account the parity of the coproduct states we arrive at formula (5.3).
In order to find the coefficients W s;N,k we shall derive a 3-term recurrence relation for them. Taking into account relation (5.5), we see that the eigenvalue equationQΦ N,k = q k Φ N,k can be presented in the form From the expression for the Casimir operator it is seen that Q 0 is tri-diagonal in the basis e s ⊗ e N −s . Hence, the Clebsch-Gordan coefficients W s;N,k satisfy the 3-term recurrence relation where the recurrence coefficients A s , B s are easily expressed in terms of the known representation matrix elements for sl −1 (2): where we adopt the notation (2.4).
Note that the expression for the coefficient B s can be simplified to: Thus the CGC are expressed in terms of some orthogonal polynomials P s (x) These orthogonal polynomials satisfy the 3-term recurrence relation with initial conditions P −1 = 0, P 0 = 1. From the above expressions for A s , B s we can conclude that the polynomials P s (x) coincide with the generic dual −1 Hahn polynomials [19]. Indeed, it is convenient to present the polynomials P n (x) in monic form Then the polynomialsP n (x) = x n + O(x n−1 ) satisfy on the one hand Note that u n > 0, n = 1, 2, . . . , N and u N +1 = 0. On the other hand, the dual −1 Hahn polynomials [19] R (−1) n (x; α, β; N ) depend on 3 parameters α, β and N = 1, 2, . . . and obey the recurrence relation where the recurrence coefficients are [19] The parameters ξ, η, ζ are related to the parameters α, β, N . When N is even When N is odd Comparing the recurrence coefficients of the polynomialsP n (x) with the corresponding coefficients of the dual −1 Hahn polynomials we conclude that where the parameters α, β are found from formulas (5.7) and (5.8) with ξ = µ 1 , η = µ 2 . The shift parameter x 0 can also be expressed in terms of µ 1 , µ 2 in an obvious way. We thus expressed the Clebsch-Gordan coefficients of the sl −1 (2) algebra in terms of the dual −1 Hahn polynomials R (−1) n (x; α, β; N ). The remaining problem is to find an explicit expression for the coefficient W 0;N,k in (5.6). This can be done using the following observation. The vectors ψ s = e s ⊗ e N −s form an orthonormal basis in the N + 1-dimensional linear space. There is thus a scalar product such that (ψ s , ψ t ) = δ st .
The vectors Φ N,k form another orthonormal basis on the same space and so: Hence, the matrix W s;N,k is orthogonal, i.e. it obeys N k=0 W n;N,k W m;N,k = δ nm .
Taking into account formula (5.6) we thus have on the one hand N k=0 W 2 0;N,k P n (q k )P m (q k ) = δ nm .
On the other hand, the orthonormal dual −1 Hahn polynomials P n (x) satisfy the orthogonality property [19] N k=0 w k P n (q k )P m (q k ) = δ nm , where w k are positive discrete weights (concentrated masses) localized at the spectral points q k .
(The positivity property w k > 0 follows from the positivity of the recurrence coefficients u n > 0, n = 1, 2, . . . , N [19].) We thus have Explicit expressions for the weights were found in [19]. This solves the problem of finding the Clebsch-Gordan coefficients W s;N,k up to sign factors ±1.
The result is not surprising. We have seen that the sl −1 (2) algebra is a q → −1 limit of the sl q (2) algebra and for the latter algebra, the CGC are expressed in terms of the dual q-Hahn polynomials [9].
Also, when µ 1 = µ 2 = 0 the dual −1 Hahn polynomials coincide with the ordinary Krawtchouk polynomials. This result is also expected: the case µ 1 = µ 2 = 0 corresponds to the case when both sl −1 (2) algebras in the product are equivalent to oscillator algebras whose Clebsch-Gordan coefficients are expressed in terms of Krawtchouk polynomials [21]. Note nevertheless, that even if we start with pure oscillator algebras (i.e. µ 1 = µ 2 = 0), the addition rule is nonstandard: it involves the reflection operator. Hence even in this simplest case the composed algebra will not be a pure oscillator algebra.

The Clebsch-Gordan problem in the Fock-Bargmann picture
The Clebsch-Gordan problem can be considered also in the Fock-Bargmann picture. This leads to a generating function for the Clebsch-Gordan coefficients.
The representation space for the coproduct is the space of polynomials in two variables f (x, y) which are homogeneous of degree N : f (x, y) = y N Φ(x/y), (6.1) where Φ(z) is a polynomial of degree N in the variable z. For fixed N the action of the operator operatorJ 0 is diagonal: it has the eigenvalue N + µ 1 + µ 2 + 1 (due to Euler's theorem on homogeneous polynomials).
Using the representation (6.1), we obtain the eigenvalue equatioñ Qf (x, y) = q k f (x, y), (6.2) where the eigenvalues q k are given by (5.3). Substituting f (x, y) expressed as in (6.1) into (6.2) we obtain a differential-difference equation for the function Φ(z): LΦ k (z) = q k Φ k (z), (6.3) where the operator L is and where R acts according to RΦ(z) = Φ(−z) and I is the identity operator. The operator L preserves the linear space of polynomials of degree ≤ N and belongs to a class of Dunkl type operators of the first order considered in [22,23,24]. More precisely, the operator L is a linear combination (with coefficients depending on z) of the operators I, R and ∂ z R. The main difference with respect to the Dunkl type operators used in the papers mentioned above is that the operator (6.4) does not preserve the whole space of polynomials of a given arbitrary degree. Moreover, it is seen that the operator (6.4) is 3-diagonal in the monomial basis z n , n = 0, 1, . . . , N .
Using the decomposition of the function Φ(z) = Φ e (z) + Φ o (z) into its even Φ e (z) and odd Φ o (z) parts we can reduce the equation (6.3) to standard hypergeometric equations for Φ e (z) and Φ o (z).
The explicit form of the solution will depend on the parity of the integers N and k.