Racah Problems for the Oscillator Algebra, the Lie Algebra $$\mathfrak {sl}_n$$ sl n , and Multivariate Krawtchouk Pol
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Annales Henri Poincar´ e
Racah Problems for the Oscillator Algebra, the Lie Algebra sln, and Multivariate Krawtchouk Polynomials Nicolas Cramp´e, Wouter van de Vijver
and Luc Vinet
Abstract. The oscillator Racah algebra Rn (h) is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra h. An embedding of the Lie algebra sln−1 into Rn (h) is presented. It relates the representation theory of the two algebras. We establish the connection between recoupling coefficients for h and matrix elements of sln -representations which are both expressed in terms of multivariate Krawtchouk polynomials of Griffiths type.
Contents 1. 2. 3.
Introduction The Oscillator Algebra h and the Racah Algebra Embedding of sln−1 into Rn (h) 3.1. Embedding of sl2 into R3 (h) 3.2. Embedding of sl3 into R4 (h) 3.3. Embedding of sln−1 into Rn (h) 3.4. Labeling Abelian Algebras 4. Connection Between Recoupling Coefficients for h and sln−1 Representations 4.1. sl2 and the Krawtchouk Polynomials 4.2. sln−1 and the Multivariate Krawtchouk Polynomials 4.3. 6j- and 9j-Symbols 4.4. Automorphisms of Rn (h) and sln−1 5. Conclusion Acknowledgements Appendix A: Calculation of overlap coefficients References
N. Cramp´e et al.
Ann. Henri Poincar´e
1. Introduction This paper studies the oscillator Racah algebra Rn (h) viewed as the centralizer of the diagonal action of the oscillator algebra h [45] in the n-fold tensor product of its universal algebra. We shall find that it admits an embedding of sln−1 . Building upon that result, we shall connect the facts that the multivariate Krawtchouk polynomials of Griffiths arise as 3(n − 1)j symbols of h as well as matrix elements of the restriction to the group O(n + 1) of the symmetric representations of SU(n + 1). There is growing interest in Racah algebras. These are, in particular, identified in the framework of Racah problems where one looks at the recouplings of tensor products of certain Lie algebras. We shall denote by n the number of factors. The cases with n = 3 for the Lie algebra su(2) (or su(1, 1)), the quantum algebra Uq (sl2 ) and the Lie superalgebra osp(1|2) have first been examined. They have led, respectively, to the (universal versions of the) Racah algebra R(3) [21,23,28] the Askey–Wilson algebra AW(3) [29,32] and the Bannai-Ito algebra BI(3) [22]. In this picture, where there is an implicit map from the abstract Racah algebra onto the centralizer of the diagonal action of say, su(2), Uq (sl2 ) or osp(1|2) on their triple product, the images of the three generators of the Racah algebra are expressed in terms of the intermediate Casimir elements. The representations of these algebras encompass the bispectral properties of the orthogonal polynomials bearing the same name that are essentially the Racah or 6j-coefficients of the corresponding algebras whose triple tensor products are considered. In fact this is how the AW(3) was first identified [52] through its realization in terms of the recurrence and q-difference operators of the Askey–Wilson polynomia
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