R-matrix and Mickelsson algebras for orthosymplectic quantum groups

Thomas Ashton,Andrey Mudrov

doi:10.1063/1.4927582

Abstract

Let 𝔤 be a complex orthogonal or symplectic Lie algebra and 𝔤′ ⊂ 𝔤 the Lie subalgebra of rank rk 𝔤′ = rk 𝔤 − 1 of the same type. We give an explicit construction of generators of the Mickelsson algebra Zq(𝔤, 𝔤′) in terms of Chevalley generators via the R-matrix of Uq(𝔤).

Highlights

In the mathematics literature, lowering and raising operators are known as generators of step algebras, which were originally introduced by Mickelsson[1] for reductive pairs of Lie algebras, g′ ⊂ g
The general theory of step algebras for classical universal enveloping algebras was developed in Refs. 2 and 4 and extended to the special linear and orthogonal quantum groups in Ref. 5
They admit a natural description in terms of extremal projectors,[4] introduced for classical groups in Refs. 6 and 7 and generalized to the quantum group case in Refs. 8 and 9

Summary

INTRODUCTION

In the mathematics literature, lowering and raising operators are known as generators of step algebras, which were originally introduced by Mickelsson[1] for reductive pairs of Lie algebras, g′ ⊂ g These algebras naturally act on g′-singular vectors in U(g)-modules and are important in representation theory.[2,3]. It is known that the step algebra Z(g,g′) is generated by the image of the orthogonal complement g ⊖ g′ under the extremal projector of the g′ Another description of lowering/raising operators for classical groups was obtained in Refs. Extending[3] to orthogonal and symplectic quantum groups is not straightforward, since there are no nilpotent triangular Lie subalgebras g± in Uq(g) but only their deformed associative envelope We suggest such a generalization, where the lack of g± is compensated by the entries of the universal R-matrix with one leg projected to the natural representation. Explicit form of these generators is useful in quantization of conjugacy classes, because they are related to singular vectors generating certain submodules involved.[13,14]

Quantized universal enveloping algebra

R-MATRIX OF NON-EXCEPTIONAL QUANTUM GROUPS

MICKELSSON ALGEBRAS

Lowering operators

Raising operators