Strongly multiplicity free modules for Lie algebras and quantum groups

Abstract

Let U be either the universal enveloping algebra of a complex semisimple Lie algebra g or its Drinfel'd–Jimbo quantisation over the field C ( z ) of rational functions in the indeterminate z. We define the notion of “strongly multiplicity free” (smf) for a finite-dimensional U -module V, and prove that for such modules the endomorphism algebras End U ( V ⊗ r ) are “generic” in the sense that in the classical (unquantised) case, they are quotients of Kohno's infinitesimal braid algebra T r while in the quantum case they are quotients of the group ring C ( z ) B r of the r-string braid group B r . In the classical case, the generators are generalisations of the quadratic Casimir operator C of U , while in the quantum case, they arise from R-matrices, which may be thought of as square roots of a quantum analogue of C in a completion of U ⊗ r . This unifies many known results and brings some new cases into their context. These include the irreducible 7-dimensional module in type G 2 and arbitrary irreducibles for sl 2 . The work leads naturally to questions concerning non-semisimple deformations of the relevant endomorphism algebras, which arise when the ground rings are varied.