Strongly Multiplicity Free Modules for Lie Algebras and Quantum Groups

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 EndU (V ⊗r) are “generic” in the sense that they are quotients of Kohno’s infinitesimal braid algebra Tr (in the classical (unquantised) case) and of the group ring C(z)Br of the r-string braid group Br in the quantum case. 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 G2 and arbitrary irreducibles for sl2. The work leads naturally to questions concerning non-semisimple deformations of the relevant endomorphism algebras, which arise when the ground rings are varied.