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 in the classical (unquantised) case, they are quotients of Kohno's infinitesimal braid algebra Tr while in the quantum case they are quotients of the group ring C(z)Br of the r-string braid group Br. 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.