Gus Lehrer (University of Sydney)

Friday 5th May, 12.05-12.55pm, Carslaw 159

Endomorphism algebras of tensor powers

Let g be a complex semisimple Lie algebra and U_q its Drinfel'd-Jimbo quantisation over the field C(q) of rational functions in the indeterminate q. If V and V_q are corresponding irreducible modules for g and U_q, it is known that in several cases, End_Uq(V⊗r) is a deformation of End_g (V⊗r), and both algebras have a cellular structure, which in principle permits one to study non-semisimple deformations of either.

We present a framework ("strongly multiplicity free" modules) where the endomorphism algebras 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(q)B_r of the r-string braid group B_r. In addition to the well known cases above, these include the irreducible 7-dimensional module in type G₂ and arbitrary irreducibles for sl₂.

This is joint work with Ruibin Zhang.