Gus Lehrer (University of Sydney)
Friday 5th May, 12.05-12.55pm, Carslaw 159Endomorphism algebras of tensor powersLet 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 G_{2} and arbitrary irreducibles for sl_{2}. This is joint work with Ruibin Zhang. |