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 Uq its Drinfel'd-Jimbo quantisation over the field C(q) of rational functions in the indeterminate q. If V and Vq are corresponding irreducible modules for g and Uq, it is known that in several cases, EndUq(V⊗r) is a deformation of Endg (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 Tr, while in the quantum case, they are quotients of the group ring C(q)Br of the r-string braid group Br. 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.