Sacha Blumen (Australian Council for Educational Research)

Friday 19th May, 12.05-12.55pm, Carslaw 159

The Birman-Wenzl-Murakami algebra, Hecke algebra, and representations of U_q(osp(1|2n))

In this talk, I will present results from the first third of my PhD thesis, along with some further results I have since obtained.

It is well know that representations of the Birman-Wenzl-Murakami algebra BW_f(r,q) can be defined in the centraliser algebras of tensor products of the fundamental modules of the quantum algebras U_q(so(2n+1)) and U_q(sp(2n)) for different values of r.

We show that a representation of BW_f(-q²ⁿ,q) exists in the centraliser algebra End_{Uq(osp(1|2n))}(V^⊗f) of V^⊗f, where V is the irreducible (2n+1)-dimensional fundamental (or vector) representation of the quantum superalgebra U_q(osp(1|2n)). This representation of BW_f(-q²ⁿ,q) is obtained essentially upon defining a representation of the braid group B_f on f strings in End_{Uq(osp(1|2n))}(V^⊗f) using the "permuted" R-matrices acting on V^⊗f.

We show that there exists an algebra homomorphism from a quotient of BW_f(-q²ⁿ,q) onto End_{Uq(osp(1|2n))}(V^⊗f) that is also a surjection, and that the permuted R-matrices in fact generate End_{Uq(osp(1|2n))}(V^⊗f). We do this using a set of projector and intertwiner matrix units in BW_f(-q²ⁿ,q). This work also sheds light on the relationship between tensorial irreducible representations of U_q(osp(1|2n)) and U_-q(so(2n+1)) noted by R. B. Zhang, and may fit into his recent joint work with G. Lehrer that was presented at the algebra seminar on 5th May 2006.

Finally, we show that a representation of the Hecke algebra H_f(-q) can be defined in the centraliser algebra of f-fold tensor products of the two dimensional irreducible spinor representation of U_q(osp(1|2)). This representation is obtained upon defining a representation of the braid group B_f given by the "permuted" R-matrices in this centraliser algebra.