School of Mathematics and Statistics
Algebra Seminar
The University of Sydney

University of Sydney Algebra Seminar


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 Uq(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 BWf(r,q) can be defined in the centraliser algebras of tensor products of the fundamental modules of the quantum algebras Uq(so(2n+1)) and Uq(sp(2n)) for different values of r.

We show that a representation of BWf(-q2n,q) exists in the centraliser algebra EndUq(osp(1|2n))(V⊗f) of V⊗f, where V is the irreducible (2n+1)-dimensional fundamental (or vector) representation of the quantum superalgebra Uq(osp(1|2n)). This representation of BWf(-q2n,q) is obtained essentially upon defining a representation of the braid group Bf on f strings in EndUq(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 BWf(-q2n,q) onto EndUq(osp(1|2n))(V⊗f) that is also a surjection, and that the permuted R-matrices in fact generate EndUq(osp(1|2n))(V⊗f). We do this using a set of projector and intertwiner matrix units in BWf(-q2n,q). This work also sheds light on the relationship between tensorial irreducible representations of Uq(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 Hf(-q) can be defined in the centraliser algebra of f-fold tensor products of the two dimensional irreducible spinor representation of Uq(osp(1|2)). This representation is obtained upon defining a representation of the braid group Bf given by the "permuted" R-matrices in this centraliser algebra.