The BMW algebra is a deformation of the Brauer algebra, and has the Hecke algebra of type A as a quotient. Its specializations play a role in types B, C, D akin to that of the symmetric group in Schur-Weyl duality.
One can enlarge these algebras by a commutative subalgebra to an affine, or annular, version. Unlike the affine Hecke algebra, the affine BMW algebra is not of finite rank as a right -module, so induction functors are ill-behaved, and many of the classical Hecke-theoretic constructions of simple modules fail. However, the affine BMW algebra still has a nice class of -semisimple, or calibrated, representations, that don't necessarily factor through the affine Hecke algebra.
I will discuss Walker's TQFT-motivated 1-handle construction of the -semisimple, or calibrated, representations of the affine BMW algebra. While the construction is topological, the resulting representation has a straightforward combinatorial description in terms of Young diagrams. This is joint work with Kevin Walker.
Representations of the affine BMW algebra Sponsored by the Meyer Fund
Ever since the Banach-Tarski Paradox was discovered, people have wondered about eliminating the Axiom of Choice in favor of a system in which the paradox evaporates. But, as discovered by Mycielski and Sierpinski, the most common way of doing this -- taking all sets to be Lebesgue measurable -- leads to a different sort of paradox that is just as disturbing. I will also present a construction that shows how congruent regular tetrahedra can be chained together face-to-face so as to make a loop that is closed to within any preassigned error. This settles a question first raised and studied 60 years ago by Steinhaus and Swierczkowski.
A Paradox Due to the Absence of the Axiom of Choice; and a Proof of the Tetrahedral Loop Conjecture
Apr. 11, 2017 3pm (Math 220)
Functional Analysis
Marty Walter (CU Boulder)
X
An interesting connection between functional analysis and number theory.