Markov traces on the Birman-Wenzl-Murakami algebras

Abstract

We classify the Markov traces factoring through the Birman-Wenzl-Murakami (BMW) algebras. For this purpose, we define a common `cover' for the two variations of the BMW-algebra originating from the quantum orthogonal/symplectic duality, which are responsible for the so-called `Dubrovnik' variation of the Kauffman polynomial. For generic values of the defining parameters of the BMW algebra, this cover is isomorphic to the BMW algebra itself, and this fact provides a shorter defining relation for it, in the generic case. For a certain 1-dimensional family of special values however, it is a non-trivial central extension of the BMW-algebra which induces a central extension of the Temperley-Lieb algebra. Inside this 1-dimensional family, exactly two values provide possibly additional Q-valued Markov traces. We describe both of these potential traces on the (extended) Temperley-Lieb subalgebra. While we only conjecture the existence of one of them, we prove the existence of the other by introducing a central extension of the Iwahori-Hecke algebra at q=-1 for an arbitrary Coxeter system, and by proving that this extension indeed admits an exotic Markov trace. These constructions provide several natural non-vanishing Hochschild cohomology classes on these classical algebras.