2004/11/07 by Frederick M. Goodman, Goodman, Frederick M., Holly Hauschild +2 · 2 citations
Mathematics · #17B37 #57M25 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.GT #math.QA #math.RT #msc:17B37 #msc:57M25
paper · pdf · doi:10.48550/arxiv.math/0411155
Corrected error in axioms for affine BMW algebra. Many minor corrections throughout. Added references
openalex publication_date 2004/11/07 · arxiv created 2005/12/21 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The ordinary (or classical) Birman-Wenzl-Murakami algebras were initially conceived as an algebraic framework for the Kauffman link invariant. They also appear as centralizer algebras for representations of quantum universal enveloping algebras of orthogonal or symplectic types. It was shown by Morton and Wassermann that the BMW algebras are isomorphic to algebras of tangles in (disc × interval), modulo Kauffman skein relations. This isomorphism allows one to see very clearly certain properties of the BMW algebras -- for example existence of Markov traces and conditional expectations and a good basis. In the present work, we establish the correponding results for affine BMW algebras, showing they are isomorphic to algebras of tangles in (annulus × interval), modulo Kaufffman skein relations. Again, this provides an important foundation for studying these algebras, as the existence of Markov traces and conditional expectations are consequences of the isomorphism. Moreover, we obtain a basis of the affine BMW algebras analogous to a well known basis of the affine Hecke algebras.