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# A Remark on BMW algebra, $q$-Schur Algebras and Categorification

Published:2013-08-10
Printed: Apr 2014
• Pedro Vaz,
CAMGSD, Instituto Superior Técnico , Avenida Rovisco Pais, 1049-001 Lisboa, Portugal
• Emmanuel Wagner,
Institut de Mathématiques de Bourgogne UMR 5584 du CNRS, 7 avenue Alain Savary BP 47870 21078 Dijon Cedex, France
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## Abstract

We prove that the 2-variable BMW algebra embeds into an algebra constructed from the HOMFLY-PT polynomial. We also prove that the $\mathfrak{so}_{2N}$-BMW algebra embeds in the $q$-Schur algebra of type $A$. We use these results to suggest a schema providing categorifications of the $\mathfrak{so}_{2N}$-BMW algebra.
 Keywords: tangle algebras, BMW algebra, HOMFLY-PT Skein algebra, q-Schur algebra, categorification
 MSC Classifications: 57M27 - Invariants of knots and 3-manifolds 81R50 - Quantum groups and related algebraic methods [See also 16T20, 17B37] 17B37 - Quantum groups (quantized enveloping algebras) and related deformations [See also 16T20, 20G42, 81R50, 82B23] 16W99 - None of the above, but in this section