A cellular basis of the q-Brauer algebra related with Murphy bases of Hecke algebras

A new basis of the [Formula: see text]-Brauer algebra is introduced, which is a lift of Murphy bases of Hecke algebras of symmetric groups. This basis is shown to be a cellular basis in the sense of Graham and Lehrer. Using combinatorial tools we prove that the non-isomorphic simple [Formula: see text]-Brauer modules are indexed by the [Formula: see text]-restricted partitions of [Formula: see text] where [Formula: see text] is an integer, [Formula: see text]. When the [Formula: see text]-Brauer algebra has low dimension a criterion of semisimplicity is given, which is used to show that the [Formula: see text]-Brauer algebra is in general not isomorphic to the BMW-algebra.

Topological basis realization for BMW algebra and Heisenberg XXZ spin chain model

In this paper, we study three-dimensional (3D) reduced Birman–Murakami–Wenzl (BMW) algebra based on topological basis theory. Several examples of BMW algebra representations are reviewed. We also discuss a special solution of BMW algebra, which can be used to construct Heisenberg XXZ model. The theory of topological basis provides a useful method to solve quantum spin chain models. It is also shown that the ground state of XXZ spin chain is superposition state of topological basis.

A Remark on BMW Algebra, q-Schur Algebras and Categorification

We prove that the two-variable BMW algebra embeds into an algebra constructed from the HOMFLY-PT polynomial. We also prove that the so2N-BMW algebra embeds in the q-Schur algebra of type A. We use these results to suggest a schema providing categorifications of the 2N-BMW algebra.

SYMMETRIZERS AND ANTISYMMETRIZERS FOR THE BMW-ALGEBRA

Let n ∈ ℕ and Bn(r, q) be the generic Birman–Murakami–Wenzl algebra with respect to indeterminants r and q. It is known that Bn(r, q) has two distinct linear representations generated by two central elements of Bn(r, q) called the symmetrizer and antisymmetrizer of Bn(r, q). These generate for n ≥ 3 the only one-dimensional two sided ideals of Bn(r, q) and generalize the corresponding notion for Hecke algebras of type A. The main result, Theorem 3.1, in this paper explicitly determines the coefficients of these elements with respect to the graphical basis of Bn(r, q).