Yang-Baxterization of braid group representations

Abstract

For a given braid group representation (BGR), a process of the Yang-Baxterization is formulated to generate solutions of the Yang-Baxter equation (YBE). When a BGR admits the Birman-Wenzl (BW) algebraic structure, this process can be explicitly passed through and two types of trigonometric solutions of YBE are generated from such a BGR. These two solutions have the essential difference to each other and both of them preserve the crossing symmetry property if the given BGR has. By taking certain reduction on the BW algebra, the rational solution is also generated. A practical condition to judge whether a BGR satisfies the BW algebra is given, from which one finds that not only the familiar BGRs of [5,7,9], but also some new ones obtained recently in [12] have the BW structure. Thus they can be explicitly Yang-Baxterized to solutions of the YBE.

Representations of the quantum doubles of finite group algebras and spectral parameter dependent solutions of the Yang–Baxter equation

“…32 Baxterization, as coined by Jones, 33 and there is sizable literature on this topic. [34][35][36][37][38][39][40][41] We begin by studying the case of the two-dimensional irreducible representations of D͑D n ͒ and find that Baxterization can be performed successfully. Our approach is based on an ansatz taken from Ref.…”

Incompressible Representations of the Birman-Wenzl-Murakami Algebra

“…algebra on the generatorᾱ. The matrices t i (15) are such that the column indexed by the tangle π ∈ Π i 2 has only zeros except for a 1 at the intersection with the line indexed by t i π ∈ Π i 1 . Similarly, the line t i π ∈ Π i 1 has only zeros except for a 1 at the intersection with the column π ∈ Π i 2 .…”

“…In the formulas (15) , one can set a = 1, ǫ = 0, τ = 2(1 + ) and all the nonzero matrix elements of v are equal to 1.…”

Birman—Wenzl—Murakami Algebra and Topological Basis

“…BGRs of two and three eigenvalues have direct relationship with Temperley-Lieb algebra (TLA) [7] and Birman-Wenzl-Murakami algebra (BWMA) [8] respectively. TLA and BWMA have been widely used to construct the solutions of YBE [9][10][11][12].…”

“…As we know the BWMA relations [8,9,20,21] including braid relations and TLA relations satisfy the following relations,…”