Unitary R-Matrices: Representations of deformations of the Braid Group arising from the BMW algebra and racks

Deadman, Tasarla 2024. Unitary R-Matrices: Representations of deformations of the Braid Group arising from the BMW algebra and racks. PhD Thesis, Cardiff University. Item availability restricted.

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Abstract

In this thesis we look at R-matrix representations of the BMW algebra and the Bloop group, a new group arising from the quandalisation process of racks. Both the BMW algebra and the Bloop group have diagrammatic presentations involving adding loops to the braid group, although the structures of these loops are different. We define the contractive R-matrices, a restriction of which form a representation of the BMW algebra, and show that they are stable under equivalence. We show that they have a maximum of 3 eigenvalues and we classify all 2-dimensional examples. Then we utilise a Markov trace to deduce restrictions on the possible values of the contraction constant c. In particular, we show that |c| −2 is the Jones Index [ρR(B∞) : φ(ρR(B∞))] and we deduce the form of the contractive R-matrices for each value in the discrete range of the Jones Index. The rack-induced R-matrices are shown to be closely tied with the racks from which they are derived. In particular, isomorphic racks induce equivalent R-matrices (though the opposite is not necessarily true), and if two rack-induced R-matrices are equivalent then the quandalisations of their underlying racks also produce equivalent R-matrices. We show that quandle-induced R-matrices are equivalent if and only if their coloring numbers are equal for all oriented links. The quandalisation process of racks is the inspiration for the Bloop Group developed in this thesis, and we develop an R-matrix representation of this new group. These developments contribute to ongoing area of research of the classification of unitary R-matrices, which has applications in many areas including quantum groups, knot theory and topological quantum computing. Further research in this area would include the analysis of R-matrix representations of other structures.

Item Type:	Thesis (PhD)
Date Type:	Completion
Status:	Unpublished
Schools:	Mathematics
Subjects:	Q Science > QA Mathematics
Funders:	EPSRC
Date of First Compliant Deposit:	16 January 2025
Last Modified:	16 Jan 2025 12:34
URI:	https://orca.cardiff.ac.uk/id/eprint/175308

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