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Biracks and Biquandles: Theory, applications, and new perspectives

Algebra and Representation Theory in the North (ARTIN) at Leeds:

Workshop description

Biracks and biquandles arise as set-theoretical solutions of the celebrated Yang-Baxter equation, $$f_{12} \circ f_{23}\circ f_{12}=f_{23}\circ f_{12}\circ f_{23},$$ for a linear map, $f \colon V\otimes V \to V\otimes V$, where $V$ is a vector space.

The Yang-Baxter equation arose in the early 1970s in the context of integrable models for statistical mechanics. During the 1980s, strong connections were found between the Yang-Baxter equation and quantised Lie theory, leading to the quantum groups revolution, where the Yang-Baxter equation was formulated in the context of $R$-matrices in quasi-triangular Hopf algebras. This synergy between integrable systems and Lie theory had a spectacular impact on low-dimensional topology, leading to the emerging field of quantum topology, of wide application to solving classification problems in low-dimensional topology, and moreover providing a crucial component of the mathematical framework for the recent developments in the mathematical modelling of (2+1)dimensional topological phases of matter and the ensuing paradigms for topological quantum computing.



In this workshop, we will focus on set-theoretical solutions of the Yang-Baxter equation, particularly those that are bi-invertible: biracks and biquandles. This field has developed rapidly in the last two decades due to its applications to knot theory and the theory of motion groups of loops in 3-dimensional space, and the recent breakthrough notion of skew braces. The latter provides a group-theoretical underpinning for the set-theoretical Yang-Baxter equation. Furthermore, in 2016, unexpected connections between skew braces and Galois theory were discovered, providing a generalisation of the classical Galois correspondence, “the Hopf-Galois correspondence”, and, on another topic relations were found between biquandles (derived from skew-braces) and discrete higher gauge theory (2017), which provided an algebraic model for loop-excitations in 3+1-dimensional topological phases with higher gauge symmetry.



This meeting aims to put together specialists in the field of “set theoretical solutions of the Yang-Baxter equation”, particularly the bi-invertible ones, and discuss some of their different perspectives and incarnations. We intend to gather a set of invited speakers from different fields (e.g. algebra, integrable systems, low dimensional topology / knot theory, and motion groups) and we will instruct speakers to ensure their talks are accessible for researchers in the wider field, facilitating communication and collaboration between participants.

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