Solutions to generalized Yang-Baxter equations via ribbon fusion categories

Abstract

Inspired by quantum information theory, we look for representations of the braid groups $B_n$ on $V^{\otimes (n+m-2)}$ for some fixed vector space $V$ such that each braid generator ${\sigma }_i,i=1,...,n-1,$ acts on $m$ consecutive tensor factors from $i$ through $i+m-1$. The braid relation for $m=2$ is essentially the Yang-Baxter equation, and the cases for $m>2$ are called generalized Yang-Baxter equations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case $m=3$. Examples are given from the Ising theory (or the closely related $SU(2{)}_2$), $SO(N{)}_2$ for $N$ odd, and $SU(3{)}_3$. The solution from the Jones-Kauffman theory at a $6^{th}$ root of unity, which is closely related to $SO(3{)}_2$ or $SU(2{)}_4$, is explicitly described in the end.

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Cited by in corpus (7)

• Braiding quantum gates from partition algebras
• Local invariants of braiding quantum gates -- associated link polynomials and entangling power
• Quantum entanglement, supersymmetry, and the generalized Yang-Baxter equation
• Polyadic braid operators and higher braiding gates
• Parameter-dependent Gaussian $(z,N)$-generalized Yang-Baxter operators
• Generating W states with braiding operators
• From ribbon categories to generalized Yang-Baxter operators and link invariants (after Kitaev and Wang)