Cohomology of Frobenius Algebras and the Yang-Baxter Equation
J. Scott Carter (Univ. of South Alabama), Alissa S. Crans (Loyola, Marymount Univ), Mohamed Elhamdadi (Univ. of South Fla.), Enver Karadayi, (Univ. of South Fla.), Masahico Saito (Univ. of South Fla.)
TL;DR
This paper develops a cohomology theory for Frobenius algebras' operations, providing concrete examples and linking to solutions of the Yang-Baxter equation through skein theory and deformations.
Contribution
It introduces a low-dimensional cohomology framework for Frobenius algebras and connects it to the Yang-Baxter equation via skein theoretic constructions.
Findings
Concrete computations for key Frobenius algebra examples
Construction of solutions to the Yang-Baxter equation from Frobenius algebra operations
Deformations of R-matrices using 2-cocycles
Abstract
A cohomology theory for multiplications and comultiplications of Frobenius algebras is developed in low dimensions in analogy with Hochschild cohomology of bialgebras based on deformation theory. Concrete computations are provided for key examples. Skein theoretic constructions give rise to solutions to the Yang-Baxter equation using multiplications and comultiplications of Frobenius algebras, and 2-cocycles are used to obtain deformations of R-matrices thus obtained.
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology