Outdated Archival Version

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Cohomology of Categorical Self-Distributivity

J. Scott Carter, Alissa S. Crans, Mohamed Elhamdadi and Masahico Saito

We define self-distributive structures in the categories of coalgebras and cocommutative coalgebras. We obtain examples from vector spaces whose bases are the elements of finite quandles, the direct sum of a Lie algebra with its ground field, and Hopf algebras. The self-distributive operations of these structures provide solutions of the Yang--Baxter equation, and, conversely, solutions of the Yang--Baxter equation can be used to construct self-distributive operations in certain categories.

Moreover, we present a cohomology theory that encompasses both Lie algebra and quandle cohomologies, is analogous to Hochschild cohomology, and can be used to study deformations of these self-distributive structures. All of the work here is informed via diagrammatic computations.

Journal of Homotopy and Related Structures, Vol. 3(2008), No. 1, pp. 13-63