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Cohomology of Categorical Self-Distributivity

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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