Abstract
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.
Original language | American English |
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Pages (from-to) | 13-63 |
Journal | Journal of Homotopy and Related Structures |
Volume | 3 |
Issue number | 1 |
State | Published - 2008 |
Keywords
- Categorical internalization
- self-distributivity
- quandle
- Lie Algebra
- Hopf Algebra
- Hochschild Cohomology
- coalgebra
- Yang Baxter equation
- trigonometric coalgebras
- rack
Disciplines
- Algebra