Distributivity versus associativity in the homology theory of algebraic structures
By Józef H. Przytycki. 2011
52 citations recorded by the source corpus.
Abstract While homology theory of associative structures, such as groups and rings, has been extensively studied in the past beginning with the work of Hopf, Eilenberg, and Hochschild, homology of non-associative distributive structures, such as quandles, were neglected until recently. Distributive structures have been studied for a long time. In 1880, C.S. Peirce emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for these universal algebras was introduced only sixteen years ago by Fenn, Rourke, and Sanderson. We develop this theory in the historical context and propose a general framework to study homology of distributive structures. We illustrate the theory by computing some examples of 1-term and 2-term homology, and then discussing 4-term homology for Boolean algebras. We outline potential relations to Khovanov homology, via the Yang-Baxter operator.
Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/9ce7e776df1925511bc208ff7df5827eafb3987e
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Citations in the corpus
- Homology of left non-degenerate set-theoretic solutions to the Yang–Baxter equation (2015, methodology): “Przytycki [33] noticed the resemblance between this equivalence of differentials and the corresponding phenomenon in the self-distributive situation.”
- Homologies of Algebraic Structures via Braidings and Quantum Shuffles (2012, background): “[23] for details and [31] for weak simplicial notions; note that our definition is a shifted version of theirs, and that our definition of bisimplicial vector spaces is different from the one in [23]): Definition 3.”