George Washington University
Distributivity versus Associativity in Homology Theories
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 15-20 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, 2-term, and 3-term homology, and then discussing 4-term homology for Boolean algebras and distributive lattices. We outline potential relations to Khovanov homology, via the Yang-Baxter operator.
Friday January 27, 2012 at 3:00 PM in SEO 636