Swiss Knots 2011

Knot Theory and Algebra

Lake Thun, May 23-27

Homology of distributive structures: monoid of binary operations and examples of computation

Prof. Dr. Jozef Przytycki's talk
Date: 23.05.11   Time: 17.30 - 18.20   Room: Gwatt Zentrum

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, the non-associative structures, such as quandles, were neglected until recently. The distributive structures have been studied for a long time and already C.S. Peirce in 1880 emphasized the importance of (right) self-distributivity in algebraic structures. However, homology for such universal algebras was introduced only fifteen years ago by Fenn, Rourke and Sanderson. I will develop this theory in the historical context and describe relations to topology and similarity with some structures in logic. In particular, we will compute 4-term distributive homology for some Boolean algebras. We will also speculate how to define homology for Yang-Baxter operators and how to relate our work to Khovanov homology and categorification. We use here the fact that Yang Baxter equation can be thought of as a generalization of self-distributivity.