Friday, 3:20-4:10pm, Carver 401
Speaker: Anna Zamojska-Dzienio
Title: Quasigroup language for set-theoretical solutions of the Yang–Baxter equation
Abstract: Translating set-theoretical non-degenerate solutions of the Yang–Baxter equation into universal algebra language, we obtain an algebra called a birack. Biracks appeared in low-dimensional topology in 1999 when Louis Kauffman introduced virtual knot theory (with two sorts of crossings). They play there a similar role as racks in classical knot theory. Biracks consist of two one-sided quasigroup structures connected by additional conditions (identities) which give a natural algebraic counterpart of Reidemeister moves. Such an equational description (i.e., via identities) allows one to characterize solutions of the Yang–Baxter equation using universal algebra tools. In particular, this is convenient when one of the two operations is either trivial (racks), or can be reconstructed from the other one (e.g., Rump’s cycle sets), or is a full (two-sided) quasigroup operation (Latin solutions). In all these situations, one can apply knowledge and methods coming from quasigroup theory. The aim of this talk is to present such an approach.
This will be our final seminar of the semester. We plan to keep the same schedule for the fall semester so let me know if you or a guest would like to claim a speaking spot.