We present three case studies of enumeration in nonassociative algebra: quandles, code loops, and Bruck loops. Quandles are algebraic structures designed to color arcs of oriented knots and they also form a well-studied class of set-theoretical solutions of the Yang-Baxter equation. Their classification is based on a representation in symmetric groups and we extend existing results by several orders of magnitude in complexity. Code loops first appeared in Conway's construction of the Monster group and the general case was introduced by Griess. Their classification is based on trilinear alternating forms, and we again significantly improve upon existing results. Finally, Bruck loops provide framework for Einstein's relativistic addition of vectors and we use central extensions to count Bruck loops of small odd prime power orders. All three results require extensive computer calculations. This is joint work with Seung Yeop Yang, Eamonn O'Brien and Izabella Stuhl, in order of appearance.
In joint work with Radha Kessar, we show that the Morita Frobenius numbers of the -blocks of quasi-simple finite groups are less than or equal to four, and hence the basic algebras of such -blocks are defined over the finite field of elements. I will explain the connection between this result and Donovan's Conjecture and illustrate how recent results of Bonnafè—Dat—Rouquier played a crucial role in our methods.
Rationality of blocks of quasi-simple finite groups Sponsored by the Meyer Fund
Apr. 17, 2018 3pm (MATH 350)
Topology
Markus Pflaum (CU Boulder) Hochschild Homology and Cyclic Homology of Convolution Algebras
Enumeration in nonassociative algebra