Combinatorics/Algebra Seminar
Spring 2020 Archive:

April 13

Alex Nowak: Linear aspects of quasigroup triality.

March 23, 30

Oyeyemi Oyebola (Federal University of Agriculture, Abeokuta, Nigeria): Inverse property polyquasigroups and polyloops.

Abstract:

In this work (joint with Temitope G. Jaiyeola and Olusola J. Adeniran), we study some newly introduced non-associative algebraic hyperstructures known as polyquasigroups and polyloops, which generalize quasigroups and loops. We will focus on the left, right, and (two-sided) inverse properties (LIP, RIP, IP). We also consider autotopisms of polyquasigroups and polyloops.

March 9

Kuei-Nuan Lin (Penn State Greater Allegheny): Algebraic invariants of weighted oriented graphs.

Abstract:

We start with the definitions of a weighted oriented graph, D, and its edge ideal, I(D), in a polynomial ring R. We give the formula of Castelnuovo-Mumford regularity of R/I(D) when D is a weighted oriented path or cycle. Additionally, we compute the projective dimension for this class of graphs. This is joint work with S. Beyarslan, J. Biermann, and A. O’Keefe.

February 24, March 2

Jonathan Smith: The algebraic structure of the qubit.

Abstract:

The qubit, or quantum bit, is the two-level quantum mechanical system that forms the basis for quantum information science. It is usually handled in terms of spherical coordinates, which introduce singularities, and do not lend themselves well to repeated manipulations. In these talks, we will discuss various algebraic approaches to the analysis of a qubit.

February 10, 17

Alex Nowak: Quasigroup-graded algebras via oriented Hopf algebra action.

Abstract:

In these expository talks, we'll review Chris Plyley and David Riley's paper on Hopf algebras with a Z₂-graded algebra structure. (Note this is weaker than the super Hopf algebra concept, which requires the grading to respect the coalgebra). Such a grading allows the authors to study Hopf algebras which act on algebras by both automorphisms and anti-automorphisms. Our interests lie mainly in the paper's concluding section, in which Plyley and Riley apply oriented Hopf algebra actions to furnish an example of a Lie algebra graded by a nonassociative quasigroup operation. What's more, the grading cannot be realized via associative, semigroup coaction. According to the authors, such "exotic" gradings were once conjectured not to exist.

The first talk will provide a review of group algebras, their duals, Hopf algebras, and H-module algebras. We will end with an introduction to oriented Hopf algebras.

The second talk will examine the duality between oriented Hopf algebra actions and the nonassociative oriented convolution of functionals. We will then describe Plyley and Riley's Lie algebra and its nonassociative grading. We conclude with a discussion of the prospects for generalizing this example and widening the scope of quasigroup-gradings permitted by Hopf algebra action.

February 3

Anna Romanowska (Warsaw University of Technology): Identities in Mal’tsev products of varieties.

Abstract:

Let Q and R be (quasi)varieties of algebras of the same type. Assume additionally that algebras in R are idempotent. Then the Mal'tsev product Q o R of Q and R consists of algebras A with a congruence r such that the quotient A/r is in R and each congruence class a/r is in Q.

It is known that the Mal'tsev product of two varieties V and W of similar algebras is a quasivariety, but not necessarily a variety. A recent result shows how to find identities true in the Mal’tsev product V o W, if the identities true in V and in W are known. When V o W is a variety this method provides an equational basis for V o W. There are not many results known providing conditions under which the product V o W is a variety. One positive result of this type (joint with Bergman and Penza) concerns the situation where V is a certain irregular variety and W is a variety of the same type as V equivalent to the variety of semilattices. In this talk I will provide some results concerning varieties V and W of groupoids such that the Mal’tsev product V o W is never a variety.

January 27

Jonathan Smith: A quasigroup with modular data.

Abstract:

Modular data, with an action of the modular group

    PSL(2,Z) = < S, T | S² = (ST)³ = 1 >,

is a feature of certain tensor categories (modular categories) and quantum field theories. Terry Gannon has emphasized its connection with algebraic combinatorics.

In this talk, we will exhibit perhaps the most elementary non-trivial instance of modular data, arising from the combinatorial character theory of the quasigroup of integers modulo 4 under subtraction. Along the way, we will discuss how this character theory arises, in particular placing emphasis on the unitary character table.

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