Combinatorics/Algebra Seminar
Summer/Fall 2022 Archive:

November 14

Connor Depies: An alternative analog of the exterior algebra, and related topics.

November 7

Mitchell Ashburn: Vector-matrix algebras over Lie algebras.

Abstract: Zorn's vector-matrices are a convenient representation of the split octonions, allowing them to have the feel of a typical algebra of matrices. However, since this representation seems like an outlier among non-associative algebras, it would be nice to have further examples of similar constructions. In this talk, we will discuss a generalization of Zorn's vector-matrices to what we call vector-matrix algebras, and compare them to Zorn's example. We will also explore some connections between our vector-matrix algebras and the Lie algebras on which they depend.

October 31

Jonathan Smith: Modules and derivations.

Abstract: Derivations are available for each particular kind of algebra, and are related to the concept of module appropriate for those algebras. For associative algebras and Lie algebras, the defining equation for a derivation reflects Leibniz' Product Rule from calculus, while for groups, derivations take the form of crossed homomorphisms. We present derivations in a general setting (related to Kähler differentials), with a view to understanding Clifford algebras and more general alternative algebras in terms of derivations.

October 24

No seminar.

October 17

Gavin Nop: Linear poset embeddings and representations.

October 10

Justin Stevenson: Character quasigroups for the dihedral groups.

Abstract: To every finite abelian group, the set of linear characters forms an abelian group under pointwise multiplication called the character group, which turns out to be isomorphic to the original group. In this talk I will discuss recent developments in combinatorics that allow us to generalize this setup to character quasigroups of not necessarily abelian groups. In particular we will talk about the character quasigroups for the dihedral groups, and which groups arise as a character quasigroup for them.

October 3

Connor Depies: Octonions, Clifford algebras, and twisted algebras

September 12, 19, 26

Jonathan Smith: Wigner functions and web geometry

Abstract: At a level accessible to graduate students, the talks will review two topics from the nineteen-thirties, and discuss their relevance for questions in present-day quantum information theory.

• I: Quasigroups
• II: Quantum mechanics
• III: Qubits

August 1

Jonathan Smith: Augmented quasigroups and Heyting algebras

July 25

Connor Depies: The Clifford adjunction

July 18

Jonathan Smith: A gentle introduction to Hopf algebras and quantum quasigroups

June 27, July 11

Justin Stevenson: Character quasigroups for the dihedral groups

June 13, 20

Connor Depies: An introduction to Clifford algebras

May 30, June 6

Mitchell Ashburn: Sub-Riemannian geodesics on SL(2,R), after D'Alessandro and Cho

Archive of earlier seminars

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