Spring 2025
Time & Location: All talks are on TBA in ___ at 1:00 P.M. unless otherwise noted.
Organizer: Mahir Can
April 8
Title: Tensor Products of Leibniz Bimodules and Grothendieck Rings
Joerg Feldvoss - University of South Alabama, Mobile
Abstract: Leibniz algebras were introduced by Blo(k)h and Loday as non-anticommutative analogues of Lie algebras. Many results for Lie algebras and their modules have been proven to hold for Leibniz algebras and Leibniz bimodules, but there are also several results that are not true in this more general context. In this talk we will define three different notions of tensor products for Leibniz bimodules. The "natural" tensor product of Leibniz bimodules is not always a Leibniz bimodule. In order to fix this, we will introduce the notion of a weak Leibniz bimodule and show that the "natural" tensor product of weak bimodules is again a weak bimodule. Moreover, it turns out that weak Leibniz bimodules are modules over a cocommutative Hopf algebra canonically associated to the Leibniz algebra. Therefore, the category of all weak Leibniz bimodules is symmetric monoidal and the full subcategory of finite-dimensional weak Leibniz bimodules in addition is rigid and pivotal. On the other hand, we introduce two truncated tensor products of Leibniz bimodules which are again Leibniz bimodules. These tensor products induce a non-associative multiplication on the Grothendieck group of the category of finite-dimensional Leibniz bimodules. In particular, we prove that in characteristic zero for a finite-dimensional solvable Leibniz algebra over an algebraically closed field this Grothendieck ring is an alternative power-associative commutative Jordan ring, but for a finite-dimensional non-zero semi-simple Leibniz algebra it is neither alternative nor a Jordan ring. We also expect it not to be power-associative in the semi-simple case, but at the moment we are neither able to prove nor to disprove this.
This is joint work with Friedrich Wagemann from Nantes Universit\'e.
Location: Lindy Boggs Energy Center - BO-242
Time: 3:00 PM
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April 10
Title: Linear Codes Associated to the Grassmanian
Cyrus Jonathan Young - UCLA Host: (Mahir Bilen Can)
Abstract: The Grassmannian $G(k, V)$, which is the set of $k$-dimensional subspaces of a given vector space $V$, is an algebraic variety with rich algebraic, geometric, and combinatorial structure. When $V$ is defined over a finite field, $G(k, V)$ can be studied using techniques from algebraic coding theory. In this talk, we will give an introduction to linear codes and the Grassmanian, and discuss the properties of certain linear codes related to the Grassmanian. This talk will be designed for an undergraduate audience, and throughout the talk we will be primarily using undergraduate methods.
Location: Gibson 126
Time: 1:00 PM
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April 22
Title: A Comparison of Two Supercharacter Theories
Julianne Bilen Rainbolt - Saint Louis University Host: (Mahir Bilen Can)
Abstract: In 2008, P. Diaconis and I.M. Isaacs introduced a generalization of classical character theory, which they called supercharacter theory. Let X be a partitioning of the irreducible characters of a finite group G such that the trivial character is in its own block of this partition. Let Y be a partitioning of the conjugacy classes of G such that identity element of G is in its own block of this partition. Define the supercharacters of G to be the sums of the irreducible characters in each block in X, weighted by their degree. Define the superclasses of G to be the unions of the elements in each block in Y. If the values of the supercharacters are constant on the superclasses, this is called a supercharacter theory of G. As a supercharacter is determined by these partitions, the structure of a group allows for multiple supercharacter theories. In this talk we will compare the construction of two different supercharacter theories, one based on the degrees of the irreducible characters of G and one based on the size of the conjugacy classes of G. In particular, we will demonstrate necessary and sufficient conditions that will force these two supercharacter theories to coincide when the groups considered are semidirect products of cyclic groups.
Location: Richardson Memorial - RM-304
Time: 11:00 AM
