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
