Tensor products of Leibniz bimodules and Grothendieck rings

TL;DR

This paper explores tensor products of Leibniz bimodules, introduces weak bimodules with a monoidal structure, and studies the resulting algebraic structures on Grothendieck rings, revealing characteristic-dependent properties.

Contribution

It defines new tensor product notions for Leibniz bimodules, introduces weak bimodules with monoidal properties, and analyzes their Grothendieck rings in different algebraic contexts.

Findings

Weak Leibniz bimodules form a symmetric monoidal category.

Tensor products induce a non-associative multiplication on Grothendieck groups.

Grothendieck ring is a Jordan ring for solvable Leibniz algebras in characteristic zero.

Abstract

In this paper we 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 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 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 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.