Tensor products of Leibniz bimodules and Grothendieck rings
Jörg Feldvoss, Friedrich Wagemann
TL;DR
The paper develops a tensor-product framework for Leibniz bimodules by first introducing weak bimodules to restore a natural tensor product that remains closed. It then constructs the weak universal enveloping algebra $\mathrm{UL}_{\mathrm{weak}}(\mathfrak{L})$, endowing the weak bimodules with a symmetric monoidal, rigid, and pivotal structure, and showing a reflective relation to ordinary bimodules. Building on this, two truncated tensor products yield genuine Leibniz bimodules, and their nonassociative Grothendieck ring $\mathrm{Gr}^{\mathrm{bi}}(\mathfrak{L})$ is described as a unital commutative product of two copies of $\mathrm{Gr}(\mathfrak{L}_{\mathrm{Lie}})$. In characteristic zero, finite-dimensional solvable Leibniz algebras have $\mathrm{Gr}^{\mathrm{bi}}(\mathfrak{L})$ that is alternative and Jordan, whereas finite-dimensional semisimple cases fail these identities. The paper provides explicit instances with $\mathfrak{e}$ and $\mathfrak{sl}_{2}(\mathbb{C})$, showing the general phenomenon that the Grothendieck ring is governed by the canonical Lie algebra, and it highlights open questions about irreducible weak bimodules and associativity in broader settings.
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.