Functors between representation categories. Universal modules

TL;DR

This paper constructs functors between categories of modules over universal algebras associated with Lie algebras, establishing adjoint functors and universal modules that generalize classical universal coacting objects.

Contribution

It introduces functors linking categories of modules over universal algebras and Lie algebra modules, and constructs universal modules as representation-theoretic analogs of known universal coacting objects.

Findings

Established a natural Lie algebra module structure on tensor products

Constructed functors between module categories with adjoints under finite dimensionality

Developed universal modules as representation-theoretic counterparts of Manin-Tambara objects

Abstract

Let $\mathfrak{g}$ and $\mathfrak{h}$ be two Lie algebras with $\mathfrak{h}$ finite dimensional and consider ${\mathcal A} = {\mathcal A} (\mathfrak{h}, \, \mathfrak{g})$ to be the corresponding universal algebra as introduced in \cite{am20}. Given an ${\mathcal A}$-module $U$ and a Lie $\mathfrak{h}$-module $V$ we show that $U \otimes V$ can be naturally endowed with a Lie $\mathfrak{g}$-module structure. This gives rise to a functor between the category of Lie $\mathfrak{h}$-modules and the category of Lie $\mathfrak{g}$-modules and, respectively, to a functor between the category of ${\mathcal A}$-modules and the category of Lie $\mathfrak{g}$-modules. Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal ${\mathcal A}$-modules and universal Lie $\mathfrak{h}$-modules as the representation theoretic counterparts of Manin-Tambara's universal coacting objects \cite{Manin, Tambara}.