Functors between representation categories. Universal modules

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}.

Continues arXiv:2006.00711