Functors between representation categories. Universal modules

Let g and h be two Lie algebras with h finite dimensional and consider A=A(h,g) to be the corresponding universal algebra as introduced in [4]. Given an A-module U and a Lie h-module V we show that U⊗V can be naturally endowed with a Lie g-module structure. This gives rise to a functor between the category of Lie h-modules and the category of Lie g-modules and, respectively, to a functor between the category of A-modules and the category of Lie g-modules. Under some finite dimensionality assumptions, we prove that the two functors admit left adjoints which leads to the construction of universal A-modules and universal Lie h-modules as the representation theoretic counterparts of Manin-Tambara's universal coacting objects [11,16].