Gorenstein homological algebra for rngs and Lie superalgebras

Abstract

We generalise notions of Gorenstein homological algebra for rings to the context of arbitrary abelian categories. The results are strongest for module categories of rngs with enough idempotents. We also reformulate the notion of Frobenius extensions of noetherian rings into a setting which allows for direct generalisation to arbitrary abelian categories. The abstract theory is then applied to the BGG category O for Lie superalgebras, which can now be seen as a "Frobenius extension" of the corresponding category for the underlying Lie algebra and is therefore "Gorenstein". In particular we obtain new and more general formulae for the Serre functors and instigate the theory of Gorenstein extension groups.

References in corpus (1)

• Crystal structures arising from representations of $GL(m|n)$

Cited by in corpus (4)

• Whittaker modules for classical Lie superalgebras
• Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras
• Some homological properties of category $\mathcal{O}$ for Lie superalgebras
• On semisimplicity of Jantzen middles for the periplectic Lie superalgebra