Odd Verma's Theorem

TL;DR

This paper explores properties of Verma supermodules over symmetrizable Kac--Moody Lie superalgebras, revealing their $rak{gl}(1|1)$-structure, and investigates how changing Borel subalgebras affects their category structure and homomorphisms.

Contribution

It introduces a new perspective on Verma modules via Borel subalgebra variations, connecting to $rak{gl}(1|1)$-structure and semibricks, and refines understanding of their homomorphisms and associated varieties.

Findings

Realization of the principal block of $rak{gl}(1|1)$ as an extension-closed subcategory

Description of homomorphism compositions via odd reflections

Refined results on associated varieties and projective dimensions

Abstract

We formulate several basic properties of Verma supermodules over regular symmetrizable Kac--Moody Lie superalgebras, exhibiting $\mathfrak{gl}(1|1)$-nature as revealed through changing Borel subalgebras. We investigate variants of Verma modules obtained by changing Borel subalgebras, which enable us to realize the principal block of $\mathfrak{gl}(1|1)$ as an extension-closed abelian subcategory of category $\mathcal{O}$. This phenomenon is precisely formulated in terms of semibricks. On the other hand, by applying the exchange property of odd reflections, we describe compositions of homomorphisms between Verma modules associated with different Borel subalgebras that share the same character. As an application, we refine existing results on the associated varieties and projective dimensions of Verma modules.