Odd Verma's Theorem
Shunsuke Hirota
TL;DR
This work develops a unifying framework for Verma modules over regular symmetrizable Kac–Moody Lie superalgebras by exploiting odd reflections and edge-colored graphs. It shows that the local interaction of Verma modules across adjacent Borel subalgebras is governed by a $gl(1|1)$-type structure encoded via semibricks, and it builds a categorified picture of the odd reflection graph OR(g) that connects Verma homomorphisms to combinatorial walks. Central constructions include adjusted Borels, hypercube decompositions, and realizations of the principal block of $\mathcal{O}$ for $\mathfrak{gl}(1|1)^{\oplus n}$ as filtrations of semibrick-closures, culminating in a Morita equivalence to the infinite zigzag algebra $K_1^\infty$. The framework yields refined statements about associated varieties and projective dimensions across type I and certain exceptional cases, aided by Duflo–Serganova functors. Overall, the paper provides a cohesive, structural view of Verma-module networks under odd reflections and their representation-theoretic consequences.
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.