In the context of Kac-Moody Lie superalgebras and modular Kac-Moody Lie algebras, it has long been known that the choice of Borel subalgebra is not essentially unique. The Weyl groupoids introduced by Heckenberger and Yamane (2008) provide a unified generalization of such “pseudo root systems” and frieze patterns, offering a Lie-theoretic framework for the classification of Nichols algebras of diagonal type, which are universal objects in the theory of Hopf algebras.

In this talk, I present a natural “odd reflection version” of Verma’s theorem—which is well known for describing homomorphisms between Verma modules in the Lie algebra setting—obtained in the context of finite-dimensional Kac-Moody Lie superalgebras and, more generally, for Nichols algebras of diagonal type, by utilizing properties of the Weyl groupoid. Applications include criteria for the finiteness of the projective dimension of Verma modules, as well as brick decompositions of Verma modules in terms of quivers and relations.