Odd Verma's Theorem

Abstract: 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 realize the module category of certain finite-dimensional algebras containing Verma modules, such as the preprojective algebra of type ( A_2 ), as extension-closed abelian subcategories of category ( \mathcal{O} ). This allows us to establish lower bounds on the length of socle of Verma modules in terms of combinatorial invariants of graphs, such as finite Young lattices. In addition, we investigate variants of Verma modules obtained by changing Borel subalgebras. These variants enable us to realize the principal block of ( \mathfrak{gl}(1|1) ) as an extension-closed abelian subcategory of category ( \mathcal{O} ). We further apply the exchange property of odd reflections to refine existing results on the associated varieties and projective dimensions of Verma modules. As explained in our companion manuscript, many of our results extend to the setting of regular symmetrizable Kac Moody Lie superalgebras and Nichols algebras of diagonal type, in view of the Weyl groupoids framework in the sence of Heckenberger and Yamane.