On Category $\mathcal{O}$ over triangular Generalized Weyl Algebras

TL;DR

This paper studies the structure and homological properties of Category  over a broad class of triangular generalized Weyl algebras, revealing their quasi-hereditary, Koszul, and combinatorial features.

Contribution

It provides a detailed analysis of Category  over triangular GWAs, including classification of modules and a new combinatorial connection with Young tableaux.

Findings

Endomorphism algebra of projective generators is quasi-hereditary and Koszul.

Classified all tilting modules and submodules within the blocks.

Established a novel link between blocks of GWAs and Young tableaux.

Abstract

We analyze the BGG Category $\mathcal{O}$ over a large class of generalized Weyl algebras (henceforth termed GWAs). Given such a "triangular" GWA for which Category $\mathcal{O}$ decomposes into a direct sum of subcategories, we study in detail the homological properties of blocks with finitely many simples. As consequences, we show that the endomorphism algebra of a projective generator of such a block is quasi-hereditary, finite-dimensional, and graded Koszul. We also classify all tilting modules in the block, as well as all submodules of all projective and tilting modules. Finally, we present a novel connection between blocks of triangular GWAs and Young tableaux, which provides a combinatorial interpretation of morphisms and extensions between objects of the block.