PBW deformations of quadratic monomial algebras

TL;DR

This paper provides an efficient method, including graphical tools, to determine PBW deformations of quadratic monomial algebras, extending previous theoretical results and demonstrating that all such algebras admit nontrivial PBW deformations.

Contribution

It establishes equivalent conditions to the Braverman-Gaitsgory Theorem for quadratic monomial algebras and introduces graphical interpretations to simplify the verification process.

Findings

Graphical interpretation of PBW deformation conditions

Conditions for PBW deformations can often be simplified

Every quadratic monomial algebra admits a nontrivial PBW deformation

Abstract

A result of Braverman and Gaitsgory from 1996 gives necessary and sufficient conditions for a filtered algebra to be a Poincar\'e-Birkhoff-Witt (PBW) deformation of a Koszul algebra. The main theorem in this paper establishes conditions equivalent to the Braverman-Gaitsgory Theorem to efficiently determine PBW deformations of quadratic monomial algebras. In particular, a graphical interpretation is presented for this result, and we discuss circumstances under which some of the conditions of this theorem need not be checked. Several examples are also provided. Finally, with these tools, it is then shown that each quadratic monomial algebra admits a nontrivial PBW deformation.