# PBW deformations of graded algebras

SPEAKER: Brad Shelton

TITLE: PBW deformations of graded algebras (joint work with T. Cassidy).

ABSTRACT We consider deformations U of a graded algebra A in which the relations of A have been deformed by the addition of lower degree terms. Such a nonhomogeneous deformation is a PBW deformation if the associated graded algebra grU is isomorphic to A. There are many special situations (such as Koszul algebras and so-called N-Koszul algebras) where it is known that the PBW property can be inferred from a finite amount of linear algebra. We prove a completely general theorem about this situation in terms of what we call the complexity of the algebra A. In particular our theorem applies to graded algebras with defining relations in more than one degree.