Classification And Construction Of PBW-deformations Of Connected Graded Algebras

Connected graded algebras are a class of fundamental and basic algebraic objects,and they extensively appear in many branches of algebra such as noncommutative algebras,Hopf algebras and algebra representation theory.In recent thirties years,the PBW-deformation theory of connected graded algebras is extensively focused and investigated.In particular,the algebras that have good homological properties such as Koszul algebras,d-Koszul al-gebras have been deeply studied.The methods arising from A-infinity algebra theory are proved to be very valid in processing some problems about connected graded algebras.For Koszul algebras and d-Koszul algebras,there exists a one-to-one corresponding duality be-tween their augmented PBW-deformations and some A∞-structures on their Ext-algebras,which can be considered as a classification result of all augmented PBW-deformations of Koszul algebras and d-Koszul algebras via A-infinity algebras.For general connected graded algebras,Cassidy and Shelton generalized the results about Koszul algebras and d-Koszul algebras,and established the more extensive Jacobi condition,which is a sufficient and nec-essary condition for determining PBW-deformations.Motivated by the above facts,we focus on the classification and construction of PBW deformations of connected graded algebra in this present paper.On one hand,we investigate the PBW-deformations of multi-Koszul algebras,which is an important generalization of the notions of Koszul algebra and d-Koszul algebra.By using PBW-deformation theory of connected graded algebras,we prove that all the PBW-deformations of a restricted multi-Koszul algebra A can be given by the PBW-ones of some d-Koszul algebras induced from A.As an application,we obtain a one-to-one corre-sponding duality between all the augmented PBW-deformations of A and the A∞-structures on its Ext-algebra.On the other hand,we introduce the notion of PBW-deformations with different levels of connected graded algebras.Then by using the Jacobi condition for deter-mining PBW-deformations,we construct all the PBW-deformations of level 1 for a connected graded algebra with two generators.