PBW-deformations Of A Class Of K₂ Algebras

As a generalization of a Koszul algebra and a d-Koszul algebra,the notion of a K2 algebra was introduced by Cassidy and Shelton in 2007.Different from a Koszul algebra and a d-Koszul algebra,a K2 algebra is allowed to have homogenous defining relations in different degrees.The notion of an A-infinity algebra was introduced by Stasheff in 1963.The Koszul property and d-Koszul property of algebras can be characterized by the A-infinity algebra structures on their Yoneda algebras.In 2011,Conner and Gotez defined a class of interesting K2 algebras Bn(n ?N)and investigated the A-infinity algebra structures on their Yoneda algebras.For Koszul algebras and d-Koszul algebras,Braverman et al.successively obtained some suficient and necessary conditions for determining PBW-deformations.Moreover,for any d-Koszul algebra A,Fl(?)ystad and Vatne also gave a one-to-one correspondence between augmented PBW-deformations of A and A-infinity algebra structures on its Yoneda algebra E(A)with certain properties.For a general connected graded algebra,Cassidy and Shelton established a broader sufficient and necessary condition for determining PBW-deformations,which is called Jacobi condition including an important homological invariant—complexity of an algebra.In this dissertation,we mainly focus on the class of K2 algebras Bnn(n ?N)defined by Conner and Gotez.By using the minimal projective resolution for the trivial module of Bn we firstly compute the complexity of Bn.Then in the framework of PBW-deformation theory of connected graded algebras established by Cassidy and Shelton,we explicitly characterize all the PBW-deformations of K2 algebras and B2,and present some concrete examples.