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.