On 2-Nakayama Algebras And Almost Kosuzlity

Recently,Iyama established and developed high-dimensional algebraic representation theory or high-dimensional Auslander-Reiten theory[27-33].In the context of high-dimensional rep-resentation theory,this paper mainly discusses two aspects:First,we study the almost Koszul properties of 2-Nakayama algebras,the dual Koszul dual of 2-Nakayama algebras and the twisted trivial extension algebras of the Koszul dual of 2-Nakayama algebras.The other is to discuss the relation between Loewy matrix and almost Koszul algebra.The details are as follows:We first restate the definition of high-dimensional Nakayama algebras,We prove that the quiver of the Koszul dual algebra of d-Nakayama algebras is a connected,d-cubic closed d-cubic quiver,and the necessary condition of the quiver of the Koszul dual algebra of d-Nakayama algebra to be(d-1)-translation quiver is given.We obtain the minimal projective resolution of the simples of the 2-Nakayama algebra and its Koszul dual algebra,and from this we can get that the Koszul dual of 2-Nakayama algebra is a piecewise Koszul algebra,and all its simples S(i,j)are all(2,l_j-1)-Koszul simples.We discuss the minimal projective resolution of simples of each point on the twisted trivial extension algebra of the Koszul dual of the 2-Nakayama algebra with(?)=(3,3,2,1)as Kupisch series,we find that the twisted trivial extension algebras of Koszul duals of d-Nakayama algebras are stable d-pre-translational algebras,but not necessarily almost Koszul algebras.In the last chapter,we discuss Loewy matrix and d-linear module,Loewy matrix and almost Koszul algebra,and use Loewy matrix to give a necessary and sufficient condition that a graded algebra is almost Koszul algebra...