| Koszul algebras have pretty properties and are very important. However, only very limited special classes of Koszul algebras are known. Thus, it’s very interesting to construct new Koszul algebras from a given Koszul algebra.In this paper, we consider the question:for a given Koszul algebra A, whether its dual extension algebra A(A) is Koszul algebra or not? Through a special subcategory of A(A), we get a positive answer. This paper is divided into four parts:in the first part, we introduce the background and basic concepts; in the second part, we first give preliminary knowledge then introduce the basic concepts and recall basic properties of Koszul algebras; in the third part, we introduce the dual extension algebras and n-extension algebras and their basic properties; in the fourth part, we prove that for a given Koszul algebra its dual extension algebra A(A) is also Koszul. We construct a certain full subcategory of modules generated in degree zero of its dual extension algebra which is part of the full subcategory of linear modules of the dual extension algebra. The dual extension algebra’s maximal semisimple subalgebra belongs to the full subcategory. Thus, the dual extension algebra of a Koszul algebra is also Koszul. Therefore, for a given Koszul algebra, by dual extensions, many Koszul algebras are obtained. |
