On the multi-Koszul property for connected algebras.

In this article we introduce the notion of emphmulti-Koszul algebra for the case of a locally finite dimensional nonnegatively graded connected algebra, as a generalization of the notion of (generalized) Koszul algebras defined by R. Berger for homogeneous algebras. This notion also extends and generalizes the one recently introduced by the author and A. Rey, which was for the particular case of algebras further assumed to be finitely generated in degree 1 and with a finite dimensional space of relations. The idea of this new notion for this generality, which should be perhaps considered as a probably interesting common property for several of these algebras, was to find a grading independent description of some of the more appealing features shared by all generalized Koszul algebras. It includes several new interesting examples, textite.g. the super Yang-Mills algebras introduced by M. Movshev and A. Schwarz, which are not generalized Koszul or even multi-Koszul for the previous definition given by the author and Rey in any natural manner. On the other hand, we provide an equivalent description of the new definition in terms of the textrmTor (or textrmExt) groups, similar to the existing one for homogeneous algebras, and we show that several of the typical homological computations performed for the generalized Koszul algebras are also possible in this more general setting. In particular, we give a very explicit description of the A∞​-algebra structure of the Yoneda algebra of a multi-Koszul algebra, which has a similar pattern as for the case of generalized Koszul algebras. We also show that a finitely generated multi-Koszul algebra with a finite dimensional space of relations is a K2​ algebra in the sense of T. Cassidy and B. Shelton.