  In this article we introduce the notion of multi-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, which were in turn an extension of Koszul
algebras introduced by S. Priddy. It also extends and generalizes the
definition recently introduced by the author and A. Rey.
  In order to simplify the exposition we consider the minimal graded projective
resolution of the algebra A as a bimodule, which may be used to compute the
corresponding Hochschild (co)homology groups. This new definition includes
several new interesting examples, e.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. On the
other hand, we provide an equivalent description of the new definition in terms
of the Tor (or Ext) groups, 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 an explicit description of
the A_infinity-algebra structure of the Yoneda algebra of a multi-Koszul
algebra. We also show that a finitely generated multi-Koszul algebra with a
finite dimensional space of relations is a K_2 algebra in the sense of T.
Cassidy and B. Shelton.
