Shift associative algebras

We present a comprehensive study of algebras satisfying the identity (xy) z=y (zx), named as shift associative algebras. Our research shows that these algebras are related to many interesting identities. In particular, they are related to anti-Poisson-Jordan algebras and algebras of associative type. We study algebras of associative type to be Koszul and self-dual. A basis of the free shift associative algebra generated by a countable set X was constructed. An analog of Wedderburn-Artin's theorem was established. The algebraic and geometric classifications of complex 4-dimensional shift associative algebras are given. In particular, we proved that the first non-associative shift associative algebra appears only in dimension 5.