Torsion for Homological Complexes of Nonassociative Algebras with Metagroup Relations

The article is devoted to homological complexes and modules over nonassociative algebras with metagroup relations. Smashed tensor products of them are studied. Their torsions and homomorphisms are investigated.

Weak polynomial identities and their applications

Let R be an associative algebra over a field K generated by a vector subspace V. The polynomial f(x 1, . . . , xn ) of the free associative algebra K〈x 1, x 2, . . .〉 is a weak polynomial identity for the pair (R, V) if it vanishes in R when evaluated on V. We survey results on weak polynomial identities and on their applications to polynomial identities and central polynomials of associative and close to them nonassociative algebras and on the finite basis problem. We also present results on weak polynomial identities of degree three.

Homotopism of Homological Complexes over Nonassociative Algebras with Metagroup Relations

The article is devoted to homological complexes. Smashly graded modules and complexes are studied over nonassociative algebras with metagroup relations. Smashed tensor products of homological complexes are investigated. Their homotopisms and homologisms are scrutinized.

Reductive homogeneous spaces and nonassociative algebras

The purpose of these survey notes is to give a presentation of a classical theorem of Nomizu [21] that relates the invariant affine connections on reductive homogeneous spaces and nonassociative algebras.