arXiv subjects
Publications and source records attributed to Ivan P. Shestakov.
We prove that for every natural number n there exists a natural number N(n) such that every multilinear skew-symmetric polynomial on N(n) or more variables which vanishes in the free associative algebra vanishes as well in any n-generated alternative algebra over a field of characteristic 0. Before this was proved only for a series of such polynomials constructed by the author in [7].
We prove a coordinatization theorem for unital alternative algebras containing 2 x 2 matrix algebra with the same identity element 1. This solves an old problem announced by Nathan Jacobson on the description of alternative algebras containing a generalized quaternion algebra H with the same 1, for the case when the algebra H is split. In particular, this is the case when the basic field is finite or algebraically closed.
Relations between triple Jordan systems and integrable multi-component models of the modified Korteveg--de Vries type are established. The most general model is related to a pair consisting of a triple Jordan system and a skew-symmetric bilinear operation. If this operation is a Lie bracket, then we arrive at the Lie-Jordan algebras
We prove the Freiheitssatz for the variety of generic Poisson algebras.
It is proved that the prime degenerate (-1,1) algebra constructed in [13] (the (-1,1)-monster) generates the same variety of algebras as the Grassman (-1,1)-algebra. Moreover, the same variety is generated by the Grassmann envelope of any simple nonassociative (-1,1)-superalgebra. The variety occurs to be the smallest variety of (-1,1)-algebras that contains prime nonassociative algebras. Similar results are obtained for Jordan algebras. Thus, the Jordan monster (the prime degenerate algebra constructed in [13]) and the Grassmann envelope of the prime Jordan superalgebra of vector type have the same ideals of identities. It is also shown that the Jordan monster generates a minimal variety that contains prime degenerate Jordan algebras. All the algebras and superalgebras are considered over a field of characteristic 0.
The nilpotency degree of a relatively free finitely generated associative algebra with the identity $x^n=0$ is studied over finite fields.
We compute the orders of free commutative Moufand loops of exponent 3 with $n\leq 7$ free generators and find embeddings of such loops into a loop of invertible elements of the free commutative alternative algebra with identity $x^3=0$.