On a problem by Nathan Jacobson

Abstract

We prove a coordinatization theorem for unital alternative algebras containing 2 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.

A categories equivalence of associative bimodules

“…The description of the structure of algebras and superalgebras that contain certain finite-dimensional algebras and superalgebras has a rich history, which has important applications in representation theory and category theory (for example, see [2,3,6,7,8,9,10,11,12,14]). The classical Wedderburn Theorem says that if a unital associative algebra A contains a central simple subalgebra of finite dimension B with the same identity element, then A is isomorphic to a Kronecker product S ⊗ F B, where S is the subalgebra of the elements that commute with each b ∈ B.…”

On Malcev algebras that contain the three dimensional simple central Lie algebra

“…In the case of superalgebras, M. López-Díaz and I. Shestakov [9,8] studied the representations of simple alternative and exceptional Jordan superalgebras in characteristic 3 and through these representations, they obtained some analogues of the Kronecker Factorization Theorem for these superalgebras. Also, the first author [11] obtained analogues of the Kronecker Factorization Theorem for some central simple alternative superalgebras, where in particular the Kronecker Factorization Theorem for M (1|1) (F) answers the analogue for superalgebras of the Jacobson's problem [5], which was recently solved by the first author and I. Shestakov [12,13] in the split case. Similarly, C. Martinez and E. Zelmanov [14] obtained a Kronecker Factorization Theorem for the exceptional ten dimensional Kac superalgebra K 10 .…”