Kronecker factorization theorems for alternative superalgebras

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.…”

“…In particular, if A contains M n (F ) as a subalgebra with the same identity element, we have A ∼ = M n (S) "coordinated" by S. Kaplansky in Theorem 2 of [5] B. Jacobson in Theorem 1 of [2] gave a new proof of the result of Kaplansky using his classification of completely reducible alternative bimodules over a field of characteristic different of 2 and finally V. López-Solís in [8] proved that this result is valid for any characteristic. Using this result, Jacobson [2] proved a Kronecker Factorization Theorem for Jordan algebras that contain the Albert algebra with the same identity element.…”

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

“…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. In particular, if A contains M n (F) as subalgebra with the same identity element, we have A ∼ = M n (S) "coordinated" by S. Kaplansky in Theorem 2 of [7] generalized the Wedderburn result to the alternative algebras A and the split Cayley algebra B. Jacobson in Theorem 1 of [5] gave a new proof of the result of Kaplansky using his classification of completely reducible alternative bimodules over a field of characteristic different of 2 and finally the first author in [11] proved that this result is valid for any characteristic. Using this result, Jacobson [5] proved a Kronecker Factorization Theorem for Jordan algebras that contain the Albert algebra with the same identity element.…”

“…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 .…”