Semiassociative algebras over a field. (English) Zbl 07841511
Summary: An associative central simple algebra is a form of a matrix algebra, because a maximal étale subalgebra acts on the algebra faithfully by left and right multiplication. In an attempt to extract and isolate the full potential of this point of view, we study nonassociative algebras whose nucleus contains an étale subalgebra bi-acting faithfully on the algebra. These algebras, termed semiassociative, are shown to be the forms of skew matrix algebras, which we are led to define and investigate. Semiassociative algebras modulo skew matrix algebras compose a Brauer monoid, which contains the Brauer group of the field as a unique maximal subgroup.
MSC:
| 17A60 | Structure theory for nonassociative algebras |
| 16K20 | Finite-dimensional division rings |
| 16K50 | Brauer groups (algebraic aspects) |
Keywords:
central simple algebras; Brauer group; nonassociative algebra; maximal subfield; deformation of matrix algebrasReferences:
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