Involution Polynomial Identities Symmetric Variables Skew-Symmetric Variables Bergman Type Polynomials Characters Hilbert Series Growth Codimensions
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Serdica Mathematical Journal, Vol. 30, No 2-3, (2004), 239p-282p
The paper is a survey on involutions (anti-automorphisms of order two) of different kinds. Starting with the first systematic investigations
on involutions of central simple algebras due to Albert the author emphasizes on their basic properties, the conditions on their existence and their correspondence with structural characteristics of the algebras.
Focusing on matrix algebras a complete description of involutions of the
first kind on Mn(F) is given. The full correspondence between an involution of any kind for an arbitrary central simple algebra A over a field F of characteristic 0 and an involution on Mn(A) specially defined is studied.
The research mainly in the last 40 years concerning the basic properties of
involutions applied to identities for matrix algebras is reviewed starting with
the works of Amitsur, Rowen and including the newest results on the topic.
The cocharactes, codimensions and growth of algebras with involutions are
considered as well.