Regularity conditions on skew and symmetric elements in superalgebras with superinvolution
One of the methodologies in the study of associative algebras with involution is to get results on the algebra from properties on the set of symmetric elements (or from the set of skew-symmetric elements). In some cases the algebraic properties of the set \(H\) of symmetric elements or the set \(K\) of skew-symmetric elements, determine the algebraic structure of the algebra. Many researchers have adhered to this philosophy along the history of modern algebra. J. M. Osborn, for instance, proved in 1967 that if \(A\) is a simple algebra of characteristic not 2 in which every nonzero symmetric element is invertible, then \(A\) must either be a division algebra or the algebra of \(2\times 2\) matrices over a field. Later, he generalized this to the case of a semisimple algebra \(A\) which is \(2\)-torsion free. If every symmetric element in \(A\) is either nilpotent or invertible, then \(A\) is either a division algebra or a direct sum of a division algebra and its opposite, or a \(2 \times 2\) matrices algebra over a field. In the work under review the authors are interested in studying Osborn's theorems (and related works concerning the skew-symmetric elements) in associative superalgebras with superinvolution. More precisely they prove that if \(A\) is a semiprime superalgebra with superinvolution \(*\) such that every nonzero homogeneous symmetric element is invertible then either \(A\) is a division superalgebra or \(A=B\oplus B^*\) where \(B\) is a division superalgebra and the superinvolution in \(A\) is the exchange superinvolution. They also prove the following result on nonsemiprime superalgebras: let \(A\) be a superalgebra with superinvolution such that every nonzero homogeneous symmetric element is invertible. If \(A\) has a nonzero nilpotent ideal then there exists a maximal nilpotent ideal \(N\) such that \(N^6=0\) and \(A/N\) is either a field of the direct sum of two copies of a field. Some other interesting results include skew versions of Osborn's theorems. First, if \(A\) is a semiprime superalgebra with superinvolution in which every nonzero homogeneous skew-symmetric element is invertible, and \(A_0\) is noncommutative then either \(A\) is a division superalgebra of \(A=B\oplus B^*\) with \(B\) a division superalgebra and the superinvolution in \(A\) is the exchange superinvolution. The following is also proved: let \(A\) be a superalgebra with superinvolution such that every nonzero homogeneous skew-symmetric element is invertible. Suppose that the maximal nil ideal \(N\) of \(A\) (its nil radical) is not zero and \(A_0\) is noncommutative. Then (i) \(A/N\) is a trivial superalgebra and it is either a field or the direct sum of two copies of a field with the exchange involution. (ii) \(N^4=0\). The final result in the paper claims that if \(A\) is a superalgebra with superinvolution and \(J(A)\) its Jacobson radical, then if either (i) every nonzero homogeneous symmetric element is invertible or nilpotent, or (ii) every nonzero homogeneous skew-symmetric element is invertible or nilpotent, then \(A/J(A)\) is either a division superalgebra or the direct sum of a division superalgebra and its opposite, or is \(M_{1,1}(F)\) with the symplectic superinvolution.
associative superalgebra with superinvolution
scientific article; zbMATH DE number 5716390
scientific article