International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 386468, 12 pages
doi:10.1155/2008/386468

Existence of pseudo-superinvolutions of the first kind.

Abstract

Our main purpose is to develop the theory of existence of pseudo-superinvolutions of the first kind on finite dimensional central simple associative superalgebras over K, where K is a field of characteristic not 2. We try to show which kind of finite dimensional central simple associative superalgebras have a pseudo-superinvolution of the first kind. We will show that a division superalgebra 𝒟 over a field K of characteristic not 2 of even type has pseudo-superinvolution (i.e., K-antiautomorphism J such that (dδ)J2=(−1)δdδ) of the first kind if and only if 𝒟 is of order 2 in the Brauer-Wall group BW(K). We will also show that a division superalgebra 𝒟 of odd type over a field K of characteristic not 2 has a pseudo-superinvolution of the first kind if and only if −1∈K, and 𝒟 is of order 2 in the Brauer-Wall group BW(K). Finally, we study the existence of pseudo-superinvolutions on central simple superalgebras 𝒜=Mp+q(𝒟0).