Existence of Pseudo-Superinvolutions of the First Kind

Abstract EN

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 ? 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 ? is of order 2 in the Brauer-Wall group BW(K).

We will also show that a division superalgebra ? 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 ? is of order 2 in the Brauer-Wall group BW(K).

Finally, we study the existence of pseudo-superinvolutions on central simple superalgebras ?=Mp+q(?0).