Existence of Pseudo-Superinvolutions of the First Kind

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 dδ 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 .