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 ð ¾ , where ð ¾ 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 ð ’Ÿ over a field ð ¾ of characteristic not 2 of even type has pseudo-superinvolution (i.e., ð ¾ -antiautomorphism ð ½ such that ( ð ‘‘ ð ›¿ ) ð ½ 2 = ( − 1 ) ð ›¿ ð ‘‘ ð ›¿ ) of the first kind if and only if ð ’Ÿ is of order 2 in the Brauer-Wall group BW( ð ¾ ). We will also show that a division superalgebra ð ’Ÿ of odd type over a field ð ¾ of characteristic not 2 has a pseudo-superinvolution of the first kind if and only if √ − 1 ∈ ð ¾ , and ð ’Ÿ is of order 2 in the Brauer-Wall group BW( ð ¾ ). Finally, we study the existence of pseudo-superinvolutions on central simple superalgebras ð ’œ = ð ‘€ ð ‘ + ð ‘ž ( ð ’Ÿ 0 ) .