Existence of Pseudo-Superinvolutions of the First Kind - Ameer Jaber (International Journal of Mathematics and Mathematical Sciences)
﻿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).