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 , where 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 of characteristic not 2 of even type has pseudo-superinvolution (i.e., -antiautomorphism such that of the first kind if and only if is of order 2 in the Brauer-Wall group BW(). We will also show that a division superalgebra of odd type over a field of characteristic not 2 has a pseudo-superinvolution of the first kind if and only if and is of order 2 in the Brauer-Wall group BW(). Finally, we study the existence of pseudo-superinvolutions on central simple superalgebras .
Highlights
Let K be a field of characteristic not 2
Finite dimensional central simple associative superalgebras over a field K are isomorphic to End V ∼ Mn D, where D D0 D1 is a finite dimensional associative division superalgebra over K, that is, all nonzero elements of Dα, α 0, 1, are invertible, and V V0 V1 is an n-dimensional D superspace
Throughout my work on the existence of superinvolutions of the first kind which has yet to appear, we prove that finite dimensional central simple division superalgebras of odd or even type with nontrivial grading over a field K of characteristic not 2 have no superinvolutions of the first kind, these results were introduced in 2, Proposition 9, 3
Summary
Let K be a field of characteristic not 2. We recall a theorem of Albert which shows that a finite dimensional central simple algebra over a field k has an involution of the first kind if and only if it is of order 2 in the Brauer group Br k. The proof of this classical theorem is in many books of algebra, for example, see 1, Chapter 8, Section 8. Throughout my work on the existence of superinvolutions of the first kind which has yet to appear, we prove that finite dimensional central simple division superalgebras of odd or even type with nontrivial grading over a field K of characteristic not 2 have no superinvolutions of the first kind, these results were introduced in 2, Proposition 9 , 3. By theorem of Albert, A has an involution of the first kind, but since A is of order 2 in the Brauer-Wall group BW K , A has an antiautomorphism of the first kind respecting the grading, by 1, Chapter 8, Theorem 8.2 , A has an involution of the first kind respecting the grading, which means that A has a superinvolution which is a pseudo-superinvolution of the first kind if and only if A is of order 2 in the Brauer-Wall group BW K
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