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International Journal of Mathematics and Mathematical Sciences
Volume 2008, Article ID 386468, 12 pages
http://dx.doi.org/10.1155/2008/386468
Research Article

Existence of Pseudo-Superinvolutions of the First Kind

Department of Mathematics, The Hashemite University, Zarqa 13115, Jordan

Received 6 March 2007; Revised 8 July 2007; Accepted 2 November 2007

Academic Editor: Alexander Rosa

Copyright © 2008 Ameer Jaber. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 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 π’Ÿ over a field 𝐾 of characteristic not 2 of even type has pseudo-superinvolution (i.e., 𝐾 -antiautomorphism 𝐽 such that ( 𝑑 𝛿 ) 𝐽 2 = ( βˆ’ 1 ) 𝛿 𝑑 𝛿 ) of the first kind if and only if π’Ÿ is of order 2 in the Brauer-Wall group BW( 𝐾 ). We will also show that a division superalgebra π’Ÿ of odd type over a field 𝐾 of characteristic not 2 has a pseudo-superinvolution of the first kind if and only if √ βˆ’ 1 ∈ 𝐾 , and π’Ÿ is of order 2 in the Brauer-Wall group BW( 𝐾 ). Finally, we study the existence of pseudo-superinvolutions on central simple superalgebras π’œ = 𝑀 𝑝 + π‘ž ( π’Ÿ 0 ) .