Central simple superalgebras with superantiautomorphism of order two of the second kind

Our main purpose is to develop the theory of existence of superantiautomorphisms of order two of the second kind (which are caled superinvolutions of the second kind) on finite dimensional central simple superalgebras A=Mn(D), where D is a finite dimensional division superalgebra with nontrivial grading over K, where K is a field of any characteristic. We determine which finite dimensional central simple superalgebras posses a superinvolution of the second kind and put these results in the context of the Albert-Reihm Theorem on the existence of involutions of the second kind.

Anahtar Kelimeler:

Central simple superalgebras, Superantiautomorphisms, Superinvolutions, Brauer-Wall Groups

Central simple superalgebras with superantiautomorphism of order two of the second kind

Keywords:

Central simple superalgebras, Superantiautomorphisms, Superinvolutions, Brauer-Wall Groups,

PDF

___

Jaber, A. : Existence of Pseudo-Superinvolutions of The First Kind, International Journal of Mathematics and Mathematical Sciences, vol. 2008, Article ID 386468, 12 pages, 2008.
Jaber, A. : Superinvolutions of Associative Superalgebras, P.h. D. thesis, University of Ottawa, Ottawa, Canada, Knus, M.-A., Merkurjev, A., Rost, M. and Tignol, J.-P. : The Book of Involutions, Amer. Math. Soc., 1998.
Lam, T. Y. : The Algebraic Thoery of Quadratic Forms, the Benjamin/Cummings Publishing Company, 1973.
Lewis, D. W. : Involutions and Anti-automorphisms of Algebras, London Math. Soc. 38, 529-545(2006).
Racine, M. L. : Primitive superalgebras with superinvolution, J. Algebra 206, 588-614(1998).
Racine, M. L. and Zelmanov, E.I. : Simple Jordan superalgebras with semisimple even part, J. Algebra 270, (2003)374-444.
Riehm, C. : The Corestriction of Algebriac Structure, Invent. Math. 11, 73-98(1970).
Scharlau, W. : Quadratic and Hermitian Forms, Springer-Verlag, Heidelberg, 1985. Ameer JABER
Department of Mathematics, The Hashemite University Zarqa, 13115, JORDAN e-mail: ameerj@hu.edu.jo