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The Asymmetry Of An Anti-Automorphism

By Anne Cortella and Jean-pierre Tignol


. The asymmetry of a nonsingular pairing on a vector space is an endomorphism of the space on which the classification of arbitrary pairings (not necessarily symmetric or skew-symmetric) is based. A general notion of asymmetry is defined for arbitrary anti-automorphisms on a central simple algebra, and conditions are given to characterize the elements which are the asymmetries of some anti-automorphism. The asymmetry is used to define the determinant of an anti-automorphism. Introduction The asymmetry of an arbitrary nonsingular pairing (not necessarily symmetric or skew-symmetric) on a finite-dimensional vector space V is an invertible endomorphism of V which is an important invariant of the pairing. It is 1 if and only if the pairing is symmetric and -1 if and only if it is skew-symmetric. This invariant was first considered by Williamson [9], and more recently by Riehm [6]. In the present paper, we determine under which conditions a linear map a # GL(V ) is the asymmetry of som..

Year: 2000
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