Relationship Between Symmetric and Skew-Symmetric Bilinear Forms on V=kn and Involutions of SL(n,k) and SO(n,k,beta)
No Thumbnail Available
Files
Date
2003-10-22
Authors
Journal Title
Series/Report No.
Journal ISSN
Volume Title
Publisher
Abstract
In this paper, we show how viewing involutions on matrix groups as having been induced by a given non-degenerate symmetric or skew-symmetric bilinear form on the vector space of corresponding dimension can lead to a classification up to isomorphism of the resulting reductive symmetric space in the group setting. We establish a direct link between bilinear algebraic properties of the vector space V=kˆn for an arbitrary field k of characteristic not 2 and involutions of the matrix group G, where G is a subgroup of GL(n,k). Symmetric spaces are defined in terms of involutions, and the development of this classification theory which classifies the involutions also classifies the symmetric spaces coming from these involutions.
We classify all involutions on SL(n,k) and develop important foundations for a full classification of the involutons of SO(n,k,beta) where beta is any non-degenerate symmetric bilinear form. We prove that all involutions of SO(n,k,beta) are inner when n is odd. Additionally, we provide criteria for the matrix which gives the conjugation that is the inner involution of SO(n,k,beta), which covers all involutions when n is odd and all involutions which can be written as conjugations when n is even, a fact which is proven in this thesis.
Description
Keywords
involutions, Lie groups, bilinear forms, symmetric spaces
Citation
Degree
PhD
Discipline
Mathematics
