Hodge Operators and Exceptional Isomorphisms between Unitary Groups

We give a generalization of the Hodge operator to spaces \((V,h)\) endowed with a hermitian or symmetric bilinear form \(h\) over arbitrary fields, including the characteristic two case. Suitable exterior powers of \(V\) become free modules over an algebra \(K\) defined using such an operator. This...

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Bibliographic Details
Published in:arXiv.org 2022-09
Main Authors: Kramer, Linus, Stroppel, Markus J
Format: Article
Language:English
Subjects:
Algebra
Group theory
Homomorphisms
Isomorphism
Operators (mathematics)
Online Access:Get full text
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Summary:We give a generalization of the Hodge operator to spaces \((V,h)\) endowed with a hermitian or symmetric bilinear form \(h\) over arbitrary fields, including the characteristic two case. Suitable exterior powers of \(V\) become free modules over an algebra \(K\) defined using such an operator. This leads to several exceptional homomorphisms from unitary groups (with respect to \(h\)) into groups of semi-similitudes with respect to a suitable form over some subfield of \(K\). The algebra \(K\) depends on \(h\); it is a composition algebra unless \(h\) is symmetric and the characteristic is two.
ISSN:2331-8422