Journal of Lie Theory 33 (2023), No. 1, 329--360
Copyright Heldermann Verlag 2023
Hodge Operators and Exceptional Isomorphisms between Unitary Groups
Linus Kramer
Mathematisches Institut, Fachbereich Mathematik und Informatik, Universität Münster, Germany
linus.kramer@uni-muenster.de
Markus J. Stroppel
Mathematische Fakultät, Universität Stuttgart, Germany
stroppel@mathematik.uni-stuttgart.de
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.
Keywords: Hermitian form, symmetric bilinear form, exterior product, Pfaffian form, Hodge operator, exceptional isomorphism, composition algebra, quaternion algebra.
MSC: 20G15, 20E32, 20G20, 20G40, 22C05, 11E39, 11E57.
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