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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

Markus J. Stroppel
Mathematische Fakultät, Universität Stuttgart, Germany

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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