
Journal of Lie Theory 33 (2023), No. 1, 329360 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@unimuenster.de Markus J. Stroppel Mathematische Fakultät, Universität Stuttgart, Germany stroppel@mathematik.unistuttgart.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 semisimilitudes 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. [ Fulltextpdf (981 KB)] for subscribers only. 