A. V. Kondrateva, M. I. Kuznetsov

National Research Lobachevsky State University (Nizhny Novgorod, Russia)

Abstract. In the paper the action of the orthogonal Lie algebra \frako (V ) on the exterior powers of a space V is considered for n-dimensional vector space V over a perfect field K of characteristic two with a given nondegenerate orthogonal. The exterior algebra is identified with the algebra of truncated polynomials in n variables. The exterior powers of V taken as modules over \frako (V ) are identified with homogeneous subspaces of non-alternating Hamiltonian Lie algebra P(n) with respect to the Poisson bracket corresponding to an orthonormal basis of the space V of variables. It is proved that the exterior powers of the standard representation for Lie algebra \frako (V ) are irreducible and pairwise nonequivalent. With respect to subalgebra so(V ), n = 2l+1 or n = 2l, there exist l pairwise nonequivalent fundamental representations in the spaces \Lambda rV , r = 1, . . . , l. All of them admit a nondegenerate invariant orthogonal form, being irreducible when n = 2l + 1. When n = 2l the representations of so(V ) in \Lambda rV , r = 1, . . . , l - 1 are irreducible and the space \Lambda lV possesses the only non-trivial proper invariant subspace M, which is a maximal isotropic subspace with respect to an invariant form. Two exceptional simple Lie subalgebras P1(6), P2(6) of P(n), of dimension 25 - 1 and 26 - 1, correspondingly, containing the submodule M, and exising only in the case of 6 variables, are found.

Key Words: perfect field of characteristic two, non-altenating Hamiltonian Lie algebras, fundamental representations

For citation: A. V. Kondrateva, M. I. Kuznetsov. Fundamental representations of orthogonal Lie algebra and new simple subalgebras of nonalternating Hamiltonian Lie algebras. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 27:1(2025), 25–33. DOI: https://doi.org/10.15507/2079-6900.27.202501.25-33

Information about the authors:

Alisa V. Kondrateva, Assistant at the Departments of Algebra, Geometry and Discrete Mathematics, National Research Lobachevsky State University (23 Gagarin Ave., Nizhny Novgorod, 603022, Russia), ORCID: https://orcid.org/0009-0006-7722-870X, alisakondr@mail.ru

Michael I. Kuznetsov, D. Sc. in Phys. and Math., Professor of the Departments of Algebra, Geometry and Discrete Mathematics, National Research Lobachevsky State University (23 Gagarin Ave., Nizhny Novgorod, 603022, Russia), ORCID: https://orcid.org/0000-0001- 9231-301X, kuznets-1349@yandex.ru