On an embedding theorem for filtered deformations of graded nonalternating Hamiltonian Lie algebras

A. V. Kondrat'eva ; M. I. Kuznetsov

Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 26 (2024) no. 4, pp. 392-403.

Résumé

It is proved that for graded non-alternating Hamiltonian Lie algebras over a perfect field of characteristic two corresponding to a flag of the variables’ space the condition of the embedding theorem of filtered deformations is fulfilled. The group of one-dimensional homology of the first member of the standard filtration for a graded non-alternating Hamiltonian Lie algebra is described. In the case when the number of variables $n\neq 4$, the estimate is obtained for multiplicity of the standard module over an orthogonal Lie algebra in a composition series of the homology group with respect to the natural structure of a module over the null-member of the grading. For $n=4$ the estimate is true if a set of variables coordinated with the flag contains a variable of height greater than 1 which is non-isotropic with respect to Poisson bracket, corresponding to the non-alternating Hamiltonian form. The homology computation employs the normal shape of non-alternating Hamiltonian form, corresponding to its class of equivalence. The monomials of the divided power algebra included into the commutant of the filtration’s first member are found. The multiplicity of the standard module over an orthogonal Lie algebra in a composition series of the first member of grading of the homology group is calculated. This calculation is based on the structure of weights with respect to a special maximal torus of the $p$-closure of the null-member of the standard grading in the Lie algebra of linear operators acting on the negative part of the grading of a non-alternating Hamiltonian Lie algebra.

Détail

Keywords: perfect field of characteristic two, non-altenating Hamiltonian Lie algebras, filtered deformations, embedding theorem

Bibliographie

[1] H. Strade, Simple Lie algebras over fields of positive characteristic. I: Structure theory. de Gruyter Expositions in Mathematics, Walter de Gruyter Co., Berlin, 2004 | DOI | MR

[2] H. Strade, Simple Lie algebras over fields of positive characteristic. II: Classifying the absolute toral rank two case. de Gruyter Expositions in Mathematics, Walter de Gruyter Co., Berlin, 2009 | MR

[3] H. Strade, Simple Lie algebras over fields of positive characteristic. III: Completion of the classification. de Gruyter Expositions in Mathematics, Walter de Gruyter Co., Berlin, 20013 | MR

[4] M. I. Kuznetsov, “Usechennye indutsirovannye moduli nad tranzitivnymi algebrami Li kharakteristiki $p$”, Izv. AN SSSR. Ser.matem., 34:3 (1990), 575–608 (In Russ.) | MR | Zbl

[5] M. I. Kuznetsov, “Teorema vlozheniya dlya tranzitivnykh filtrovannykh algebr Li kharakteristiki $p$”, Izv. vuzov. Matem., 10 (1991), 43–45 (In Russ.) | Zbl

[6] A. V. Kondrateva, M. I. Kuznetsov, “Filtrovannye deformatsii graduirovannykh nealterniruyuschikh gamiltonovykh algebr Li”, Izv. vuzov. Matem., 9 (2024), 100–105 (In Russ.) | DOI

[7] A. I. Kostrikin, I. R. Shafarevich, “Graded Lie algebras of finite characteristic”, Math. USSR-Izv., 33:2 (1969), 251–322 (In Russ.) | MR

[8] M. I. Kuznetsov, A. V. Kondrateva, N. G. Chebochko, “O gamiltonovykh algebrakh Li kharakteristiki 2”, Matematicheskii zhurnal, 16:2 (2016), 54–65 (In Russ.) | Zbl

[9] A. V. Kondrateva, M. I. Kuznetsov, “Nealterniruyuschie gamiltonovy formy nad algebroi razdelennykh stepenei v kharakteristike 2”, Izv. vuzov. Matem., 67:6 (2023), 82–87 (In Russ.) | DOI | MR | Zbl

[10] A. V. Kondrateva, “Non-alternating Hamiltonian Lie algebras in three variables”, Lobachevskii Journal of Mathematics, 42:12 (2021), 2841–2853 | DOI | MR | Zbl