On an Embedding Theorem for Filtered Deformations of Graded Nonalternating Hamiltonian Lie Algebras

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.