Fundamental representations of orthogonal Lie algebra and new simple subalgebras of nonalternating Hamiltonian Lie algebras

In the paper the action of the orthogonal Lie algebra $\mathfrak{o}(V)$ 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 $\mathfrak{o}(V)$ 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 $\mathfrak{o}(V)$ 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 $\Lambda^{r}V$, $r= 1, \ldots, l$. All of them admit a nondegenerate invariant orthogonal form, being irreducible when $n= 2l+1$. When $n= 2l$ the representations of $so(V)$ in $\Lambda^{r}V$, $r= 1, \ldots, l-1$ are irreducible and the space $\Lambda^{l}V$ 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 $P_{1}(6)$, $P_{2}(6)$ of $P(n)$, of dimension $2^{5}-1$ and $2^{6}-1$, correspondingly, containing the submodule $M$, and exising only in the case of $6$ variables, are found.