On symmetric invariants of some modular Lie algebras. (English. Russian original) Zbl 0821.17021

Bedratyuk, L. P.

Russ. Acad. Sci., Sb., Math. 80, No. 1, 125-135 (1995); translation from Mat. Sb. 184, No. 9, 149-160 (1993).

For Lie algebras \(L\) of Cartan type \(W(n : {\mathbf m})\), \(S(n : {\mathbf m})\), \(H^ 0 (n : {\mathbf m})\) over a field of characteristic \(p\) the structure of invariants of the symmetric algebra \(S(L)\) is investigated. Applying the inclusion \(S(L)^ L \subset S(L)^{L_{-1}}\) and a special realization of \(L^*\) in the space of differential forms the author aims at proving the fact that any nontrivial invariant is the leading coefficient of some polynomial on \(L^*\). This follows from the assertion (Theorem 1) that any nontrivial \(L_{-1}\)-invariant in \(S(L)\) is of the form \(\partial^{\delta}y\) for some \(y \in S(L)\) where \(\partial^{\delta} = \partial^{\delta_ 1}_ 1 \cdots \partial^{\delta_ n}_ n\), \(\delta_ i = p^{m_ i} -1\).

Reviewer: M.Kuznetsov (Nizhnij Novgorod)

MSC:

17B50	Modular Lie (super)algebras
17B35	Universal enveloping (super)algebras

Keywords:

Lie algebra of Cartan type; symmetric invariant