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On symmetric invariants of some modular Lie algebras. (English. Russian original) Zbl 0821.17021
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 $$\partial^{\delta} = \partial^{\delta_ 1}_ 1 \cdots \partial^{\delta_ n}_ n$$, $$\delta_ i = p^{m_ i} -1$$.
##### MSC:
 17B50 Modular Lie (super)algebras 17B35 Universal enveloping (super)algebras
##### Keywords:
Lie algebra of Cartan type; symmetric invariant
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