Symmetrical invariants of some modular Lie algebras of Cartan type

Abstract

Let $L$ be one of the finite dimensional Lie algebras $W_n(\mathbf{m}),$ $S_n(\mathbf{m}),$ $H_n(\mathbf{m})$ of Cartan type over an algebraically closed field of prime characteristic $p>0.$ For an elements $F$ of the symmetrical algebra $S(L)$ we found necessary and sufficient condition in order to the element $ad({\partial }_1{)}^{p^{m_1}-1}ad({\partial }_2{)}^{p^{m_2}-1}...ad({\partial }_n{)}^{p^{m_n}-1}(F)$ belongs to the symmetrical invariants algebra $S(L{)}^L.$ Also, for $p=3,5$ the algebra of symmetrical invariants $S(H_2{)}^{H_2}$ is calculated in explicit way.

7 pages in Ukrainian