Lie algebras of triangular polynomial derivations and an isomorphism criterion for their Lie factor algebras

The ideals of the Lie algebras of unitriangular polynomial derivations are classified. An isomorphism criterion is given for the Lie factor algebras of the Lie algebras of unitriangular polynomial derivations.Comment: 33 page

The group of automorphisms of the algebra of polynomial integro-differential operators

The group \rG_n of automorphisms of the algebra \mI_n:=K..., x_n, \frac{\der}{\der x_1}, ... ,\frac{\der}{\der x_n}, \int_1, >..., \int_n> of polynomial integro-differential operators is found: \rG_n=S_n\ltimes \mT^n\ltimes \Inn (\mI_n) \supseteq S_n\ltimes \mT^n \ltimes \underbrace{\GL_\infty (K)\ltimes... \ltimes \GL_\infty (K)}_{2^n-1 {\rm times}}, \rG_1\simeq \mT^1 \ltimes \GL_\infty (K), where $S_n$ is the symmetric group, \mT^n is the $n$-dimensional torus, \Inn (\mI_n) is the group of inner automorphisms of \mI_n (which is huge). It is proved that each automorphism \s \in \rG_n is uniquely determined by the elements \s (x_i)'s or \s (\frac{\der}{\der x_i})'s or \s (\int_i)'s. The stabilizers in \rG_n of all the ideals of \mI_n are found, they are subgroups of {\em finite} index in \rG_n. It is shown that the group \rG_n has trivial centre, \mI_n^{\rG_n}=K and \mI_n^{\Inn (\mI_n)}=K, the (unique) maximal ideal of \mI_n is the {\em only} nonzero prime \rG_n-invariant ideal of \mI_n, and there are precisely $n+2$ \rG_n-invariant ideals of \mI_n. For each automorphism \s \in \rG_n, an {\em explicit inversion formula} is given via the elements \s (\frac{\der}{\der x_i}) and \s (\int_i).Comment: 27 page

Dixmier's Problem 5 for the Weyl Algebra

A proof is given to the Dixmier's 5'th problem for the Weyl algebra.Comment: 19page