121 research outputs found
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 is the symmetric group, \mT^n is the -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
\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
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