2009/12/13 by V. V. Bavula, Bavula, V. V.
Mathematics · #14E07 #14H37 #14R10 #14R15 #16W20 #Algebraic Geometry (math.AG) #FOS: Mathematics #Rings and Algebras (math.RA) #math.AG #math.RA #msc:14E07 #msc:14H37 #msc:14R10 #msc:14R15 #msc:16W20
paper · pdf · doi:10.48550/arxiv.0912.2537
27 pages
arxiv created 2010/11/13 · arxiv updated 2010/11/16
The group \rGn of automorphisms of the algebra \mIn:=K< x1, >..., xn, (\der)/(\der x1), ... ,(\der)/(\der xn), ∫1, >..., ∫n> of polynomial integro-differential operators is found: \rGn=Sn\ltimes \mTn\ltimes \Inn (\mIn) ⊇ Sn\ltimes \mTn \ltimes \underbrace\GL_∞ (K)\ltimes... \ltimes \GL_∞ (K)_2n-1 \rm times, \rG1≃ \mT1 \ltimes \GL_∞ (K), where Sn is the symmetric group, \mTn is the n-dimensional torus, \Inn (\mIn) is the group of inner automorphisms of \mIn (which is huge). It is proved that each automorphism \s ∈ \rGn is uniquely determined by the elements \s (xi)'s or \s ((\der)/(\der xi))'s or \s (∫i)'s. The stabilizers in \rGn of all the ideals of \mIn are found, they are subgroups of \em finite index in \rGn. It is shown that the group \rGn has trivial centre, \mIn\rGn=K and \mIn\Inn (\mIn)=K, the (unique) maximal ideal of \mIn is the \em only nonzero prime \rGn-invariant ideal of \mIn, and there are precisely n+2 \rGn-invariant ideals of \mIn. For each automorphism \s ∈ \rGn, an \em explicit inversion formula is given via the elements \s ((\der)/(\der xi)) and \s (∫i).