2011/04/03 by V. V. Bavula, Bavula, V. V.
Mathematics · #16D60 #16S32 #FOS: Mathematics #Representation Theory (math.RT) #Rings and Algebras (math.RA) #math.RA #math.RT #msc:16D60 #msc:16S32
paper · pdf · doi:10.48550/arxiv.1104.0423
10 pages
arxiv created 2011/04/03 · arxiv updated 2011/04/05
In contrast to its subalgebra An:=K<x1, ..., xn, (\der)/(\der x1), ...,(\der)/(\der xn)> of polynomial differential operators (i.e. the n'th Weyl algebra), the algebra \mIn:=K<x1, ..., xn, (\der)/(\der x1), ...,(\der)/(\der xn), ∫1, ..., ∫n> of polynomial integro-differential operators is neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that \mIn is a left (right) coherent algebra iff n=1; the algebra \mIn is a \em holonomic An-bimodule of length 3n and has multiplicity 3n, and all 3n simple factors of \mIn are pairwise non-isomorphic An-bimodules. The socle length of the An-bimodule \mIn is n+1, the socle filtration is found, and the m'th term of the socle filtration has length n\choose m2n-m. This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra \mIn is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra \mIn.