The algebra of polynomial integro-differential operators is a holonomic bimodule over the subalgebra of polynomial differential operators
arXiv:1104.0423
Abstract
In contrast to its subalgebra $A_n:=K<x_1, ..., x_n, \frac{\der}{\der x_1}, ...,\frac{\der}{\der x_n}>$ of polynomial differential operators (i.e. the 'th Weyl algebra), the algebra $\mI_n:=K<x_1, ..., x_n, \frac{\der}{\der x_1}, ...,\frac{\der}{\der x_n}, \int_1, ..., \int_n>$ 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 $\mI_n$ is a left (right) coherent algebra iff ; the algebra $\mI_n$ is a {\em holonomic -bimodule} of length and has multiplicity , and all simple factors of $\mI_n$ are pairwise non-isomorphic -bimodules. The socle length of the -bimodule $\mI_n$ is , the socle filtration is found, and the 'th term of the socle filtration has length . This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra $\mI_n$ is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra $\mI_n$.
10 pages