多项式积分-微分算子代数是多项式微分算子子代数上的完整双模
The algebra of polynomial integro-differential operators is a holonomic bimodule over the subalgebra of polynomial differential operators
摘要：
与多项式微分算子的子代数$A_n:=K<x_1, ..., x_n, \frac{\der}{\der x_1}, ...,\frac{\der}{\der x_n}>$（即$n$的第Weyl代数）不同，多项式积分-微分算子的代数$\mI_n:=K<x_1, ..., x_n, \frac{\der}{\der x_1}, ...,\frac{\der}{\der x_n}, \int_1, ..., \int_n>$既不是左Noether代数，也不是右Noether代数，而且它包含非零左理想和非零右理想的无穷多个直和.证明了$\mI_n$是左（右）凝聚代数当且仅当$n=1$ ;代数$\mI_n$是长度为$3^n$的{ \em完整$A_n$ -双模}，且有重数$3^n$，$\mI_n$的所有$3^n$单因子是两两非同构的$A_n$ -双模. $A_n$ -双模$\mI_n$的柱脚长度为$n+1$，找到了柱脚过滤，柱脚过滤的第$m$项的长度为${n\choose m}2^{n-m}$ .这一事实给出了每一个多项式积分微分算子的一个新的标准型.证明了代数$\mI_n$是极大左（或极大左）代数.左为最大左为最大左为最大左最大左为最右）代数$\mI_n$的商环。
英文摘要：
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 $n$'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 $n=1$; the algebra $\mI_n$ is a {\em holonomic $A_n$-bimodule} of length $3^n$ and has multiplicity $3^n$, and all $3^n$ simple factors of $\mI_n$ are pairwise non-isomorphic $A_n$-bimodules. The socle length of the $A_n$-bimodule $\mI_n$ is $n+1$, the socle filtration is found, and the $m$'th term of the socle filtration has length ${n\choose m}2^{n-m}$. 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 $\mI_n$.
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