Abstract
In contrast to its subalgebra \(A_n:=K\langle x_1, \ldots , x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}\rangle \) of polynomial differential operators (i.e. the n’th Weyl algebra), the algebra \({\mathbb{I}}_n:=K\langle x_1, \ldots ,\) \( x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}, \int_1, \ldots , \int_n\rangle \) 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 \({\mathbb{I}}_n\) is a left (right) coherent algebra iff n = 1; the algebra \({\mathbb{I}}_n\) is a holonomic A n -bimodule of length 3n and has multiplicity 3n with respect to the filtration of Bernstein, and all 3n simple factors of \({\mathbb{I}}_n\) are pairwise non-isomorphic A n -bimodules. The socle length of the A n -bimodule \({\mathbb{I}}_n\) 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 \({\mathbb{I}}_n\) is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra \({\mathbb{I}}_n\).
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Bavula, V.V. The Algebra of Polynomial Integro-Differential Operators is a Holonomic Bimodule over the Subalgebra of Polynomial Differential Operators. Algebr Represent Theor 17, 275–288 (2014). https://doi.org/10.1007/s10468-012-9398-3
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DOI: https://doi.org/10.1007/s10468-012-9398-3