Asian Journal of Mathematics

Volume 25 (2021)

Number 5

Explicit description of generalized weight modules of the algebra of polynomial integro-differential operators $\mathbb{I}_n$

Pages: 727 – 756

DOI: https://dx.doi.org/10.4310/AJM.2021.v25.n5.a6

Authors

V. V. Bavula (Department of Pure Mathematics, University of Sheffield, United Kingdom)

V. Bekkert (Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)

V. Futorny (Instituto de Matemática e Estatística, Universidade de São Paulo, Brazil; and International Center for Mathematics, SUSTech, Shenzhen, China)

Abstract

For the algebra $\mathbb{I}_n = K {\langle x_1, \dotsc, x_n, \partial_1, \dotsc, \partial_n, \int_1, \dotsc, \int_n \rangle}$ of polynomial integrodifferential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $\mathbb{I}_n$‑modules is given. It is proven that the category of weight $\mathbb{I}_n$‑modules is semisimple. An explicit description of generalized weight $\mathbb{I}_n$‑modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight $\mathbb{I}_n$‑modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight $\mathbb{I}_n$‑modules is given. In the wild case ‘natural‘ tame subcategories are considered with explicit description of indecomposable modules. For an arbitrary ring $R$, we introduce the concept of absolutely prime $R$‑module (a nonzero $R$‑module $M$ is absolutely prime if all nonzero subfactors of $M$ have the same annihilator). It is proven that every generalized weight $\mathbb{I}_n$‑module is a unique sum of absolutely prime modules. It is also shown that every indecomposable generalized weight $\mathbb{I}_n$‑module is equidimensional. A criterion is given for a generalized weight $\mathbb{I}_n$‑module to be finitely generated.

Keywords

algebra of polynomial integro-differential operators, weight and generalized weight modules, indecomposable module, simple module, finite representation type, tame and wild

2010 Mathematics Subject Classification

16D60, 16D70, 16P50, 16U20

Received 28 May 2019

Accepted 23 June 2021

Published 6 July 2022