Indecomposable generalized weight modules over the algebra of polynomial integro-differential operators
HTML articles powered by AMS MathViewer
- by V. Bavula, V. Bekkert and V. Futorny PDF
- Proc. Amer. Math. Soc. 146 (2018), 2373-2380 Request permission
Abstract:
For the algebra $\mathbb {I}_1= K\langle x, \frac {d}{dx}, \int \rangle$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of indecomposable, generalized weight $\mathbb {I}_1$-modules of finite length is given. Each such module is an infinite dimensional uniserial module. Ext-groups are found between indecomposable generalized weight modules; it is proven that they are finite dimensional vector spaces.References
- V. V. Bavula, Finite-dimensionality of $\textrm {Ext}^n$ and $\textrm {Tor}_n$ of simple modules over a class of algebras, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 80–82 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 229–230 (1992). MR 1139880, DOI 10.1007/BF01085496
- V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz 4 (1992), no. 1, 75–97 (Russian); English transl., St. Petersburg Math. J. 4 (1993), no. 1, 71–92. MR 1171955
- Vladimir Bavula and Viktor Bekkert, Indecomposable representations of generalized Weyl algebras, Comm. Algebra 28 (2000), no. 11, 5067–5100. MR 1785490, DOI 10.1080/00927870008827145
- V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. Lond. Math. Soc. (2) 83 (2011), no. 2, 517–543. MR 2776649, DOI 10.1112/jlms/jdq081
- V. V. Bavula, The group of automorphisms of the algebra of polynomial integro-differential operators, J. Algebra 348 (2011), 233–263. MR 2852239, DOI 10.1016/j.jalgebra.2011.09.025
- V. V. Bavula, The algebra of integro-differential operators on an affine line and its modules, J. Pure Appl. Algebra 217 (2013), no. 3, 495–529. MR 2974228, DOI 10.1016/j.jpaa.2012.06.024
- V. V. Bavula, The algebra of polynomial integro-differential operators is a holonomic bimodule over the subalgebra of polynomial differential operators, Algebr. Represent. Theory 17 (2014), no. 1, 275–288. MR 3160724, DOI 10.1007/s10468-012-9398-3
- V. V. Bavula and T. Lu, The quantum Euclidean algebra and its prime spectrum, Israel J. Math. 219 (2017), no. 2, 929–958. MR 3649612, DOI 10.1007/s11856-017-1503-1
- V. V. Bavula, The global dimension of the algebras of polynomial integro-differential operators $\mathbb {I}_n$ and the Jacobian algebras $\mathbb {A}_n$, arXiv:1705.05227.
- Viktor Bekkert, Georgia Benkart, and Vyacheslav Futorny, Weight modules for Weyl algebras, Kac-Moody Lie algebras and related topics, Contemp. Math., vol. 343, Amer. Math. Soc., Providence, RI, 2004, pp. 17–42. MR 2056678, DOI 10.1090/conm/343/06182
- Richard E. Block, The irreducible representations of the Lie algebra ${\mathfrak {s}}{\mathfrak {l}}(2)$ and of the Weyl algebra, Adv. in Math. 39 (1981), no. 1, 69–110. MR 605353, DOI 10.1016/0001-8708(81)90058-X
- Vyacheslav Futorny, Dimitar Grantcharov, and Volodymyr Mazorchuk, Weight modules over infinite dimensional Weyl algebras, Proc. Amer. Math. Soc. 142 (2014), no. 9, 3049–3057. MR 3223361, DOI 10.1090/S0002-9939-2014-12071-5
- Vyacheslav Futorny and Uma N. Iyer, Representations of $D_q({\Bbbk }[x])$, Israel J. Math. 212 (2016), no. 1, 473–506. MR 3504334, DOI 10.1007/s11856-016-1305-x
- Li Guo, Georg Regensburger, and Markus Rosenkranz, On integro-differential algebras, J. Pure Appl. Algebra 218 (2014), no. 3, 456–473. MR 3124211, DOI 10.1016/j.jpaa.2013.06.015
- Jonas T. Hartwig, Locally finite simple weight modules over twisted generalized Weyl algebras, J. Algebra 303 (2006), no. 1, 42–76. MR 2253653, DOI 10.1016/j.jalgebra.2006.05.036
- Jonas T. Hartwig, Pseudo-unitarizable weight modules over generalized Weyl algebras, J. Pure Appl. Algebra 215 (2011), no. 10, 2352–2377. MR 2793941, DOI 10.1016/j.jpaa.2010.12.015
- Volodymyr Mazorchuk and Lyudmyla Turowska, Simple weight modules over twisted generalized Weyl algebras, Comm. Algebra 27 (1999), no. 6, 2613–2625. MR 1687329, DOI 10.1080/00927879908826584
- Rencai Lü, Volodymyr Mazorchuk, and Kaiming Zhao, Simple weight modules over weak generalized Weyl algebras, J. Pure Appl. Algebra 219 (2015), no. 8, 3427–3444. MR 3320227, DOI 10.1016/j.jpaa.2014.12.003
- J. C. McConnell and J. C. Robson, Homomorphisms and extensions of modules over certain differential polynomial rings, J. Algebra 26 (1973), 319–342. MR 342566, DOI 10.1016/0021-8693(73)90026-4
- Ian Shipman, Generalized Weyl algebras: category $\scr O$ and graded Morita equivalence, J. Algebra 323 (2010), no. 9, 2449–2468. MR 2602389, DOI 10.1016/j.jalgebra.2010.02.027
Additional Information
- V. Bavula
- Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
- MR Author ID: 293812
- Email: v.bavula@sheffield.ac.uk
- V. Bekkert
- Affiliation: Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, Av. Antônio Carlos, 6627, CP 702, CEP 30123-970, Belo Horizonte-MG, Brasil
- MR Author ID: 240772
- ORCID: 0000-0002-3629-4181
- Email: bekkert@mat.ufmg.br
- V. Futorny
- Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, São Paulo, CEP 05315-970, Brasil
- MR Author ID: 238132
- Email: futorny@ime.usp.br
- Received by editor(s): January 28, 2017
- Received by editor(s) in revised form: August 27, 2017
- Published electronically: January 29, 2018
- Communicated by: Kailash Misra
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 146 (2018), 2373-2380
- MSC (2010): Primary 16D60, 16D70, 16P50, 16U20
- DOI: https://doi.org/10.1090/proc/13985
- MathSciNet review: 3778141