Symplectic Differential Reduction Algebras and Generalized Weyl Algebras
Given a map Ξ: () → of associative algebras, with () the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra g, the restriction functor from -modules to ()-modules is intimately tied to the representation theory of an -subquotient known as the reduction algebra with...
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| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
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| Дата: | 2025 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2025
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| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/212890 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Symplectic Differential Reduction Algebras and Generalized Weyl Algebras. Jonas T. Hartwig and Dwight Anderson Williams II. SIGMA 21 (2025), 001, 15 pages |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860276816902619136 |
|---|---|
| author | Hartwig, Jonas T. Williams II, Dwight Anderson |
| author_facet | Hartwig, Jonas T. Williams II, Dwight Anderson |
| citation_txt | Symplectic Differential Reduction Algebras and Generalized Weyl Algebras. Jonas T. Hartwig and Dwight Anderson Williams II. SIGMA 21 (2025), 001, 15 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Given a map Ξ: () → of associative algebras, with () the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra g, the restriction functor from -modules to ()-modules is intimately tied to the representation theory of an -subquotient known as the reduction algebra with respect to (, , Ξ). Herlemont and Ogievetsky described differential reduction algebras for the general linear Lie algebra () as algebras of deformed differential operators. Their map Ξ is a realization of () in the -fold tensor product of the -th Weyl algebra tensored with (()). In this paper, we further the study of differential reduction algebras by finding a presentation in the case when is the symplectic Lie algebra of rank two, and Ξ is a canonical realization of inside the second Weyl algebra tensor the universal enveloping algebra of g, suitably localized. Furthermore, we prove that this differential reduction algebra is a generalized Weyl algebra (GWA), in the sense of Bavula, of a new type we term skew-affine. It is believed that symplectic differential reduction algebras are all skew-affine GWAs; then their irreducible weight modules could be obtained from standard GWA techniques.
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| first_indexed | 2026-03-14T20:11:16Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-212890 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T20:11:16Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Hartwig, Jonas T. Williams II, Dwight Anderson 2026-02-13T13:51:27Z 2025 Symplectic Differential Reduction Algebras and Generalized Weyl Algebras. Jonas T. Hartwig and Dwight Anderson Williams II. SIGMA 21 (2025), 001, 15 pages 1815-0659 2020 Mathematics Subject Classification: 16S15; 16S32; 17B35; 17B37; 81R05 arXiv:2403.15968 https://nasplib.isofts.kiev.ua/handle/123456789/212890 https://doi.org/10.3842/SIGMA.2025.001 Given a map Ξ: () → of associative algebras, with () the universal enveloping algebra of a (complex) finite-dimensional reductive Lie algebra g, the restriction functor from -modules to ()-modules is intimately tied to the representation theory of an -subquotient known as the reduction algebra with respect to (, , Ξ). Herlemont and Ogievetsky described differential reduction algebras for the general linear Lie algebra () as algebras of deformed differential operators. Their map Ξ is a realization of () in the -fold tensor product of the -th Weyl algebra tensored with (()). In this paper, we further the study of differential reduction algebras by finding a presentation in the case when is the symplectic Lie algebra of rank two, and Ξ is a canonical realization of inside the second Weyl algebra tensor the universal enveloping algebra of g, suitably localized. Furthermore, we prove that this differential reduction algebra is a generalized Weyl algebra (GWA), in the sense of Bavula, of a new type we term skew-affine. It is believed that symplectic differential reduction algebras are all skew-affine GWAs; then their irreducible weight modules could be obtained from standard GWA techniques. The authors thank the referees for their comments, which have informed the current paper and provided suggestions for continued research. J.T.H. is partially supported by the Army Research Office grant W911NF-24-1-0058. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Symplectic Differential Reduction Algebras and Generalized Weyl Algebras Article published earlier |
| spellingShingle | Symplectic Differential Reduction Algebras and Generalized Weyl Algebras Hartwig, Jonas T. Williams II, Dwight Anderson |
| title | Symplectic Differential Reduction Algebras and Generalized Weyl Algebras |
| title_full | Symplectic Differential Reduction Algebras and Generalized Weyl Algebras |
| title_fullStr | Symplectic Differential Reduction Algebras and Generalized Weyl Algebras |
| title_full_unstemmed | Symplectic Differential Reduction Algebras and Generalized Weyl Algebras |
| title_short | Symplectic Differential Reduction Algebras and Generalized Weyl Algebras |
| title_sort | symplectic differential reduction algebras and generalized weyl algebras |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212890 |
| work_keys_str_mv | AT hartwigjonast symplecticdifferentialreductionalgebrasandgeneralizedweylalgebras AT williamsiidwightanderson symplecticdifferentialreductionalgebrasandgeneralizedweylalgebras |