The classification of simple complex Lie superalgebras of polynomial vector fields and their deformations
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866916298945986560 |
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| author | Leites, Dimitry Shchepochkina, Irina |
| author_facet | Leites, Dimitry Shchepochkina, Irina |
| contents | We overview classifications of simple infinite-dimensional complex $\mathbb{Z}$-graded Lie (super)algebras of polynomial growth, and their deformations. A subset of such Lie (super)algebras consist of vectorial Lie (super)algebras whose elements are vector fields with polynomial, or formal power series, or divided power coefficients. A given vectorial Lie (super)algebra with a (Weisfeiler) filtration corresponding to a maximal subalgebra of finite codimension is called W-filtered; the associated graded algebra is called W-graded. Here, we correct several published results: (1) prove our old claim "the superization of É. Cartan's problem (classify primitive Lie algebras) is wild", (2) solve a tame problem: classify simple W-graded and W-filtered vectorial Lie superalgebras, (3) describe the supervariety of deformation parameters for the serial W-graded simple vectorial superalgebras, (4) conjecture that the exceptional simple vectorial superalgebras are rigid. We conjecture usefulness of our method in classification of simple infinite-dimensional vectorial Lie (super)algebras over fields of positive characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_16760 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The classification of simple complex Lie superalgebras of polynomial vector fields and their deformations Leites, Dimitry Shchepochkina, Irina Representation Theory Primary 17B20, 17B66, Secondary 58A30 We overview classifications of simple infinite-dimensional complex $\mathbb{Z}$-graded Lie (super)algebras of polynomial growth, and their deformations. A subset of such Lie (super)algebras consist of vectorial Lie (super)algebras whose elements are vector fields with polynomial, or formal power series, or divided power coefficients. A given vectorial Lie (super)algebra with a (Weisfeiler) filtration corresponding to a maximal subalgebra of finite codimension is called W-filtered; the associated graded algebra is called W-graded. Here, we correct several published results: (1) prove our old claim "the superization of É. Cartan's problem (classify primitive Lie algebras) is wild", (2) solve a tame problem: classify simple W-graded and W-filtered vectorial Lie superalgebras, (3) describe the supervariety of deformation parameters for the serial W-graded simple vectorial superalgebras, (4) conjecture that the exceptional simple vectorial superalgebras are rigid. We conjecture usefulness of our method in classification of simple infinite-dimensional vectorial Lie (super)algebras over fields of positive characteristic. |
| title | The classification of simple complex Lie superalgebras of polynomial vector fields and their deformations |
| topic | Representation Theory Primary 17B20, 17B66, Secondary 58A30 |
| url | https://arxiv.org/abs/2406.16760 |