Maximal graded subalgebras of simple vectorial Lie superalgebras with polynomial coefficients
Abstract
S. Lie was the first to try to classify maximal subalgebras of simple vectorial (i.e., consisting of vector fields) Lie algebras with polynomial coefficients. In Appendix it is shown that, as stated, the problem is wild: describing all subalgebras contains the classification of pairs of matrices up to simultaneous conjugation, so there are too many such subalgebras. Several researchers distinguished various classes of maximal subalgebras important in applications. Here, in simple (or in Cartan prolongations close to simple) vectorial Lie superalgebras with polynomial coefficients considered with gradings associated with Weisfeiler filtrations, we describe maximal simple (and close to simple) graded subalgebras --- a superization of what Sophus Lie tackled. Together with I. Shchepochkina's description of the maximal subalgebras of matrix Lie superalgebras, see arXiv:hep-th/9702122, this paper solves the super and infinite-dimensional version of Dynkin's problem --- the description of the maximal subalgebras of simple finite-dimensional Lie algebras --- except for the 3 exceptional ambients considered elsewhere.
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BibTeX
@misc{leites2026maximal,
title = {{Maximal graded subalgebras of simple vectorial Lie superalgebras with polynomial coefficients}},
author = {Dimitry Leites and Irina Shchepochkina},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03340},
eprint = {2610.03340},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03340},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximal graded subalgebras of simple vectorial Lie superalgebras with polynomial coefficients
AU - Dimitry Leites
AU - Irina Shchepochkina
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03340
UR - https://arxiv.org/abs/2610.03340
AB - S. Lie was the first to try to classify maximal subalgebras of simple vectorial (i.e., consisting of vector fields) Lie algebras with polynomial coefficients. In Appendix it is shown that, as stated, the problem is wild: describing all subalgebras contains the classification of pairs of matrices up to simultaneous conjugation, so there are too many such subalgebras. Several researchers distinguished various classes of maximal subalgebras important in applications. Here, in simple (or in Cartan prolongations close to simple) vectorial Lie superalgebras with polynomial coefficients considered with gradings associated with Weisfeiler filtrations, we describe maximal simple (and close to simple) graded subalgebras --- a superization of what Sophus Lie tackled. Together with I. Shchepochkina's description of the maximal subalgebras of matrix Lie superalgebras, see arXiv:hep-th/9702122, this paper solves the super and infinite-dimensional version of Dynkin's problem --- the description of the maximal subalgebras of simple finite-dimensional Lie algebras --- except for the 3 exceptional ambients considered elsewhere.
ER -