Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras
Abstract
Dynkin classified the maximal subalgebras of the simple finite-dimensional Lie algebras. The super version of his problem is solved, for the matrix Lie superalgebras, by Shchepochkina, and, for the simple vectorial ones with polynomial coefficients taken with their Weisfeiler gradings, in the companion paper by Leites and Shchepochkina --- with three exceptions left open: the three simple finite-dimensional Lie superalgebras ospa(4|2), ag(2) and ab(3), which are vectorial in more than one way and are not covered by the general argument. Here we settle these three cases for \textbf{graded} subalgebras. For each of the three Lie superalgebras, we list all the ℤ-gradings to which the method of that paper applies --- two for ospa(4|2), five for ag(2), eight for ab(3) --- and describe, for every one of them, all the maximal graded subalgebras, the semi-simple ones explicitly. A simple criterion upgrades most of the answers from ``maximal graded\rq\rq\ to ``maximal\rq\rq. Every answer is computed over ℚ. Two by-products: the classical list of \textbf{five} cases into which the graded subalgebras of a vectorial Lie superalgebra split is incomplete for ambients of depth >1 --- a sixth case has to be added, and it is non-empty precisely here; and the parameter a of the copies of ospa(4|2) sitting inside ag(2), resp. ab(3), is computed intrinsically.
Related
- Maximal graded subalgebras of simple vectorial Lie superalgebras with polynomial coefficients
- Simple finite-dimensional binary Lie superalgebras
- Schur--Weyl duality for coideal subalgebras of quantum affine superalgebras of type A
- Fusion rules and rigidity for weight modules over the simple admissible affine sl(2) and N=2 superconformal vertex operator superalgebras
- Finite free position of maximal abelian ∗-subalgebras of the matrix algebra
BibTeX
@misc{leites2026maximal,
title = {{Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras}},
author = {Dimitry Leites and Oleksandr Lozhechnyk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03354},
eprint = {2610.03354},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03354},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras
AU - Dimitry Leites
AU - Oleksandr Lozhechnyk
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03354
UR - https://arxiv.org/abs/2610.03354
AB - Dynkin classified the maximal subalgebras of the simple finite-dimensional Lie algebras. The super version of his problem is solved, for the matrix Lie superalgebras, by Shchepochkina, and, for the simple vectorial ones with polynomial coefficients taken with their Weisfeiler gradings, in the companion paper by Leites and Shchepochkina --- with three exceptions left open: the three simple finite-dimensional Lie superalgebras osp_a(4|2), ag(2) and ab(3), which are vectorial in more than one way and are not covered by the general argument. Here we settle these three cases for \textbf{graded} subalgebras. For each of the three Lie superalgebras, we list all the ℤ-gradings to which the method of that paper applies --- two for osp_a(4|2), five for ag(2), eight for ab(3) --- and describe, for every one of them, all the maximal graded subalgebras, the semi-simple ones explicitly. A simple criterion upgrades most of the answers from ``maximal graded\rq\rq\ to ``maximal\rq\rq. Every answer is computed over ℚ. Two by-products: the classical list of \textbf{five} cases into which the graded subalgebras of a vectorial Lie superalgebra split is incomplete for ambients of depth >1 --- a sixth case has to be added, and it is non-empty precisely here; and the parameter a of the copies of osp_a(4|2) sitting inside ag(2), resp. ab(3), is computed intrinsically.
ER -