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 $\mathfrak{osp}_a(4|2)$, $\mathfrak{ag}(2)$ and $\mathfrak{ab}(3)$, 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 $\mathbb{Z}$-gradings to which the method of that paper applies --- two for $\mathfrak{osp}_a(4|2)$, five for $\mathfrak{ag}(2)$, eight for $\mathfrak{ab}(3)$ --- 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 $\mathbb{Q}$. 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 $\mathfrak{osp}_a(4|2)$ sitting inside $\mathfrak{ag}(2)$, resp. $\mathfrak{ab}(3)$, is computed intrinsically.