Irreducible branch in the non-contact gradings

Settle the irreducible branch of case 3^o for the 11 non-contact gradings of the exceptional Lie superalgebras \mathfrak{ag}(2) and \mathfrak{ab}(3), by enumerating the almost simple subalgebras of the corresponding degree-zero components that act irreducibly on \mathfrak{g}_{-1}.

Background

The paper classifies maximal graded subalgebras for the gradings of \mathfrak{osp}_a(4|2), \mathfrak{ag}(2), and \mathfrak{ab}(3) satisfying condition (\ref{cond3exc}). In case 3o, the degree-zero subalgebra is a proper subalgebra of \mathfrak{g}_0 while the negative degree-one component is unchanged. The computational sweep treats subalgebras obtained as stabilizers of subspaces, corresponding to the reducible branch.

For the two contact gradings, the authors settle the irreducible branch using Dynkin’s classification and show that the relevant first prolongations vanish. The remaining 11 non-contact gradings have degree-zero components not of the form central extension of a simple algebra, so the required almost simple irreducible subalgebras have not been enumerated.

References

Settle the irreducible branch of the case $3o$ for the $11$ non-contact gradings of tables tabAgGr, tabAbGr.

— Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras  (2610.03354 - Leites et al., 2 Oct 2026) in Section 8, Problem \ref{Pirr}, “The irreducible branch in the 11 non-contact gradings”

Are the two gradings of depth $4$ of Statement \ref{StGr3exc} to be included? They violate the condition cond3exc --- the $\fg_0$-action on $\fg_{-k}$ is reducible for some $k$ --- but they are W-gradings in the weaker sense, and the ambients they produce are of interest in their own right.

— Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras  (2610.03354 - Leites et al., 2 Oct 2026) in Section 8, Problem \ref{Pdepth4}, “Are the two gradings of depth 4 to be included?”

Are the embeddings $\fosp_a(4|2)\subset\fag(2)$ with $j(a)=\nfrac{133}{144}$, $\fsl(3|1)\subset\fag(2)$, and $\fsl(4|1)\subset\fab(3)$ already in the literature?

— Maximal subalgebras in the three exceptional simple finite-dimensional Lie superalgebras  (2610.03354 - Leites et al., 2 Oct 2026) in Section 8, Question \ref{Qlit}, “Are the three embeddings already in the literature?”