Signatures above the Witt-index bound for fixed semisimple ideals

Determine which signatures occur for semisimple Lie-algebraic solutions in case (a) of Theorem 2, once the fixed simple ideals and their couplings are specified, beyond the lower bound \(\min(p,q)\ge d_i\).

Background

The paper classifies signatures of solutions to the algebraic constraint hab[ta,[tb,tc]]=0h_{ab}[t^a,[t^b,t^c]]=0 for semisimple Lie algebras. When the reality involution fixes at least one simple ideal, that ideal is totally isotropic, which imposes the lower bound min⁡(p,q)≥di\min(p,q)\ge d_i. The authors do not determine the complete set of signatures compatible with particular collections of fixed ideals and their off-diagonal couplings.

References

In case (a) of Theorem~\ref{thm:signature}, the fixed ideals force the lower bound $\min(p,q)\ge d_i$, but which signatures above this bound occur, for a given set of ideals and couplings, remains undetermined.

— Lie Algebra Saddles in the IKKT Matrix Model and Criteria for the Emergence of Time  (2609.40183 - Liao, 30 Sep 2026) in Section 6, Discussion, item (i)

How does the classification change under mass and Myers deformations? They add source terms to Eq.~eq:master, so the Casimir obstruction becomes a solvability condition (Appendix~\ref{app:E}) and semisimple algebras re-enter; which signatures the deformed equation then allows, and where time can reside, is open.

eq:master:

hab [ta,[tb,tc]]=0∀ c,h=CTη C.h_{ab}\,[t^a,[t^b,t^c]]=0\quad\forall\,c, \qquad h=C^{T}\eta\,C .

— Lie Algebra Saddles in the IKKT Matrix Model and Criteria for the Emergence of Time  (2609.40183 - Liao, 30 Sep 2026) in Section 6, Discussion, item (vi)