Inductive formulas for the index of seaweed Lie algebras (Q2762317)

The index of a finite-dimensional Lie algebra \(\mathfrak a\) is the minimal dimension of the annihilator of an element in the dual space of \(\mathfrak a\). If \(\mathfrak a\) is reductive, then the index of \(\mathfrak a\) equals the rank of \(\mathfrak a\). The index of Borel subalgebras of all simple Lie algebras was computed by \textit{V. V. Trofimov} [Sel. Math. Sov. 8, 31-56 (1989; Zbl 0659.17009)] and the index of parabolic subalgebras was determined by \textit{A. G. Elashvili} [preprint, 1990]. More recently, \textit{V. Dergachev} and \textit{A. Kirillov} [J. Lie Theory 10, 331-343 (2000; Zbl 0980.17001)] introduced the notion of a seaweed subalgebra in the general linear Lie algebra \({\mathfrak gl}(V)\) as a generalization of a parabolic subalgebra and showed that the index of any seaweed subalgebra \(\mathfrak s\) in \({\mathfrak gl}(V)\) is equal to the number of the connected components plus the number of circuits in a certain graph attached to \(\mathfrak s\). In particular, it follows that the index of \(\mathfrak s\) is at most \(\dim V\), i.e., the rank of \({\mathfrak gl}(V)\).NEWLINENEWLINENEWLINEIn the paper under review, the author generalizes the notion of a seaweed subalgebra to any reductive Lie algebra. The main result is an inductive procedure for computing the index of seaweed subalgebras of classical Lie algebras. As a consequence, the author proves that in the cases \({\mathfrak g} = {\mathfrak gl}_n\), \({\mathfrak sl}_n\), or \({\mathfrak sp}_{2n}\) the index of a seaweed subalgebra \(\mathfrak s\) is at most the rank of \(\mathfrak g\) and it equals the rank of \(\mathfrak g\) if and only if \(\mathfrak s\) is a Levi subalgebra of \(\mathfrak g\). Moreover, he also obtains lower bounds for the index of seaweed subalgebras in \({\mathfrak gl}_n\) or \({\mathfrak sl}_n\) and especially a necessary condition for a seaweed subalgebra in \({\mathfrak sl}_n\) to be a Frobenius Lie algebra (i.e., having index zero), which was observed earlier by A. G. Elashvili. For symplectic and orthogonal Lie algebras the computation of the index of seaweed subalgebras is reduced to the special case of parabolic subalgebras. In the symplectic case the author derives an explicit formula for the index of parabolic subalgebras. In the orthogonal case he only obtains a formula for the index of parabolic subalgebra in some special cases, which confirms that the rank should also be an upper bound for the index of any seaweed subalgebra as in the other cases. It should be mentioned that the formulas for the index of parabolic subalgebras in symplectic and orthogonal Lie algebras correct those obtained by A. G. Elashvili in the unpublished preprint cited above.