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The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer. (English) Zbl 1041.17022
Let \(\mathfrak q\) be a Lie algebra and \(\xi \in {\mathfrak q}^*\). The index of \(\mathfrak q\), \(\text{ind}(\mathfrak q)\), is the minimum dimension of \({\mathfrak q}_\xi = \{ x\in \mathfrak q \mid \xi([\mathfrak q, x])=0 \}\) as \(\xi\) varies over \({\mathfrak q}^*\). The index of a representation is defined similarly with the specified representation replacing the coadjoint representation. This paper mainly deals with computing the index for centralizers of elements and related algebras. However, a few other results are presented such as showing that \(\text{ind}(\mathfrak p) + \text{ind}(\mathfrak u) \leq \dim(\mathfrak l)\) where \(\mathfrak p\) is a parabolic subalgebra and \(\mathfrak p = \mathfrak l + \mathfrak u\) is its Levi decomposition.
Let \(\mathfrak g\) be a semisimple Lie algebra over an algebraically closed field of characteristic \(0\). For \(x\in\mathfrak g\), write \(z_{\mathfrak g}(x)\) for the centralizer of \(x\) in \(\mathfrak g\), \(n_{\mathfrak g}(x)\) for the normalizer of \(x\) in \(\mathfrak g\), and \({\mathfrak d}_{\mathfrak g}(x)\) for the double centralizer of \(x\) in \(\mathfrak g\). A conjecture of Elashvili states \(\text{ind}(z_{\mathfrak g}(x))=\text{rk}(\mathfrak g)\) and reduces to the case of nilpotent elements. In this paper, the conjecture is proved for regular and subregular orbits as well as for \(x\) satisfying \(\text{ad}(x)^3=0\). Furthermore, the author states a conjecture that is more general than the one by Elashvili. In particular, it is conjectured that \(\text{ind}(n_{\mathfrak g}(x)) = \text{rk}(\mathfrak g) - \dim({\mathfrak d}_{\mathfrak g}(x))\) which is proved for \(x\) regular or satisfying \(\text{ad}(x)^3=0\).
Subject to a mild condition, which is satisfied for all \(x\) when \(\mathfrak g \in \{ {\mathfrak {sl}}_n, {\mathfrak {sp}}_{2n}, {\mathfrak {so}}_{2n+1} \}\), it is shown that \(\text{ind}(n_{\mathfrak g}(x)) = \text{ind}(z_{\mathfrak g}(x)) - \dim({\mathfrak d}_{\mathfrak g}(x))\) with respect to the natural representation of \(n_{\mathfrak g}(x)\) on \({\mathfrak d}_{\mathfrak g}(x)\). Finally, for regular \(x\), it is shown that \(n_{\mathfrak g}(x)\) is a Frobenius Lie algebra, i.e., that \(\text{ind}(n_{\mathfrak g}(x)) = 0\). This in turn shows that a regular nilpotent orbit yields a solution of the classical Yang-Baxter equation.

MSC:
17B45 Lie algebras of linear algebraic groups
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