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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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