In algebra, let g be a Lie algebra over a field K. Let further \( \xi \in {\mathfrak {g}}^{*} \) be a one-form on g. The stabilizer gξ of ξ is the Lie subalgebra of elements of g that annihilate ξ in the coadjoint representation. The index of the Lie algebra is
\( {\displaystyle \operatorname {ind} {\mathfrak {g}}:=\min \limits _{\xi \in {\mathfrak {g}}^{*}}\dim {\mathfrak {g}}_{\xi }.} \)
Examples
Reductive Lie algebras
If g is reductive then the index of g is also the rank of g, because the adjoint and coadjoint representation are isomorphic and rk g is the minimal dimension of a stabilizer of an element in g. This is actually the dimension of the stabilizer of any regular element in g.
Frobenius Lie algebra
If ind g=0, then g is called Frobenius Lie algebra. This is equivalent to the fact that the Kirillov form \( K_{\xi }\colon {\mathfrak {g\otimes g}}\to {\mathbb {K}}:(X,Y)\mapsto \xi ([X,Y]) \) is non-singular for some ξ in g*. Another equivalent condition when g is the Lie algebra of an algebraic group G, is that g is Frobenius if and only if G has an open orbit in g* under the coadjoint representation.
Lie algebra of an algebraic group
If g is the Lie algebra of an algebraic group G, then the index of g is the transcendence degree of the field of rational functions on g* that are invariant under the action of G.[1]
References
Panyushev, Dmitri I. (2003). "The index of a Lie algebra, the centralizer of a nilpotent element, and the normalizer of the centralizer". Mathematical Proceedings of the Cambridge Philosophical Society. 134 (1): 41–59. doi:10.1017/S0305004102006230.
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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