In algebra, let g be a

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

K. Let further

\xi\in\mathfrak^*

be a

one-form In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...

on g. The stabilizer g_''ξ'' of ''ξ'' is the Lie subalgebra of elements of g that annihilate ''ξ'' in the

coadjoint representation In mathematics, the coadjoint representation K of a Lie group G is the dual of the adjoint representation. If \mathfrak denotes the Lie algebra of G, the corresponding action of G on \mathfrak^*, the dual space to \mathfrak, is called the coadj ...

. The index of the Lie algebra is :

\operatorname\mathfrak:=\min\limits_ \dim\mathfrak_\xi.

Examples

Reductive Lie algebras

If g is reductive then the index of g is also the rank of g, because the

adjoint In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type :(''Ax'', ''y'') = (''x'', ''By''). Specifically, adjoin ...

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\to \mathbb:(X,Y)\mapsto \xi(,Y

is non-singular for some ''ξ'' in g^*. Another equivalent condition when g is the Lie algebra of an

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

''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

''G'', then the index of g is the

transcendence degree In abstract algebra, the transcendence degree of a field extension ''L'' / ''K'' is a certain rather coarse measure of the "size" of the extension. Specifically, it is defined as the largest cardinality of an algebraically independent subset of ...

of the field of rational functions on g^* that are invariant under the (co)adjoint action of ''G''.

References

{{DEFAULTSORT:Index Of A Lie Algebra Lie algebras