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# LieIdeal -- the class of all Lie ideals

## Description

This type represents Lie ideals. It is a subtype of LieSubAlgebra and it has FGLieIdeal as a subtype.

    i1 : L = lieAlgebra{a,b}

    o1 = L

    o1 : LieAlgebra

    i2 : I=lieIdeal{a a b}

    o2 = I

    o2 : FGLieIdeal

    i3 : Q=L/I

    o3 = Q

    o3 : LieAlgebra

    i4 : f=map(Q,L)

    o4 = f

    o4 : LieAlgebraMap

    i5 : J=kernel f

    o5 = J

    o5 : LieIdeal

    i6 : I===J

    o6 = false

    i7 : describe I

    o7 = generators => { - (a b a)}
         lieAlgebra => L

The kernel of $f$ is defined as the inverse image under $f$ of the zero ideal.

    i8 : describe J

    o8 = inverse => {f, finitely generated ideal of Q}
         lieAlgebra => L

    i9 : J#inverse_1===zeroIdeal Q

    o9 = true

## See also

- FGLieIdeal -- the class of all finitely generated Lie ideals
- FGLieSubAlgebra -- the class of all finitely generated Lie subalgebras

## Types of LieIdeal:

- FGLieIdeal -- the class of all finitely generated Lie ideals

## Methods that use an object of class LieIdeal:

- LieAlgebra / LieIdeal -- make a quotient Lie algebra
- quotient(LieIdeal,FGLieSubAlgebra) -- make the quotient of a Lie ideal by a finitely generated Lie subalgebra

## For the programmer

The object LieIdeal is a type, with ancestor classes LieSubAlgebra < LieSubSpace < VectorSpace < HashTable < Thing.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc.m2:458:0.