Construction of Subalgebras, Ideals and Quotients

If the coefficient ring R of a Lie algebra L is a Euclidean domain, then submodules and ideals can be constructed in Magma; if R is a field. then quotients can be constructed in Magma. Note that left, right, and two-sided ideals are identical in a Lie algebra.

sub< L | A > : AlgLie, List -> AlgLie, Map

Create the subalgebra S of the Lie algebra L that is generated by the elements defined by L, where L is a list of one or more items of the following types:

(a)

An element of L;

(b)

A set or sequence of elements of L;

(c)

A subalgebra or ideal of L;

(d)

A set or sequence of subalgebras or ideals of L.

As well as the subalgebra S itself, the constructor returns the inclusion homomorphism f : S -> L.

ideal< L | A > : AlgLie, List -> AlgLie, Map

lideal< L | A > : AlgLie, List -> AlgLie, Map

rideal< L | A > : AlgLie, List -> AlgLie, Map

Create the ideal I of the Lie algebra L generated by the elements defined by L, where L is a list of one or more items of the following types:

(a)

An element of L;

(b)

A set or sequence of elements of L;

(c)

A subalgebra or ideal of L;

(d)

A set or sequence of subalgebras or ideals of L.

As well as the ideal I itself, the constructor returns the inclusion homomorphism f : I -> L.

quo< L | A > : AlgLie, List -> AlgLie, Map

Create the quotient algebra Q = L / I, where I is the two-sided ideal of L generated by the elements defined by L, where L is a list of one or more items of the following types:

(a)

An element of L;

(b)

A set or sequence of elements of L;

(c)

A subalgebra or ideal of L;

(d)

A set or sequence of subalgebras or ideals of L.

As well as the quotient Q itself, the constructor returns the natural homomorphism f : A -> Q.

L / S : AlgLie, AlgLie -> AlgLie

The quotient of the Lie algebra L by the ideal closure of the subalgebra S.