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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> : AlgMatLie, List -> AlgMatLie, Map
Creates the subalgebra S of the Lie algebra L that is
generated by the elements defined by A,
where A 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> : AlgMatLie, List -> AlgMatLie, Map
Creates the ideal I of the Lie algebra L
generated by the elements defined by A,
where A 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> : AlgMatLie, List -> AlgMatLie, Map
Forms the quotient algebra L / I, where I is the two-sided ideal of
L generated by the elements defined by A,
where A 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 L/I itself, the constructor returns the natural
homomorphism f : L -> L/I.
L / S : AlgMatLie, AlgMatLie -> AlgLie
The quotient of the Lie algebra L by the ideal closure of the subalgebra S.
We construct the quotient of the matrix Lie algebra of 2 x 2 matrices,
by the ideal spanned by the identity matrix.
> L := MatrixLieAlgebra( Rationals(), 2 );
> Dimension(L);
4
> I := ideal< L | L!Matrix([[1,0],[0,1]]) >;
> Dimension(I);
1
> K := L/I;
> Dimension(K);
3
> SemisimpleType( K );
A1
Given a Lie algebra L and an ideal I of L, this intrinsic returns four
values: the quotient Q = L/I, the natural homomorphism μ : L to Q
and two functions, σ and Σ with domain Q. The function σ
is a section of μ and also returns the kernel of μ. That is, for y ∈Q,
σ(y) returns x and V, such that σ(x) = y and where V is the
underlying vector space of I. For y∈Q, Σ(y) is the subalgebra of
L generated by I and x.
We consider an ideal of the Lie algebra of type G 2 over the field with 3 elements.
> R := RootDatum("G2");
> L := LieAlgebra(R, GF(3));
> pos,neg,cart := StandardBasis(L);
> shrt := [ i : i in [1..NumPosRoots(R)] | IsShortRoot(R, i) ];
> shrt;
[ 1, 3, 4 ]
> I := ideal<L | pos[shrt]>;
> _, str1 := ReductiveType(I); str1;
The 7-dim simple constituent of a Lie algebra of type A2
So apparently I is isomorphic to the 7-dimensional simple
constituent of a Lie algebra of type A 2. We will now use
QuotientWithPullback to construct L/I.
> LI, proj, pb, pbsub := QuotientWithPullback(L, I);
> _, str2 := ReductiveType(LI); str2;
The 7-dim simple constituent of a Lie algebra of type A2
So apparently I simeq L/I! Finally, we will demonstrate
the use of the additional return values. First, we verify
that an element of I maps to 0 in L/I:
> proj(pos[1]);
(0 0 0 0 0 0 0)
And then we consider the preimage in L of a randomly chosen
element of L/I.
> y := LI![0,1,1,1,1,0,1];
> y;
(0 1 1 1 1 0 1)
> x, V := pb(y);
> x;
(0 1 0 0 1 0 0 1 0 1 0 0 0 1)
> #V;
2187
> assert #V eq #I;
> {* proj(x + v) eq y : v in V *};
{* true^^2187 *}
So indeed x + v is a preimage of y for all v ∈V.
> M := pbsub(y);
> M, M meet I;
Lie Algebra of dimension 8 with base ring GF(3)
Lie Algebra of dimension 7 with base ring GF(3)
> _,str3 := ReductiveType(M);
> str3;
Twisted Lie algebra of type 2A2 [Ad]
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