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Operations on Subalgebras and Ideals






Subsections



DirectSum(A, B) : AlgGen, AlgGen -> AlgGen



Construct a structure constant algebra of dimension n + m where n and m are
the dimensions of L and M, respectively. The basis of the new algebra is
the concatenation of the bases of L and M and the products a * b where
a in A and b in B are defined to be 0.



DirectSumDecomposition(L) : AlgLie -> [ AlgLie ]



Given a Lie algebra L, return the direct sum decomposition of L as
a sequence of ideals of L whose sum is L and each of which cannot be
further decomposed into a direct sum of ideals.





Example AlgLie_DirectSumDecomposition (H90E7)

We compute the direct sum decomposition of
the simple Lie algebra of type D_2 over the rational field.





> L := SimpleLieAlgebra("D", 2, RationalField());
> L;
Lie Algebra of dimension 6 with base ring Rational Field
> D := DirectSumDecomposition(L);
> D;
[
    Lie Algebra of dimension 3 with base ring Rational Field,
    Lie Algebra of dimension 3 with base ring Rational Field
]
> Morphism(D[1], L); 
[ 0  1  0  0  0  0]
[ 0  0  1 -1  0  0]
[ 0  0  0  0  1  0]
> Morphism(D[2], L);
[1 0 0 0 0 0]
[0 0 1 1 0 0]
[0 0 0 0 0 1]





Standard Ideals and Series





Centre(L) : AlgLie -> AlgLie


Center(L) : AlgLie -> AlgLie



Given a Lie algebra L, return the centreof L.



Centraliser(L, K) : AlgLie, AlgLie -> AlgLie, Map


Centralizer(L, K) : AlgLie, AlgLie -> AlgLie, Map



Given a Lie algebra L and a subalgebra K of L,
return the centraliserof K in L, 
and its injection into L.



Centraliser(L, x) : AlgLie, AlgLieElt -> AlgLie, Map


Centralizer(L, x) : AlgLie, AlgLieElt -> AlgLie, Map



Given a Lie algebra L and an element x of L,
return the centraliserof x in L, 
and its injection into L.



Normaliser(L, K) : AlgLie, AlgLie -> AlgLie


Normalizer(L, K) : AlgLie, AlgLie -> AlgLie



Given a Lie algebra L and a subalgebra K of L,
return the normaliserof 
K in L, and its injection into L.



SolubleRadical(L) : AlgLie -> AlgLie


SolvableRadical(L) : AlgLie -> AlgLie



Given a Lie algebra L, return the soluble radicalof L.



NilRadical(L) : AlgLie -> AlgLie



Given a Lie algebra L, return the nilradicalof L.





Example AlgLie_Operations (H90E8)

We demonstrate the functions for performing basic operations with Lie algebras
like centre, etc.





> L := SimpleLieAlgebra("D", 4, RationalField());
> L;
Lie Algebra of dimension 28 with base ring Rational Field
> Centre(L);
Lie Algebra of dimension 0 with base ring Rational Field
> K := sub< L | [L.1, L.2, L.3] >;
> Centralizer(L, K);
Lie Algebra of dimension 10 with base ring Rational Field
> Normalizer(L, K);
Lie Algebra of dimension 19 with base ring Rational Field
> M := Centralizer(L, K);
> S := SolvableRadical(M);
> S;
Lie Algebra of dimension 10 with base ring Rational Field
> Morphism(S, L);
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 0 0 0 0 -1 0 0 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
> NilRadical(M);
Lie Algebra of dimension 9 with base ring Rational Field





CartanSubalgebra(L) : AlgLie -> AlgLie



Given a Lie algebra L, return a Cartan subalgebraof L.  
The algorithm works for Lie algebras L defined over a field F
such that |F| > dim L
and for restricted Lie algebras of characteristic p.
If the Lie algebra does not fit into one of these classes then the
correctness of the output is not guaranteed.





Example AlgLie_CartanSubalgebra (H90E9)

We compute a Cartan subalgebra of the simple Lie algebra of type A_4 over
the rational field.





> L := SimpleLieAlgebra("F", 4, RationalField());
> L;
Lie Algebra of dimension 52 with base ring Rational Field
> H := CartanSubalgebra(L);
Lie Algebra of dimension 4 with base ring Rational Field
> H*H;
Lie Algebra of dimension 0 with base ring Rational Field
> Normalizer(L, H);
Lie Algebra of dimension 4 with base ring Rational Field





KillingMatrix(L) : AlgLie -> AlgMatElt



Given a Lie algebra L such that
{x_1, ... , x_n} is a basis of L, return the Killing matrixof L, which is defined to be the matrix
(( Tr) (( ad)x_i.( ad) x_j)).



AdjointMatrix(L, x) : AlgLie, AlgLieElt -> AlgMatElt



Given a Lie algebra L and an element x of a subalgebra or ideal
of L, return the matrix of ( ad) x.





Example AlgLie_Other (H90E10)

We construct the killing matrix of a Lie algebra and the adjoint matrix
of one of its elements.





> L:=SimpleLieAlgebra("B",2,RationalField());
> KillingMatrix(L);
[ 0  0  0 -6  0  0  0  0  0  0]
[ 0  0 -6  0  0  0  0  0  0  0]
[ 0 -6  0  0  0  0  0  0  0  0]
[-6  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  6]
[ 0  0  0  0  0  0  0  0  6  0]
[ 0  0  0  0  0  0  6  0  0  0]
[ 0  0  0  0  0  0  0  6  0  0]
[ 0  0  0  0  0  6  0  0  0  0]
[ 0  0  0  0  6  0  0  0  0  0]
> AdjointMatrix( L, L.1 );
[ 0  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0 -1  0  0  0  0  0]
[ 0  0  0  0  0 -1  0  0  0  0]
[ 0  0  0  0  0  0 -1  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0]
[-1  0  0  0  0  0  0  0  0  0]
[ 0  0  0  0  0  0  0  0  0  0]
[ 0 -1  0  0  0  0  0  0  0  0]
[ 0  0 -1  0  0  0  0  0  0  0]





CompositionSeries(L) : AlgLie -> [ Alg ], [ AlgGen ], AlgMatElt



Compute a composition series for the algebra L.
The function has three return values:



(a)
a sequence containing the composition series as an ascending chain of
subalgebras such that the successive quotients are irreducible L-modules;

(b)
a sequence containing the composition factors as structure
constant algebras;

(c)
a transformation matrix to a basis compatible with the
composition series, that is, the first basis elements form a basis of the first
term of the composition series, the next extend these to a basis for the second
term etc.




MinimalIdeals(A : parameters) : AlgGen -> [ AlgGen ], BoolElt



    Limit: RngIntElt                    Default: Infinity



Return the minimal left/right/two-sided ideals of L (in non-decreasing size).
If Limit is set to n, at most n ideals are calculated and the second
return value indicates whether all of the ideals were computed.



MaximalIdeals(A : parameters) : AlgGen -> [ AlgGen ], BoolElt



    Limit: RngIntElt                    Default: Infinity



Return the maximal left/right/two-sided ideals of L (in non-decreasing size).
If Limit is set to n, at most n ideals are calculated and the second
return value indicates whether all of the ideals were computed.



DerivedSeries(L) : AlgLie -> [ AlgLie ]



Given a Lie algebra L, this function returns
a sequence of ideals of L that form its derived
series. 



LowerCentralSeries(L) : AlgLie -> [ AlgLie ]



Given a Lie algebra L, this function returns
a sequence of ideals of L that form its lower central
series. 



UpperCentralSeries(L) : AlgLie -> [ AlgLie ]



Given a Lie algebra L, this function returns
a sequence of ideals of L that form the upper central seriesof L.
The function repeatedly uses the algorithm for computing centres
while keeping track of the pre-images of the ideals factored out.





Example AlgLie_Series (H90E11)

We compute each of the type of series of a particular subalgebra of
the simple Lie algebra of type F_4 over the rational field.





> L:=SimpleLieAlgebra("F", 4, RationalField());
> L;
Lie Algebra of dimension 52 with base ring Rational Field
> K:=sub< L | [L.1, L.12, L.23, L.34, L.45] >;
> DerivedSeries(K);
[
    Lie Algebra of dimension 20 with base ring Rational Field,
    Lie Algebra of dimension 16 with base ring Rational Field,
    Lie Algebra of dimension 7 with base ring Rational Field,
    Lie Algebra of dimension 0 with base ring Rational Field
]
> LowerCentralSeries(K);
[
    Lie Algebra of dimension 20 with base ring Rational Field,
    Lie Algebra of dimension 16 with base ring Rational Field,
    Lie Algebra of dimension 12 with base ring Rational Field,
    Lie Algebra of dimension 8 with base ring Rational Field,
    Lie Algebra of dimension 5 with base ring Rational Field,
    Lie Algebra of dimension 2 with base ring Rational Field,
    Lie Algebra of dimension 1 with base ring Rational Field,
    Lie Algebra of dimension 0 with base ring Rational Field
]
> UpperCentralSeries(K);
[
    Lie Algebra of dimension 2 with base ring Rational Field,
    Lie Algebra of dimension 3 with base ring Rational Field,
    Lie Algebra of dimension 5 with base ring Rational Field,
    Lie Algebra of dimension 8 with base ring Rational Field,
    Lie Algebra of dimension 12 with base ring Rational Field,
    Lie Algebra of dimension 16 with base ring Rational Field,
    Lie Algebra of dimension 20 with base ring Rational Field
]






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