Given a Lie algebra L, this function returns an element of L that is not nilpotent, or the zero element of <L> if no such element exists.
> L:=SimpleLieAlgebra("G",2,RationalField());
> NonNilpotentElement( L );
(0 0 0 0 0 1 0 0 0 0 0 0 0 0)
Given a Lie algebra L, return the centreof L.
Given a Lie algebra L and a subalgebra K of L, return the centraliserof K in L.
Given a Lie algebra L and a subalgebra K of L, return the normaliserof K in L.
Given a Lie algebra L, return the soluble radicalof L.
Given a Lie algebra L, return the nilradicalof L.
> 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
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.
> 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
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)).
Given a Lie algebra L and an element x of a subalgebra or ideal of L, return the matrix of ( ad) x.
> 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]
sigind a[i] : AlgLieElt, RngIntElt -> RngElt
If a is an element of a Lie algebra L of dimension n and 1 <= i <= n is a positive integer, then the i-th component of the element a is returned (as an element of the base ring R of L).
Given an element a belonging to a sLie algebra of dimension n over R, a positive integer 1 <= i <= n and an element r in R, the i-th component of the element a is redefined to be r.