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Other properties of Lie Algebras
Other properties of Lie Algebras
KillingMatrix(L) : AlgLie -> AlgMatElt
Given a Lie algebra L such that
{x_1, ... , x_n} is a basis of L, return the Killing matrix
of 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 (H49E9)
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]
NonNilpotentElement(L) : AlgLie -> AlgLieElt
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.
Example AlgLie_NonNilpotentElement (H49E10)
We construct a non-nilpotent element of a Lie algebra.
> L:=SimpleLieAlgebra("G",2,RationalField());
> NonNilpotentElement( L );
(0 0 0 0 0 1 0 0 0 0 0 0 0 0)
Example AlgLie_LieAlgebra (H49E11)
We create the Lie algebra sl_3(Q) as a structure constant algebra and
manually examine its structure.
First, we construct gl_3(Q) from the full matrix algebra M_3(Q) and
get sl_3(Q) as the derived algebra of gl_3(Q).
> gl3 := LieAlgebra(Algebra(MatrixRing(Rationals(), 3)));
> sl3 := gl3 * gl3;
> sl3;
Lie Algebra of dimension 8 with base ring Rational Field
Let's see how the first basis element acts.
> for i in [1..8] do
> print sl3.i * sl3.1;
> end for;
(0 0 0 0 0 0 0 0)
( 0 -1 0 0 0 0 0 0)
( 0 0 -2 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 -1 0 0)
(0 0 0 0 0 0 2 0)
(0 0 0 0 0 0 0 1)
Since it acts diagonally, this element lies in a Cartan subalgebra. The next
candidate seems to be the fifth basis element.
> for i in [1..8] do
> print sl3.i * sl3.5;
> end for;
(0 0 0 0 0 0 0 0)
(0 1 0 0 0 0 0 0)
( 0 0 -1 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 -2 0 0)
(0 0 0 0 0 0 1 0)
(0 0 0 0 0 0 0 2)
This also acts diagonally and commutes with sl3.1, hence we have luckily
found a full Cartan algebra in sl_3(Q).
We can now easily work out the root system. Obviously the root spaces
correspond to the pairs (sl3.2, sl3.4), (sl3.3, sl3.7) and
(sl3.6, sl3.8). The product of a positive root with its negative should
lie in the Cartan algebra.
> sl3.2*sl3.4;
( 1 0 0 0 -1 0 0 0)
> sl3.3*sl3.7;
(1 0 0 0 0 0 0 0)
> sl3.6*sl3.8;
(0 0 0 0 1 0 0 0)
Clearly some choices have to be made and we fix sl3.3 as the element
e_alpha corresponding to the first fundamental root alpha, sl3.7
as e_(-alpha) and get sl3.1 as h_alpha = e_alpha * e_(-alpha).
For the other fundamental root beta we have to find an element
e_beta such that e_alpha * e_beta is non-zero.
> sl3.3*sl3.2;
(0 0 0 0 0 0 0 0)
> sl3.3*sl3.4;
( 0 0 0 0 0 -1 0 0)
> sl3.3*sl3.6;
(0 0 0 0 0 0 0 0)
> sl3.3*sl3.8;
(0 1 0 0 0 0 0 0)
We choose sl3.8 as e_beta, sl3.6 as e_(-beta) and
consequently -sl3.5 as h_beta. This now determines
e_(alpha + beta) to be sl3.2 and e_(-alpha - beta) to be
sl3.4.
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