Returns true if, and only if, the Lie algebras L and K are equal.
Returns true if, and only if, the Lie algebras L and K are not equal.
Returns true if, and only if, the Lie algebras L is contained in the Lie algebra K.
Returns true if, and only if, the Lie algebras L is not contained in the Lie algebra K.
The intersection of the Lie algebras L and M, which must have a common superalgebra.
The algebra product LM of the algebras L and M, which must have a common superalgebra.
The (left-normed) n-th power of the Lie algebra L, i.e. (( ... (L * L) * ... ) * L).
The map giving the morphism from L to M. Either L is a subalgebra of M, in which case the embedding of L into M is returned, or M is a quotient algebra of L, in which case the natural epimorphism from L onto M is returned.
The coefficient ring (or base ring) over which the Lie algebra L is defined.
The dimension of the Lie algebra L.
The cardinality of the Lie algebra L, if R is finite.
Given a Lie algebra L with base ring R, together with a ring S, construct the Lie algebra M with base ring S obtained by coercing the coefficients of elements of L into S, together with the homomorphism from L to M.
Given a Lie algebra L with base ring R, together with a ring S and a map f: R -> S, construct the Lie algebra M with base ring S obtained by mapping the coefficients of elements of L into S via f, together with the homomorphism from L to M.
The i-th basis element of the algebra L.
The basis of the algebra L, as a sequence of elements of L.
Note that if L is of type AlgGrpSub the returned elements will be elements of the full group algebra of which L is a subalgebra.
Given a sequence Q of elements of the R-algebra L, this functions returns true if these elements are linearly independent over R; otherwise false.
Given an algebra L and either a subalgebra S of dimension m of L or a sequence Q of m linearly independent elements of L, return a sequence containing a basis of L such that the first m elements are the basis of S resp. the elements in Q.
Let L be a Lie algebra. If L has a nondegenerate Killing form, then (over some algebraic extension of the ground field) L is the direct sum of absolutely simple Lie algebras. These Lie algebras have been classified and the classes are named A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4 and G_2. This function returns a single string containing the types of the direct summands of L.
> L := SimpleLieAlgebra("D", 7, RationalField());
> L;
Lie Algebra of dimension 91 with base ring Rational Field
> K := Centralizer(L, sub<L | [L.1, L.2, L.3, L.4]>);
> K;
Lie Algebra of dimension 41 with base ring Rational Field
> _,S := HasLeviSubalgebra(K);
> S;
Lie Algebra of dimension 6 with base ring Rational Field
> SemiSimpleType(S);
A1 A1
Given a semisimple Lie algebra L with a split Cartan subalgebra, this function computes the root system (resp. root datum). This function returns two values:
- (a)
- The root system (resp. root datum) of L with respect to the Cartan subalgebra which is output by CartanSubalgebra(L).
- (b)
- A sequence of elements of L which are the root vectors corresponding to the roots in R (so the first vector corresponds to the first root and so on).
> L := SimpleLieAlgebra("G", 2, RationalField());
> R, Rv, fund, C:=RootSystem(L);
> R;
...
> Rv;
[ (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 0 0),
(0 0 1 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 1 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 1 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 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 1), (0 0 0 0 0 0 0 0 0 0 0 0 1 0) ]
Here L is a semisimple Lie algebra. This function returns the rootdatum structure D corresponding to L. We note that the order of the positive roots in D is not necessarily the same as the order in which they appear in the root system of L.
> L:= SimpleLieAlgebra( "F", 4, Rationals() ); > rd := RootDatum(L); > rd; Root datum of type F4 > CartanMatrix(rd); [ 2 0 -1 0] [ 0 2 0 -1] [-1 0 2 -1] [ 0 -1 -2 2]
Returns three lists, x, y, h of elements of L. They form a Chevalley basis of L. The first sequence gives basis elements corresponding to positive roots, the second to the negative roots and the third to basis elements in a Cartan subalgebra.
> L := SimpleLieAlgebra("A", 2, RationalField());
> x, y, h:= ChevalleyBasis(L);
> x; y; h;
[ (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 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 1 0), (0 0 0 0 0 0 0 1) ]
> L := SemisimpleLieAlgebra( "A3", Rationals() );
> print RootDatum( L ) : Maximal;
Root datum of type A3 with simple roots
[ 1 0 1]
[ 1 -2 1]
[ 0 1 -2]
and simple coroots
[ 1 1 1]
[ 0 -1 0]
[ 0 0 -1]
The Weyl group of the reductive Lie algebra L, as a permutation Coxeter group (see Chapter COXETER GROUPS AS PERMUTATION GROUPS).
The Weyl group of the reductive Lie algebra L, as a Coxeter group (see Chapter COXETER GROUPS).
The Weyl group of the reductive Lie algebra L, as a reflection group (see Chapter COXETER GROUPS).[Next][Prev] [Right] [Left] [Up] [Index] [Root]