Operations on Lie Algebras#

L eq K: AlgLie, AlgLie -> BoolElt#
L eq K: AlgMatLie, AlgMatLie -> BoolElt#: Returns true if, and only if, the Lie algebras \(L\) and \(K\) are equal.

L ne K: AlgLie, AlgLie -> BoolElt#
L ne K: AlgMatLie, AlgMatLie -> BoolElt#: Returns true if, and only if, the Lie algebras \(L\) and \(K\) are not equal.

L subset K: AlgLie, AlgLie -> BoolElt#
L subset K: AlgMatLie, AlgMatLie -> BoolElt#: Returns true if, and only if, the Lie algebra \(L\) is contained in the Lie algebra \(K\).

L notsubset K: AlgLie, AlgLie -> BoolElt#: Returns true if, and only if, the Lie algebra \(L\) is not contained in the Lie algebra \(K\).

L meet M: AlgLie, AlgLie -> AlgLie#
L meet M: AlgMatLie, AlgMatLie -> AlgMatLie#: The intersection of the Lie algebras \(L\) and \(M\) is returned. Note that \(L\) and \(M\) have a common superalgebra.

L * M: AlgLie, AlgLie -> AlgLie#
L * M: AlgMatLie, AlgMatLie -> AlgMatLie#: The Lie algebra product \([L,M]\) of the algebras \(L\) and \(M\) is returned. Note that \(L\) and \(M\) must have a common superalgebra.

L ^ n: AlgLie, RngIntElt -> AlgLie#: The (left-normed) \(n\)-th power of the (structure constant) Lie algebra \(L\), i.e., \(((\ldots (L*L)* \ldots)*L)\) is constructed.

Morphism(L, M): AlgLie, AlgLie -> Map#: The map giving the morphism from the (structure constant) Lie algebra \(L\) to \(M\) is constructed. 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.

IsIsomorphic(L, M): AlgLie, AlgLie -> BoolElt, .#

HL: AlgLie                    Default: false
HM: AlgLie                    Default: false

Returns true if the Lie algebras \(L\) and \(M\) are isomorphic. It is currently implemented for trivial cases (such as when the dimensions differ), reductive Lie algebras, solvable Lie algebras up to dimension \(4\), nilpotent Lie algebras up to dimension \(6\) (some special cases excluded). The solvable and nilpotent cases are handled using the databases for such algebras described in Section Solvable and Nilpotent Lie Algebras Classification).

In the case of reductive Lie algebras, split maximal toral subalgebras for \(L\) and \(M\) may be provided in the optional arguments \(HL\) and \(HM\), respectively. If these are not provided an attempt is made to compute them, a process which may fail, particularly in characteristic \(0\).

This intrinsic has two return values: the first a boolean describing whether \(L\) and \(M\) are isomorphic. If so, the second is an isomorphism from \(L\) to \(M\), otherwise the second is a string describing the reason for non-isomorphism.

An error is thrown if isomorphism cannot be determined.

IsKnownIsomorphic(L, M): AlgLie, AlgLie -> BoolElt, BoolElt, .#

HL: AlgLie                    Default: false
HM: AlgLie                    Default: false

Returns true if Magma can determine isomorphism between Lie algebras \(L\) and \(M\). If so, the second return value is whether \(L\) and \(M\) are isomorphic, and the third is an isomorphism or a string (describing the reason for non-isomorphism). Refer to IsIsomorphic for more details on applicability and the meanings of the return values.

IsIsomorphism(m): Map[AlgLie, AlgLie] -> BoolElt#: Returns true if the mapping \(m\) between two Lie algebras is an isomorphism of Lie algebras.

Example: Isomorphic (ex-4fbecb)#

We demonstrate that B\(_2\) and C\(_2\) are isomorphic over \({\mathbb{Q}}\).

> k := Rationals();
> L := LieAlgebra("B2", k); M := LieAlgebra("C2", k);
> b, c := IsIsomorphic(L, M);
> b;
true
> IsIsomorphism(c);
true
> c(L.1);
(0 0 1 0 0 0 0 0 0 0)

Operations on Lie Algebras#

Basic Invariants#

Changing Base Rings#

Bases#

Operations for Semisimple and Reductive Lie Algebras#