Operations on Elements#

x + y: AlgLieElt, AlgLieElt -> AlgLieElt#
x + y: AlgMatLieElt, AlgMatLieElt -> AlgMatLieElt#
x - y: AlgLieElt, AlgLieElt -> AlgLieElt#
x - y: AlgMatLieElt, AlgMatLieElt -> AlgMatLieElt#
x * y: AlgLieElt, AlgLieElt -> AlgLieElt#
x * y: AlgMatLieElt, AlgMatLieElt -> AlgMatLieElt#
IsCentral(L, M): AlgLie, AlgLieElt -> BoolElt#
IsCentral(L, M): AlgMatLie, AlgMatLieElt -> BoolElt#: Given an element \(x\) of the Lie algebra \(L\), return true if \(x\) is central in \(L\).

NonNilpotentElement(L): AlgLie -> AlgLieElt#

Given a (structure constant) 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.

The algorithm follows [de Graaf, 2000], §2.7.

Example: Non Nilpotent Element (ex-a38e8a)#

We construct a non-nilpotent element of a Lie algebra.

> L:=LieAlgebra("G2",RationalField());
> NonNilpotentElement(L);
(0 0 0 0 0 1 0 0 0 0 0 0 0 0)

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AdjointMatrix(L, x): AlgLie, AlgLieElt -> AlgMatLieElt#
RightAdjointMatrix(L, x): AlgLie, AlgLieElt -> AlgMatLieElt#: Given a (structure constant) Lie algebra \(L\) and an element \(x\) of a subalgebra or ideal of \(L\), return the matrix of \({\rm ad} x\) as an element of a matrix Lie algebra.

Example: Other (ex-c4c36f)#

> L:=LieAlgebra("B2",RationalField());
> AdjointMatrix(L, 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  0  0]
[ 0  0  0  0  0  0  0  0  0  0]
[ 1  0  0  0  0  0  0  0  0  0]
[ 2  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]
[ 0  0  0  0  0 -1  0  0  0  0]

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Indexing#

a[i]: AlgLieElt, RngIntElt -> RngElt#

a[i]: AlgMatLieElt, RngIntElt -> ModTupRngElt#

If \(a\) is an element of a structure constant Lie algebra \(L\) of dimension \(n\) and \(1 \leq i\leq n\) is a positive integer, then the \(i\)-th component of the element \(a\) is returned (as an element of the base ring \(R\) of \(L\)).

If \(a\) is an element of a matrix Lie algebra \(L\) of degree \(n\) and \(1 \leq i \leq n\) then the \(i\)th row of the matrix \(a\) is returned.

a[i] := r: AlgLieElt, RngIntElt, RngElt -> AlgLieElt#

a[i] := r: AlgMatLieElt, RngIntElt, ModTupRngElt -> AlgMatLieElt#

Given an element \(a\) belonging to a structure constant Lie algebra of dimension \(n\) over \(R\), a positive integer \(1 \leq i\leq n\) and an element \(r \in R\), the \(i\)-th component of the element \(a\) is redefined to be \(r\).

If \(a\) is an element of a matrix Lie algebra \(L\) of degree \(n\) over \(R\) and \(1 \leq i \leq n\), the \(i\)th row of the matrix \(a\) is redefined to be the vector \(r\) over \(R\).

a[i, j]: AlgMatLieElt, RngIntElt, RngIntElt -> RngElt#
a[i, j] := r: AlgMatLieElt, RngIntElt, RngIntElt, RngElt -> AlgMatLieElt#: For an element \(a\) of a matrix Lie algebra \(L\) of degree \(n\) and integers \(1 \leq i, j \leq n\) return the element in the \(i\)th row and \(j\)th column of \(a\) or set this element to be \(r\) where \(r\) is an element of the coefficient ring of \(L\).