Construction of Elements

Zero(L): AlgLie -> AlgLieElt

Zero(L): AlgMatLie -> AlgMatLieElt

L ! 0: AlgLie, RngIntElt -> AlgLieElt

L ! 0: AlgMatLie, RngIntElt -> AlgMatLieElt

The zero element of the Lie algebra \(L\).

Random(L): AlgLie -> AlgLieElt

Random(L): AlgMatLie -> AlgMatLieElt

Given a Lie algebra \(L\) defined over a finite ring, a random element is returned.

Construction of Elements of Structure Constant Algebras

elt<L | r₁, r₂, ..., rₙ>: AlgLie, RngElt, RngElt, ..., RngElt -> AlgLieElt

Given a Lie algebra \(L\) of dimension \(n\) over a ring \(R\), and ring elements \(r_1, r_2, \ldots, r_n \in R\) construct the element \(r_1 * e_1 + r_2 * e_2 + \ldots + r_n * e_n\) of \(L\).

L ! Q: AlgLie, SeqEnum[RngElt] -> AlgLieElt

Given a Lie algebra \(L\) of dimension \(n\) and a sequence \(Q = [r_1, r_2, \ldots, r_n]\) of elements of the base ring \(R\) of \(L\), the element \(r_1 * e_1 + r_2 * e_2 + \ldots + r_n * e_n\) of \(L\) is constructed.

BasisProduct(L, i, j): AlgLie, RngIntElt, RngIntElt -> AlgLieElt

Returns the product of the \(i\)-th and \(j\)-th basis element of the Lie algebra \(L\).

BasisProducts(L): AlgLie -> SeqEnum

Rep: MonStgElt                    Default: "Dense"

Returns the products of all basis elements of the Lie algebra \(L\).

The optional parameter Rep may be used to specify the format of the result. If Rep is set to “Dense”, the products are returned as a sequence \(Q\) of \(n\) sequences of \(n\) elements of \(L\), where \(n\) is the dimension of \(L\). The element \(Q[i][j]\) is the product of the \(i\)-th and \(j\)-th basis elements.

If Rep is set to “Sparse”, the products are returned as a sequence \(Q\) containing quadruples \((i,j,k,a_{ijk})\) signifying that the product of the \(i\)-th and \(j\)-th basis elements is \(\sum_{k=1}^n a_{ijk} b_k\), where \(b_k\) is the \(k\)-th basis element and \(n =\) dim(\(L\)).

Construction of Matrix Elements

Matrix Lie elements can be constructed using the functions below. For more information on constructing matrices see Section Construction of a Matrix.

elt<R | L>: AlgMatLie, [ RngElt ] -> AlgMatLieElt

elt<R | L>: AlgMatLie, RngElt, ..., RngElt -> AlgMatLieElt

R ! L: AlgMatLie, [RngElt] -> AlgMatLieElt

Create the element of the matrix Lie algebra \(R\) of degree \(n\) whose entries are the \(n^2\) elements of the sequence \(L\).

DiagonalMatrix(L, Q): AlgMatLie, [RngElt] -> AlgMatLieElt

Diagonal matrix in the matrix Lie algebra \(L\), given by the sequence \(Q\) of ring elements.

ScalarMatrix(L, r): AlgMatLie, RngElt -> AlgMatLieElt

Scalar matrix in the matrix Lie algebra \(L\), defined by the ring element \(r\).