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 \(r\).

Construction of Elements#