Subsections

• Construction of a General Algebra
• Construction of an Element of a General Algebra

The construction of an algebra depends on its category. The chapters on the individual algebra categories describe this in detail. Here only an overview is given.

Algebra< R, n | Q > : Rng, RngIntElt, SeqEnum -> AlgGen

Let R be ring, n an integer and Q a sequence of n^3 elements of R. This function creates an algebra A of dimension n over R with basis e_1, ..., e_n such that Q contains the structure constants of A, i.e. e_i * e_j = sum a_(ij)^k e_k, where a_(ij)^k is the element in position (i - 1) * n^2 + (j - 1) * n + k of Q.

    Check: BoolElt                      Default: true

This function creates the associative structure constant algebra A as returned by Algebra< R, n | Q >. By default, the algebra is checked on associativity, but this can be avoided by setting Check := false. The returned algebra is of type AlgAss.

    Check: BoolElt                      Default: true

This function creates the Lie structure constant algebra A as returned by Algebra< R, n | Q >. By default, the algebra is checked to be a Lie algebra, but this can be avoided by setting Check := false. The returned algebra is of type AlgLie.

LieAlgebra(A) : AlgAss -> AlgLie

Given an associative algebra A, create the Lie algebra generated by the elements in L using the induced Lie product (x, y) -> x * y - y * x.

Given a ring R and a group G construct the group algebra R[G] of dimension |G| over R.

Given a positive integer n and a ring R, create the full matrix algebra M_n(R) of dimension n^2 over R.

The construction of a generic element of an algebra varies for the different types of algebras and is therefore explained in the corresponding chapters.

Zero(A) : AlgGen -> AlgGenElt

A ! 0 : AlgGen, RngIntElt -> AlgGenElt

Create the zero element of the algebra A.

A ! 1 : AlgGen, RngIntElt -> AlgGenElt

If it exists, create the identity element of the algebra A; otherwise an error occurs.

Given an algebra A defined over a finite ring, return a random element.