Construction of General Algebras and their Elements

Construction 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 \(e_1, \ldots, e_n\) 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 \(Q\).

AssociativeAlgebra< R, n | Q >: Rng, RngIntElt, SeqEnum -> AlgAss
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.

QuaternionAlgebra< K | a, b >: Rng, RngElt, RngElt -> AlgQuat

This function creates the quaternion algebra \(A\) over the field \(K\) on generators \(x\) and \(y\) with relations \(x^2 = a\), \(y^2 = b\), and \(xy = -yx\).

LieAlgebra< R, n | Q >: Rng, RngIntElt, SeqEnum -> AlgLie
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 \((x,y) \rightarrow x*y - y*x\).

GroupAlgebra(R, G): Rng, Grp -> AlgGrp

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

MatrixAlgebra(R, n): Rng, RngInt -> AlgMat

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

Construction of an Element of a General Algebra

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 \(A\).

One(A): AlgGen -> AlgGenElt
A ! 1: AlgGen, RngIntElt -> AlgGenElt

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

Random(A): AlgGen -> AlgGenElt

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