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 n³ elements of R. This function creates an algebra A of dimension n over R with basis e₁, ..., e_n such that Q contains the structure constants of A, i.e. e_i * e_j = ∑a_ij^k e_k, where a_ij^k is the element in position (i - 1) * n² + (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 x² = a, y² = 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) -> 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 n² 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.