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.
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, ..., en such that Q contains the structure constants of A, i.e. ei * ej = ∑aijk ek, where aijk is the element in position (i - 1) * n2 + (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.
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.
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.
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 Mn(R) of dimension n2 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.
Create the zero element of the algebra A.
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.