Structure Operations#

Numerical Invariants#

Note that the # operator only returns a value for finite (quotients of) free algebras.

Rank(F): AlgFr -> RngIntElt#

Return the number of indeterminates of free algebra \(F\) over its coefficient ring.

Characteristic(F): AlgFr -> RngIntElt#
# F: AlgFr -> RngIntElt#

Homomorphisms#

In its most general form, a homomorphism taking a free algebra \(K\langle x_1, \ldots, x_n\rangle\) as domain requires \(n+1\) pieces of information, namely, a map (homomorphism) telling how to map the coefficient ring \(K\) together with the images of the \(n\) indeterminates. The map for the coefficient ring is optional.

hom< F -> S | f, y₁, ..., yₙ >: AlgFr, Rng -> Map#
hom< F -> S | y₁, ..., yₙ >: AlgFr, Rng -> Map#

Given a free algebra \(F=K\langle x_1, \ldots, x_n\rangle\), a ring or associative algebra \(S\) (including another FP-algebra or a matrix algebra), and a map \(f : K\rightarrow S\) and \(n\) elements \(y_1, \ldots, y_n\in S\), create the homomorphism \(g : F\rightarrow S\) by applying the rules that \(g(rx_1^{a_1}\cdots x_n^{a_n})=f(r)y_1^{a_1}\cdots y_n^{a_n}\) for monomials and linearity, that is, \(g(M+N)=g(M)+g(N)\). The coefficient ring map may be omitted, in which case the coefficients are mapped into \(S\) by the coercion map. No attempt is made to check whether the map defines a genuine homomorphism.

Example: Homomorphism (ex-15e82d)#

In this example we map an algebra \(F\) first into \(F\) itself, and then into a matrix algebra.

> K := RationalField();
> F<x,y,z> := FreeAlgebra(K, 3);
> h := hom<F -> F | x*y, y*x, z*x>;
> h(x);
x*y
> h(y);
y*x
> h(x*y);
x*y^2*x
> h(x + y + z);
x*y + y*x + z*x
> A := MatrixAlgebra(K, 2);
> M := [A | [1,1,-1,1], [-1,3,4,1], [11,7,-7,8]];
> M;
[
    [ 1  1]
    [-1  1],

    [-1  3]
    [ 4  1],

    [11  7]
    [-7  8]
]
> h := hom<F -> A | M>;
> h(x);
[ 1  1]
[-1  1]
> h(y);
[-1  3]
[ 4  1]
> h(x*y - y*z);
[ 35 -13]
[-32 -38]

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