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SageMath 9.5.beta3	sage	sagemath

Sage-Train_Algebras

Sage-Train_Algebras is a tiny library to compute with (commutative) (pre)train algebras of finite dimension. It originates from two successive CIMPA schools in Burkina Faso (Bobo Dioulasso, 2012 & 2021).

It's meant to support computer exploration of (small) examples defined by their multiplication table, and provides utilities to easily define such, test isomorphism, test whether they actually are train algebras, and so on.

Context

A pre-train algebra is a commutative non associative algebra equiped with an algebra morphism $\omega$ to its base ring satisfying an equation of the form ... [TODO]

A train algebra is a pretrain algebra satisfying an equation of the form $\alpha_0 x^n + \alpha_1 \omega(x)x^{n-2} + \ldots + \omega(x)^n =0$, where $alpha_0+\cdots+\alpha_n=0$ (wlog, we may assume $\alpha_0=1$).

TODO: references

First example

import train_algebras.examples

Let's take an example of train algebra:

A = train_algebras.examples.TrainAlgebra_2_4()

Here is an element:

A.an_element()

One may build elements by linear combinations of elements of the basis:

e = A.basis()

y = e[0] + 2*e[1] + 3*e[2] + 4*e[3]
y

Let's do a bit of arithmetic:

y*y

(y*y)*(y*y)

y*y*y*y

By Python's priority rules, the previous product is computed as:

((y*y)*y)*y

As can be seen above, the algebra is indeed not associative.

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Looking for isomorphisms between two algebras

In this section, we show how to search for isomorphisms between two examles of finite dimensional (associative or not) algebras, here $A_2$ and $A_3$.

In the first subsection we do the search step by step to explain how it works by reducing the problem to solving polynomial equations. In the second step, we use the method isomorphism_ideal to automate the process.

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Initialisation:

import train_algebras.examples

By hand

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We start by defining a generic linear morphism from $A_2$ to $A_3$ mapping each basis element $e_i$ of $A_2$ to $\sum_{j} x_{i,j} f_j$, where the $x_{i,j}$' are indeterminates and the $f_j$'s the basis of $A_3$. To achieve this, we need to extend the base ring with these indeterminates.

In our example, the basis of $A_2$ and $A_3$ are indexed by $I={e, v, t}$.

Here are the indeterminates, as a family mapping each pair (i,j) to it's indeterminate:

I = ["e", "v", "t"]
indeterminates = Family({ (i, j): f"x_{i}{j}" for i in I for j in I })
indeterminates

indeterminates["e","v"]

We extend the base ring (here the rationals QQ) as a polynomial ring to include the above indeterminates, and also $y$ which will need later:

R = QQ[tuple(indeterminates) + ("y",)]
R

Now we rebuild the indeterminates as a family $(x_{i,j})_{i\in I, j\in I}$ of elements of R:

indeterminates = indeterminates.map(R)
xev = indeterminates["e", "v"]
xev^3 + 4*xev

We define the algebras of interest over this polynomial ring:

A2 = train_algebras.examples.A2(R)
A3 = train_algebras.examples.A3(R)

and define a linear morphism $w$ between $A_2$ and $A_3$ with generic coefficients:

def w_on_basis(j):
    return A3.sum_of_terms([ (i, indeterminates[i,j]) for i in I])
w = A2.module_morphism(w_on_basis, codomain=A3)

This is the morphism with the following matrix:

w.matrix()

Let's put this morphism into action:

e,v,t = A2.basis()

w(e + 2*v)

w(e)*w(t) - w(e*t)

Now we build the equations on the coefficients stating that this is a morphism, and display them after removing duplicates:

equations = [c
             for a1 in A2.basis()
             for a2 in A2.basis()
             for c in (w(a1)*w(a2) - w(a1*a2)).coefficients() ]
set(equations)

Let's solve the equations. Alas, Sage's solve function does not accept directly a system of polynomial equations:

solve(equations, R.gens())

We need to build explicitly the ideal defined by these equations and request properties of its variety:

I = R.ideal(equations)
I.dimension()

This means that there are a 3-dimensional subvariety of morphisms between $A_2$ and $A_3$. We could recover the values of the indeterminates from I, and build explicitly the morphisms.

However we don't want just morphisms but isomorphisms. To achieve this, we force the determinant $d$ of the matrix of the morphism to be non-zero by adding an equation $dy=1$ where $y$ is a new variable. At the end, there is no isomorphism:

d = matrix(3,3,indeterminates.values()).determinant()
y = R.gens()[-1]
equations.append(d*y-1)
I = R.ideal(equations)
I.dimension()

TODO: clarify what this does: check the documenation to see if this is solving over the base ring, or over, say, its algebraic closure.

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Using the `isomorphism_ideal` method

This process is automatized by the method isomorphism_ideal of FiniteDimensionalNonAssociativeAlgebrasWithBasis:

A2.isomorphism_ideal??

A2 = train_algebras.examples.A2(QQ)
A3 = train_algebras.examples.A3(QQ)

I = A2.isomorphism_ideal(A3)
I.dimension()

A2.isomorphism_ideal(A2).dimension()