Commutative algebra by generators and quadratic relations

Question

My apologies if this has been asked before. Please also be gentle, as this is my first question here.

I am trying to define a commutative algebra by generators and relations in the following way. For a simple example, we can consider the Z[b] algebra generated by a, modulo a^2=b. I would like to introduce a rule (or a list of rules in more complicated examples):

In[1]: TheAlgebra = a*a -> b

It would be cool to evaluate products like this:

In[2]: a^3 //. TheAlgebra

However, this input simply returns the symbol a^3. I would have expected the result a*b. The following input works though:

In[2]': (a*a//.TheAlgebra)*a //. TheAlgebra

My question is this: how can I get Mathematica to simplify an expression such as a^kb^l in a smart way? There are similar questions on the site, but maybe the answer for this one is simpler, since I'm only interested in simplifying products of the various symbols together with some integers.

Maybe there is a way to force Mathematica to write a^3 in a full-form expression that I like, such as: Times[a,Times[a,a]]?

Answer 1

The general case of simplifying a sum of products according to a predefined algebra can be solved by representing each product in the form prod[a1,a2,a3,...] — i.e. a general non-commutative product. Defining your own prod like this neatly avoids any of Mathematica's built-in rules for simplification of expressions.

Define rules for expanding out any Plus that is embedded inside prod, and extracting any coefficients not involved in the algebra (here I assume the algebra has 3 elements a, b, c):

distribrule = {prod[u___, x_ + y_, v___] :> prod[u, x, v] + prod[u, y, v], 
  prod[u___, Times[x_, y : a | b | c], v___] :> x prod[u, y, v]};

Define a function for building a product rule to use in the algebra:

prodrule[x_, y_, z_] := prod[u___, x, y, v___] :> prod[u, z, v];

Define a toy algebra with only one product rule, but you can have as many product rules as you like:

algebra = Flatten@{distribrule, prodrule[a, a, b - 2 c]};

As a test of this approach, construct a random product and simplify it:

expr = prod @@ RandomChoice[{a, b}, 10]
expr //. algebra // Expand