Implementing associative algebras in Mathematica

We introduce implementation of noncommutative algebras via Mathematica. ​

Designing idea

What expressions are noncommutative? We can specify and store noncommutative operators in the list $operator and use FreeQ to determine whether an expression is a scalar or an operator.

$operator = {};

scalarQ[expr_] :=
    FreeQ[expr,Alternatives@@$operator];

The function scalarQ[expr] returns True when expr is a scalar.

We then implement the relations of noncommutative multiplication by a list of rules $relation, and simplify expressions using the function ReplaceRepeated.

$relation = {};

algebraSimplify[expr_] :=
    ReplaceRepeated[expr,$relation]//Simplify;

Noncommutative multiplication needs to satisfy the rules of linearity and distributivity, which is independent of specific algebras:

NonCommutativeMultiply//Unprotect;

NonCommutativeMultiply//Attributes = {Flat,OneIdentity};

$relationInternal = {
    x_**y_/;scalarQ[x]||scalarQ[y]:>x*y,
    (k_?scalarQ*x_)**y_:>k*x**y,
    x_**(k_?scalarQ*y_):>k*x**y,
    (x_+y_)**z_:>x**z+y**z,
    z_**(x_+y_):>z**x+z**y
};

Here we have overload the internal function NonCommutativeMultiply with operator form ** by adding the attributes {Flat,OneIdentity}.

To make the results readable, we define additional formatting rules.

$printing = {} ; 

algebraPrint[expr_] :=
    ReplaceRepeated[expr,$printing];

To ensure the convergence of ReplaceRepeated, we need to determine the ordering of operators, and a simple choice is the Poincare-Birkhoff-Witt basis.

Example

$operator = {X,Y,H,v};

$relation = {
    Splice@$relationInternal,
    X**H:>-2 X+H**X,
    H**Y:>-2 Y+Y**H,
    X**Y:>H+Y**X,
    H**v[h_]:>h v[h],
    X**v[h_]:>0
};

To-do

• algebra and module from factorization algebra