Non-commutative algebra is a generalization of matrix algebra in which matrix multiplication is replaced by a non-commutative multiplication operator in an associative algebra. This generalization finds many applications in quantum theory, special functions, differential equations, etc.

The Wolfram Language provides a variety of functions for working with non-commutative algebra, along with state-of-the-art algorithms for basic polynomial operations, Gröbner basis computations, etc.

Non-Commutative Algebra Representation

NonCommutativeAlgebra — represent a non-commutative algebra

CliffordAlgebra — construct a Clifford algebra

GrassmannAlgebra — construct a Grassmann algebra

WeylAlgebra — construct a Weyl algebra

Non-Commutative Polynomials

NonCommutativePolynomialQ — test if an expression is a non-commutative polynomial

NonCommutativeVariables — list of variables in a non-commutative polynomial

NonCommutativeMonomialList — list of monomials in a non-commutative polynomial

Basic Structural Operations

NonCommutativeExpand — expand a non-commutative polynomial

NonCommutativeCollect — collect together terms in a non-commutative polynomial

Polynomial Systems

NonCommutativeGroebnerBasis — compute a non-commutative Gröbner basis

NonCommutativePolynomialReduction — reduce a polynomial modulo a list of polynomials

NonCommutativePolynomialReduce — explicit representation of polynomial reduction

Commutator  ▪  Anticommutator  ▪  GeneralizedPower  ▪  NonCommutativeMultiply