Noncommutative algebra is a generalization of matrix algebra in which matrix multiplication is replaced by a noncommutative 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 noncommutative algebra, along with state-of-the-art algorithms for basic polynomial operations, Gröbner basis computations, etc.

NonCommutativeAlgebra — represent a noncommutative algebra

Noncommutative Polynomials

NonCommutativePolynomialQ — test if an expression is a noncommutative polynomial

NonCommutativeVariables — list of variables in a noncommutative polynomial

NonCommutativeMonomialList — list of monomials in a noncommutative polynomial

Basic Structural Operations

NonCommutativeExpand — expand a noncommutative polynomial

NonCommutativeCollect — collect together terms in a noncommutative polynomial

Polynomial Systems

NonCommutativeGroebnerBasis — compute a noncommutative Gröbner basis

NonCommutativePolynomialReduce — noncommutative polynomial reduction

Commutator  ▪  Anticommutator  ▪  GeneralizedPower  ▪  NonCommutativeMultiply