Noncommutative Algebra

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