Commutator
Commutator[x,y]
gives the commutator x**y-y**x of x and y.
Commutator[x,y,alg]
gives the commutator of x and y in the noncommutative algebra alg.
Details
- Commutator gives the commutator of two elements of a noncommutative algebra.
- If ⊗ and ⊕ denote the multiplication and addition in the algebra alg, then Commutator[x,y,alg]==(x⊗y)⊕(-y⊗x).
- alg can be a NonCommutativeAlgebra object, {Dot,n}, Composition, TensorProduct or NonCommutativeMultiply. If the algebra argument is omitted, NonCommutativeAlgebra with the default property values is used.
Examples
open all close allBasic Examples (2)
Scope (4)
The commutator of x and y over the algebra of square matrices with Dot product:
The commutator of x and y over the algebra of linear endomorphisms with Composition:
Traditional form of Commutator[x,y]:
Reducing a polynomial modulo the commutator of two variables commutes the variables:
Properties & Relations (2)
Reducing a polynomial modulo the commutator of two variables commutes the variables:
Anticommutator gives the anticommutator of two elements of an algebra:
Text
Wolfram Research (2025), Commutator, Wolfram Language function, https://reference.wolfram.com/language/ref/Commutator.html.
CMS
Wolfram Language. 2025. "Commutator." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/Commutator.html.
APA
Wolfram Language. (2025). Commutator. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Commutator.html