commutator bracket

Let A be an associative algebra over a field K. For a,bA, the element of A defined by


is called the commutatorPlanetmathPlanetmath of a and b. The corresponding bilinear operation


is called the commutator bracket.

The commutator bracket is bilinear, skew-symmetric, and also satisfies the Jacobi identityMathworldPlanetmathPlanetmath. To wit, for a,b,cA we have


The proof of this assertion is straightforward. Each of the brackets in the left-hand side expands to 4 terms, and then everything cancels.

In categorical terms, what we have here is a functorMathworldPlanetmath from the categoryMathworldPlanetmath of associative algebras to the category of Lie algebras over a fixed field. The action of this functor is to turn an associative algebra A into a Lie algebra that has the same underlying vector space as A, but whose multiplicationPlanetmathPlanetmath operationMathworldPlanetmath is given by the commutator bracket. It must be noted that this functor is right-adjoint to the universal enveloping algebra functor.


  • Let V be a vector space. Composition endows the vector space of endomorphismsPlanetmathPlanetmathPlanetmath EndV with the structureMathworldPlanetmath of an associative algebra. However, we could also regard EndV as a Lie algebra relative to the commutator bracket:

  • The algebraPlanetmathPlanetmathPlanetmath of differential operators has some interesting properties when viewed as a Lie algebra. The fact is that even though the composition of differential operators is a non-commutative operation, it is commutativePlanetmathPlanetmathPlanetmath when restricted to the highest order terms of the involved operators. Thus, if X,Y are differential operators of order p and q, respectively, the compositions XY and YX have order p+q. Their highest order term coincides, and hence the commutator [X,Y] has order p+q-1.

  • In light of the preceding comments, it is evident that the vector space of first-order differential operators is closed with respect to the commutator bracket. Specializing even further we remark that, a vector field is just a homogeneousPlanetmathPlanetmathPlanetmath first-order differential operator, and that the commutator bracket for vector fields, when viewed as first-order operators, coincides with the usual, geometrically motivated vector field bracket.

Title commutator bracket
Canonical name CommutatorBracket
Date of creation 2013-03-22 12:33:51
Last modified on 2013-03-22 12:33:51
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 8
Author rmilson (146)
Entry type Definition
Classification msc 17A01
Classification msc 17B05
Classification msc 18A40
Related topic LieAlgebra
Defines commutator Lie algebra
Defines commutator