Let $A$ be an associative algebra over a field $K$ . For $a,b \in A$ , the element of $A$ defined by $$[a,b]=ab-ba$$ is called the commutator of $a$ and $b$ . The corresponding bilinear operation $$[-,-]: A\times A\rightarrow A$$ is called the commutator bracket.

The commutator bracket is bilinear, skew-symmetric, and also satisfies the Jacobi identity. To wit, for $a,b,c\in A$ we have $$[a,[b,c]] + [b,[c,a]] + [c,[a,b]] = 0.$$ 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 functor from the category 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 multiplication operation is given by the commutator bracket. It must be noted that this functor is right-adjoint to the universal enveloping algebra functor.

Examples

• Let $V$ be a vector space. Composition endows the vector space of endomorphisms $\End V$ with the structure of an associative algebra. However, we could also regard $\End V$ as a Lie algebra relative to the commutator bracket: $$[X,Y] = XY-YX,\quad X,Y\in \End V.$$
• The algebra 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 commutative 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 homogeneous 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.