commutator bracket

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

$$[-,-]:A\times A\to A$$

is called the commutator bracket.

The commutator bracket is bilinear, skew-symmetric, and also satisfiesthe 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 inthe left-hand side expands to 4 terms, and then everything cancels.

In categorical terms, what we have here is a functor from the categoryof associative algebras to the category of Lie algebras over a fixedfield. 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 commutatorbracket. It must be noted that this functor is right-adjoint to theuniversal enveloping algebra functor.

Examples

• Let $V$ be a vector space. Composition endows the vector space ofendomorphisms $\mathrm{End}V$ with the structure of an associative algebra.However, we could also regard $\mathrm{End}V$ as a Lie algebra relative tothe commutator bracket:

  $$[X,Y]=XY-YX,X,Y\in \mathrm{End}V.$$

• The algebra of differential operators has some interestingproperties when viewed as a Lie algebra. The fact is that eventhough the composition of differential operators is anon-commutative operation, it is commutative when restricted to thehighest order terms of the involved operators. Thus, if $X,Y$ aredifferential operators of order $p$ and $q$, respectively, thecompositions $XY$ and $YX$ have order $p+q$. Their highest orderterm coincides, and hence the commutator $[X,Y]$ has order $p+q-1$.

• In light of the preceding comments, it is evident that thevector space of first-order differential operators is closed withrespect to the commutator bracket. Specializing even further weremark that, a vector field is just a homogeneous first-orderdifferential operator, and that the commutator bracket for vectorfields, when viewed as first-order operators, coincides with theusual, geometrically motivated vector field bracket.