# 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 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 $\mathrm{End}V$ with the structure of an associative algebra. However, we could also regard $\mathrm{End}V$ as a Lie algebra relative to the commutator bracket:

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

Title commutator bracket CommutatorBracket 2013-03-22 12:33:51 2013-03-22 12:33:51 rmilson (146) rmilson (146) 8 rmilson (146) Definition msc 17A01 msc 17B05 msc 18A40 LieAlgebra commutator Lie algebra commutator