commutator bracket (original) (raw)

The commutator bracket is bilinear, skew-symmetric, and also satisfiesthe Jacobi identity. To wit, for a,b,c∈A 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 functor from the categoryof associative algebras to the category of Lie algebras over a fixed field. The action of this functor is to turn an associative algebraA into a Lie algebra that has the same underlying vector space asA, but whose multiplication operation is given by the commutator bracket. It must be noted that this functor is right-adjoint to theuniversal enveloping algebra functor.

Examples

• •
• •
  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 anon-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 X⁢Y and Y⁢X 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.