Let be an associative algebra over a field . For , the element of defined by
is called the commutator bracket.
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 into a Lie algebra that has the same underlying vector space as , 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.
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 are differential operators of order and , respectively, the compositions and have order . Their highest order term coincides, and hence the commutator has order .
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.
|Date of creation||2013-03-22 12:33:51|
|Last modified on||2013-03-22 12:33:51|
|Last modified by||rmilson (146)|
|Defines||commutator Lie algebra|