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Monday, 26 January 2015

L7a. A bracket for T+T*.

The Lie bracket of two vector fields X,YC(T) is usually defined as the only vector field [X,Y] such that [X,Y](f)=X(Y(f))Y(X(f)). This works very nicely for vector fields, since they are derivations of functions. One checks that fX(Y(f))Y(X(f)) is a derivation and we call that derivation [X,Y].

The Lie bracket is also the only one such that, for any differential form ϕ, i[X,Y]ϕ=[LX,iY]ϕ=[[d,iX],iY]ϕ, where the bracket on the leftmost term is the Lie bracket that we are defining, while the other brackets are supercommutators ( [x,y]=xy(1)|x||y|yx ).

We would like now to generalize this. We do want a bracket for the sections of T+T. Our starting poing should be how they act on a differential form: (X+ξ)ϕ=iXϕ+ξϕ. If we only look at functions, we will not be succesful. Whereas X(f)=iXdf is again a function, (X+ξ)df is not. We must use the second approach. Define the bracket [X+ξ,Y+η] as the only generalized vector field satisfying i[X+ξ,Y+η]ϕ=[[d,(X+ξ)],(Y+η)]ϕ for any differential form ϕ. This is called a derived bracket and there is a general theory behind it.

Since the expression we are using to derive the bracket is bilinear on the fields, we can first compute the terms [X,Y],[X,η],[ξ,Y],[ξ,η] as we did in the class. The result is the Dorfman bracket. [X+ξ,Y+η]=[X,Y]+LXηiYdξ.

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