The Lie bracket of two vector fields X,Y∈C∞(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 f↦X(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+ξ∧ϕ. 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]+LXη−iYdξ.
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+ξ∧ϕ. 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]+LXη−iYdξ.
No comments:
Post a Comment
Please, use the comments to share any thoughts or concerns, and also to leave some anonymous feedback, which will not be published.