L7a. A bracket for T+T*.

The Lie bracket of two vector fields \( X, Y \in \mathcal{C}^\infty(T) \) 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 \(f\mapsto X(Y(f)) - Y(X(f)) \) is a derivation and we call that derivation \( [X,Y] \).

The Lie bracket is also the only one such that, for any differential form $\phi$, $$ i_{[X,Y]} \phi = [ \mathcal{L}_X, i_Y] \phi = [ [d,i_X],i_Y] \phi,$$ 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: $$(X+\xi)\cdot \phi = i_X \phi + \xi\wedge \phi .$$ If we only look at functions, we will not be succesful. Whereas $X(f)=i_X df$ is again a function, $(X+\xi)\cdot df $ is not. We must use the second approach. Define the bracket $[X+\xi,Y+\eta]$ as the only generalized vector field satisfying $$ i_{[X+\xi,Y+\eta]} \phi = [ [d,(X+\xi)\cdot],(Y+\eta)\cdot] \phi$$ for any differential form $\phi$. 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 \( [X,Y], [X,\eta], [\xi,Y], [\xi,\eta] \) as we did in the class. The result is the Dorfman bracket. $$[X+\xi,Y+\eta] = [X,Y] + \mathcal{L}_X \eta - i_Y d\xi .$$