Sergei Yakovenko's blog: on Math and Teaching

Monday, February 6, 2017

Lecture 12 (Jan 23, 2017)

Filed under: Calculus on manifolds course,lecture — Sergei Yakovenko @ 4:51
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Lie groups and Lie algebras

A Lie group is a smooth manifold with carries on it the structure of a group which is compatible with the smooth structure (i.e., the multiplication by an element of the group is a smooth self-map, necessarily a diffeomorphism, of the manifold).

This group structure means very high “homogeneity” of the manifold, in particular, existence of a flat connexion. On the other hand, there is a distinguished point on the manifold, corresponding to the group unit.

It turns out that the tangent space at the group unit is equipped with a natural operation, the antisymmetric bilinear bracket, closely related to the commutator of vector fields on the Lie group. This algebraic structure is called the Lie algebra, and it in a sense “encodes” the group structure.

The notes will be available later.

Objects that live on manifolds: functions, curves, vector fields

We discussed how one may possibly define smooth functions on manifolds, smooth curves, tangent vectors, smooth vector fields. Next we discussed how these objects can be carried between manifolds if there exists a smooth map (or diffeomorphism) between these manifolds.

Flow of vector field. Lie derivatives.

Every vector field $X$ on a compact smooth manifold $M$ defines a family of automorphisms $F^t_X$ (diffeomorphic self-maps) of $M$ which form a one-parametric group, called the flow. Any object living on $M$ can be carried by the flow by the operators $\bigl(F^t_X\bigr)^*$, $t\in\mathbb R$. The Lie derivative along $X$ is the velocity of this action at $t=0$, namely, $L_X=\frac{\mathrm d}{\mathrm dt}\big|_{t=0}\bigl(F^t_X\bigr)^*$.

We show that the Lie derivative of functions coincides with the action of the corresponding derivations, and the Lie derivation of another vector field is the Lie bracket $L_XY=[X,Y]$.

At the end of the day we establish the identities $[L_X,L_Y]=L_{[X,Y]}$ and the Leibniz rule for $L_X$ with respect to the Lie bracket, $L_X[Y,Z]=[Y,L_XZ]+[L_XY,Z]$. Both turn out to be equivalent to the Jacobi identity $[X,[Y,Z]]+[Y,[X,Z]]+[Z,[X,Y]]=0$ for the Lie bracket.

The lecture notes are available here.

* In fact, it is a slightly different Jacobi identity, not for the Lie bracket of vector fields, but for the vector product $\mathbb R^3\times\mathbb R^3\mapsto\mathbb R^3$, $u,v\mapsto [u,v]=u\times v$. But later we will see that this vector product is the commutator in the Lie algebra of vector fields on the group of orthogonal transformations of $\mathbb R^3$, thus the difference is purely technical.