Lie Groups Lecture 7

This lectue (lie7) was given on 9/17/09.


Recall ad \colon \mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}) takes X \mapsto [X, -] = ad_X.  Now for every Z \in \mathfrak{g},

[ad_X, ad_Y](Z) = ad_X(ad_Y(Z)) - ad_Y(ad_X(Z))

= [X,[Y,Z]] - [Y,[X,Z]]

Also, ad_{[X,Y]}(Z) = [[X,Y],Z].  It follows ad_{[X,Y]} = [ad_X,ad_Y] iff

[[X,Y], Z] = [X,[Y,Z] - [Y,[X,Z]]

[X,[Y,Z] + [Y,-[X,Z]] + [Z,[X,Y]] = 0

[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0

So the Jacobi identity is equivalent to the adjoint representation being a homomorphism of lie algebras.

Some results: (thm 2.40; Ado) Every finite dimensional lie algebra is isomorphic to a subalgebra of \mathfrak{gl}(n,\mathbb{R}) for some n (similar result for \mathbb{C}).

Given a basis for a lie algebra \{X_i\} you can determine the structure of the lie bracket by the structure constants [X_i, X_j] = \sum_k X^{ij}_k.

A real lie algebra \mathfrak{g} can be complexfied and the lie bracket extends uniquely \mathfrak{g}_\mathbb{C} = \mathfrak{g}\otimes \mathbb{C}.

Some of the material in Appendix C was covered.  Let G be a smooth manifold with a group structure.  The tangent space at a point p \in G is

T_pG = \{X \colon C^\infty(G) \to \mathbb{R}| X(fg) = f(p)X(g) + X(f)g(p) \ etc.\}

You also require a condition like if f = g in a nbd of p, then X(f) = X(g). Left multiplication L_g gives a map DL_g \colon T_pG \to T_{gp}G.  Fixing a vector v \in T_eG a left invariant vector field can be defined via X^v_g = DL_g(v).  This gives a vector space isomorphism between T_eG and the set of left invariant vector fields.  Also, there is a notion of taking the bracket of vector fields. A vector field is derivation C^\infty(G) \to C^\infty(G), hence they can be composed, so it makes sense to take X\circ Y - Y\circ X of two vector fields.

In this way we can define a Lie bracket on T_eG by thinking of vectors as left invariant vector fields and taking the bracket as above.

Alternatively, for every g \in G, there is an adjoint map Ad_g = L_g \circ R_g = R_g \circ L_g \colon G \to G.  Recall R_g(h) = hg^{-1}.  You can take the differential at the identity and get a representation Ad \colon G \to GL(T_eG).  Taking differential of this you get ad\colon T_eG \to \mathfrak{gl}(T_eG), then can define [X,Y] = ad_X(Y).

Some questions

  1. Is the description as left invariant vector fields consistent with the definition using adjoint representation
  2. What happens when you define things with right invariant vector fields
  3. What is the bracket of a left invaraint vector fields with a right invariant vector field.

The lower central series is the unbalanced one: G_n := [G_{n-1}, G]; the derived series is the symmetric one G_{n+1}= [G_n, G_n]. Nilpotent means the lower central series terminates, solvable means the derived series terminates and nilpotent implies solvable.

A lie algebra \mathfrak{g} is simple is it has no notrivial ideals; semisimple means no notrivial solvable ideals.


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