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\begin{document}
\begin{center}
{\bf Math 261A: Lie Groups, Fall 2015\\
Problem Set 1}
\end{center}
\bigskip
Problems from Varadarajan:
\smallskip
Chapter 1, Exercise 2.
Chapter 2, Exercises 2, 3, 4, 9(a), 11, 12(b,c), 16 (assuming as in
the hint that a Lie group with Lie algebra $\gfrak / \hfrak$ exists),
18, 19, 20, 21, 22, 27, 39.
\bigskip
Other problems:
\medskip
1. (a) For any object $X$ in a category $C$, the functor $h_{X} = \Hom
_{C}(-,X)$ from $C^{\op }$ (the category $C$ with arrows reversed) to
$\Sets $ is called the functor {\it represented by $X$}. Suppose $C$
is a category with finite products. Show that to give an object $G$
the structure of a group object in $C$ is equivalent to giving a
functor $g_{G}\colon C^{\op }\rightarrow \Groups $ such that $h_{G} =
f\circ g_{G}$, where $f\colon \Groups \rightarrow \Sets $ is the
`forgetful functor' sending a group to its underlying set.
\smallskip
(b) Show that to give an action of a group object $G$ in $C$ on an
object $X$ in $C$ (that is, an arrow $G\times X\rightarrow X$ such
that suitable diagrams corresponding to the usual definition of a
group action commute) is equivalent to giving a an action of the group
$g_{G}(T)$ on the set $h_{X}(T)$, for every object $T$ in $C$, which
is functorial in $T$.
\smallskip
(c) Show that when $C$ is the category of topological spaces, or of
smooth, analytic or holomorphic manifolds, $G$ is a group object if
and only if its underlying set is a group, in such a way that that the
group law and the map $g\mapsto g^{-1}$ are morphisms in the category,
that is, continuous, smooth, analytic or holomorphic maps.
\smallskip
(d) Show that in the categories of spaces in part (c) an action of a
group object $G$ on an object $X$ is the same as a group action of the
underlying set of $G$ on the underlying set of $X$, such that the
action map $G\times X\rightarrow X$ is a morphism in the category.
\medskip
2. Prove that if $G$ is a manifold and a group such that the group
operation is smooth (resp.\ analytic, holomorphic), then the map
$i\colon g\mapsto g^{-1}$ is automatically smooth (resp.\ analytic,
holomorphic). Hint: show that the group operation $\mu $ is a
submersion, and deduce that the graph of $i$ is a regularly embedded
submanifold of $G\times G$.
\medskip
3. (a) Show that the tangent space $T_{p}X$ of a real analytic
manifold $X$ at a point $p$ is canonically identified with the tangent
space at $p$ of $X$ considered as a real smooth manifold.
\smallskip
(b) Show that if $X$ is a complex holomorphic manifold, the tangent
space $T_{p}X$, when regarded as a real vector space, is canonically
identified with the tangent space at $p$ of $X$ considered as a real
analytic manifold, of twice its complex dimension.
\medskip
4. (a) Construct an isomorphism of Lie algebras $\sofrak _{4}(\CC
)\cong \sofrak _{3}(\CC )\times \sofrak _{3}(\CC )$.
(b) Construct a corresponding isogeny of Lie groups $SO_{4}(\CC
)\rightarrow SO_{3}(\CC )\times SO_{3}(\CC )$ and show that its kernel
is $\{\pm I_{4} \}$.
\medskip
5. (a) Show that the simply connected covering space of $SL_{2}(\RR
)$ can be described as the set $\widehat{SL_{2}(\RR )}$
of pairs $(g,\theta )$, where $g = \begin{pmatrix} a& b\\
c& d \end{pmatrix}\in SL_{2}(\RR )$, and $\theta \in \RR $ is a value
of $\operatorname{arg}(a+ib)$, the covering map being given by
$(g,\theta )\mapsto g$.
\smallskip
(b) Write down the group law on $\widehat{SL_{2}(\RR )}$ explicitly in
terms of $g$ and $\theta $. (You will probably find this impossible
to express as a simple closed formula.)
\medskip
6. Let $G$ be the group of Euclidean motions of $\RR ^{n}$, that is,
the semidirect product $SO(n,\RR )\ltimes \RR ^{n}$, where $\RR ^{n}$
acts on itself by translations. Describe the Lie algebra $\gfrak
=\Lie (G)$ and the exponential map $\exp \colon \gfrak \rightarrow G$.
\medskip
7. Prove that for any Lie group $G$, if $x,y\in \Lie (G)$ satisfy
$[x,y]=0$, then $\exp (x+y)=\exp (x)\exp (y)$. (You might do this
using either the Baker-Campbell-Hausdoff formula or Chevalley's
subgroup theorem. If you use BCH you will need to deal with the fact
that $x,y$ are not assumed to lie in a domain on which the formula
converges.)
\medskip
8. Calculate explicitly, in terms of matrix coordinates, the left and
right invariant vector fields $\lambda _{x}$ and $\rho _{x}$ on $G =
GL_{n}$ with value $x\in T_{e}G = \glfrak _{n}$ at the identity. Then
verify directly that $[\lambda _{x},\lambda _{y}] = \lambda _{[x,y]}$,
where $[x,y]$ is matrix commutator, and that $[\rho _{x},\rho
_{y}] = -\rho _{[x,y]}$.
\medskip
9. In the situation of Problem 8, express the differential $(d\,
\exp)_{x}$ of the exponential map at $x\in \glfrak _{n}$ explicitly in
terms of matrix coordinates. Use this to find the vector field $\xi
_{y}$ on $\glfrak _{n}$ which is related via $\exp $ to $\lambda _{y}$,
and verify that the result agrees with the formula
\[
\xi_{y} (x)=\frac{\ad x}{1-e^{-\ad x}}\, y.
\]
On what subset of $\gfrak _{n}$ is $\xi _{y}$ defined for all $y$?
How is this related to the locus where $d\, \exp $ is singular, and
why?
\medskip
10. Let $T$ be the tensor algebra over $\QQ $ on generators $X,Y$ and
let $F$ be the Lie subalgebra of $T$ generated by $X$ and $Y$, with
commuator in $T$ as Lie bracket. Note that $F$ is a graded subspace
of $T$, that is, $F$ is the direct sum of its degree components $F_{n}
= F\cap T_{n}$, since if $a\in T_{m}$, $b\in T_{n}$, then
$[a,b]=ab-ba\in T_{m+n}$.
For $q\in T$, let $\Theta (q)$ be the operator on $F$ given by
subsituting $(\ad X)$ for $X$ and $(\ad Y)$ for $Y$ in $q$. Define a
$\QQ $-linear map $\Psi \colon T\rightarrow F$ by $\Psi (1) = 0$ and
$\Psi (q Z)=\Theta (q)Z$ for $Z=X,Y$.
Explicitly, given a tensor monomial $Z_{1}\cdots Z_{n}\in T$, where
each $Z_{i}$ is either $X$ or $Y$, we have
\[
\Psi (Z_{1}\cdots Z_{n}) = (\ad Z_{1})\cdots (\ad Z_{n-1})Z_{n}.
\]
\smallskip
(a) Using the fact that $\ad \colon F\rightarrow \End _{\QQ }(F)$ is a
Lie algebra homomorphism, conclude that $\Theta (q) = \ad q$ if $q\in F$.
\smallskip
(b) Show that $F_{n} = (\ad X)F_{n-1}+(\ad Y)F_{n-1}$. In other
words, Lie bracket monomials $(\ad Z_{1})\cdots (\ad Z_{n-1})Z_{n}$
span $F$ (but are not linearly independent).
\smallskip
(c) Prove that $\Psi (p) = n p$ if $p\in F_{n}$, by induction on $n$,
using (a) and (b).
\smallskip
(d) Let $B(X,Y) = X+Y+\frac{1}{2}[X,Y]+\cdots $ be the
Baker-Campbell-Hausdorff series. We can consider $B(tX,tY) = (X+Y)t +
\frac{1}{2}[X,Y]t^{2}+\cdots $ as a formal power series in $t$ whose
coefficient of $t^{n}$ belongs to $F_{n}$ and thus to $T_{n}$. As
such, it is just the formal logarithm $\log (e^{tX}e^{tY})$, where
$\log (1+\phi(t) )= \sum _{k=1}^{\infty } (-1)^{k-1} \phi(t) ^{k}/k$
for any series $\phi (t)$ with coefficients in $T$ and zero constant
term.
Use this to obtain the explicit formula, due to Dynkin,
\[
B(tX,tY) = \sum _{k=1}^{\infty } \frac{(-1)^{k-1}}{k}\sum
_{p_{1}+q_{1}\geq 1,\ldots,p_{k}+q_{k}\geq 1} \frac{\Psi
(X^{p_{1}}Y^{q_{1}}\cdots X^{p_{k}}Y^{q_{k}})}{p_{1}!q_{1}!\cdots
p_{k}!q_{k}!(\sum p_{i}+\sum q_{i})}
t^{\sum p_{i}+\sum q_{i}}
\]
\medskip
11. (a) Let $\phi \colon S^{2}\rightarrow \CC \PP ^{1}$ be the map
given by stereographic projection from the north pole of $S^{2}$ to
the complex plane $\CC $, with $\phi $ mapping the south pole to $0$,
the equator to the unit circle $\{|z| = 1 \}$, and the north pole to
$\infty $. Verify that $\phi $ is an isometry between the standard
angle metric on $S^{2}$ and the Fubini-Study metric on $\CC \PP ^{1}$
given by $d(\overline{x},\overline{y}) = 2 \cos ^{-1} |(x,y)|$, where
$x,y\in \CC ^{2}$ are unit vectors.
\smallskip
(b) Work out the resulting Lie group homomorphism $\psi \colon U(2)\rightarrow
SO(3)$ in explicit coordinates, {\it i.e.}, find the entries of the
$3\times 3$ real orthogonal matrix $\psi (A)$ in terms of the entries
of the $2\times 2$ complex unitary matrix $A$.
\medskip
12. Construct an isomorphism of $GL(n,\CC )$ (as a Lie group and an
algebraic group) with a closed subgroup of $SL(n+1,\CC )$.
\medskip
13. Show that the map $\CC ^{*}\times SL(n,\CC )\rightarrow GL(n,\CC
)$ given by $(z,g)\mapsto zg$ is a surjective homomorphism of Lie and
algebraic groups, find its kernel, and describe the corresponding
homomorphism of Lie algebras.
\medskip
14. The {\it free Lie algebra} on generators $X_{i}$ over a field
$k$ is a Lie algebra $F$ over $k$ with generators $X_{i}$ and only
the relations that follow from the Lie algebra axioms. More
precisely, $F$, together with its distinguished elements $X_{i}$, is
characterized by the property that for every Lie algebra $\gfrak $
over $k$ and system of elements $x_{i}\in \gfrak $, there is a unique
Lie algebra homomorphism $F\rightarrow \gfrak $ sending $X_{i}$ to
$x_{i}$.
\smallskip
(a) Show that an $F$ module is just a vector space $V$ together with
arbitrary endomorphisms $\xi _{i} = \rho (X_{i})$.
\smallskip
(b) Deduce that the universal enveloping algebra of $F$ is the tensor
algebra $T$ over $k$ on the generators $X_{i}$. (This is slightly
subtle. It is clear from (a) that the associative algebras $T$ and
$\Afrak (F)$ have canonically equivalent categories of modules, but
not entirely obvious that this implies that $T$ and $\Afrak (F)$ are
isomorphic.)
\smallskip
(c) Using Poincar\'e-Birkhoff-Witt, deduce that $F$ is isomorphic to
the Lie subalgebra of $T$ generated by the elements $X_{i}$, where we regard $T$
as a Lie algebra with commutator as the Lie bracket.
\smallskip
(d) Assume now that the set of generators $X_{i}$ is a finite set
$\{X_{1},\ldots,X_{n} \}$, so that the graded algebras $T$ and $F$
have finite dimension in each degree. In particular, $t_{d} = \dim
(T_{d}) = n^{d}$ is the number of words (or tensor monomials) of
length $d$ in the $n$ letters $X_{i}$, with generating function
\[
\sum _{d} t_{d}\, z^{d} = \frac{1}{1-nz}.
\]
Show that $f_{d} = \dim (F_{d})$ is characterized by the identity
\[
\prod _{d}\frac{1}{(1-z^{d})^{f_{d}}} = \frac{1}{1-nz}.
\]
\smallskip
(e) Derive the explicit formula $f_{d} = (1/d)\sum _{k|d}\mu
(d/k)\, n^{k}$. Here $\mu (m)$ is the classical M\"obius function,
equal to $(-1)^{r}$ if $m$ is a product of $r$ distinct primes, or $0$
if $m$ is divisible by a square, which is characterized by the
M\"obius inversion formula $a_{d} = \sum _{k|d}\mu (k/d)b_{k}$ if
$b_{d} = \sum _{k|d}a_{k}$, for any sequence $a_{1},a_{2},\ldots$.
\smallskip
(f) A word $w$ in the letters $X_{1},\ldots,X_{n}$ is {\it aperiodic}
if all rotations of $w$ are distinct. Show that $f_{d}$ is equal to
the number of rotation classees of non-empty aperiodic words of length
$d$ in $n$ letters.
\smallskip
(g) A {\it Lyndon word} is a non-empty aperiodic word which is
lexicographically least in its rotation class. Thus $f_{d}$ is the
number of Lyndon words of length $d$ in $n$ letters. Show that every
Lyndon word $w$ of length $d>1$ can be factored (not necessarily
uniquely) as $w=uv$, where $u$ and $v$ are Lyndon. Hint: it works to
take for $v$ the right factor such that $vu$ is the lexicographically
least rotation of $w$ other than $w$ itself.
\smallskip
(h) Fix one Lyndon factorization $w=uv$ for each Lyndon word $w$ of
length $d>1$, and define a Lie bracket monomial $[w]\in F$ inductively
by $[w] = X_{i}$ if $w=X_{i}$, otherwise $[w] = [[u],[v]]$, where
$w=uv$ is the chosen factorization. Show that the lexicographically
least term of $[w]$, considered as an element of $T$, is $w$.
\smallskip
(i) Deduce that the Lie bracket monomials $[w]$ for all Lyndon words
$w$ form a basis of the free Lie algebra $F$ (for any given choice of
the factorizations $w=uv$).
\smallskip
(j) Show that if $n$ is a power of a prime, so there exists a finite
field $\FF $ of order $n$, then $f_{d}$ is equal to the number of
distinct monic irreducible polynomials $g(x)$ of degree $d$ over $\FF
$. Is this purely a numerical coincidence, or can some deeper
connection with the free Lie algebra be found?
\medskip
15. Prove that if $\gfrak $ is a solvable Lie algebra over $\RR $,
then every finite-dimensional irreducible $\gfrak $ module has
dimension at most 2.
\medskip
16. Construct an example of a solvable Lie algebra $\gfrak $ over a
field of characteristic $2$ such that the derived subalgebra $[\gfrak
,\gfrak ]$ is not nilpotent. Hint: start with the the 2-dimensional
non-nil module $V$ for the 3-dimensional Heisenberg algebra $\hfrak $,
and form the semidirect product of $\hfrak $ with $V$, regarded as an
abelian Lie algebra.
\end{document}