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\ihead{Errata to ``Lie algebras''}
\ohead{\today}
\begin{document}

\begin{center}
\textbf{Lie algebras}

\textit{Shlomo Sternberg}

\url{http://www.math.harvard.edu/~shlomo/docs/lie_algebras.pdf}

version of April 23, 2004

\textbf{Errata and addenda by Darij Grinberg}

\bigskip
\end{center}

%\setcounter{section}{}


\section*{Errata}

The numbering of pages in the following errata matches the numbering of pages
in the notes (not the numbering of pages in the PDF file).

\begin{itemize}
\item \textbf{page 8, very first formula:} Replace
\[
A+B+\dfrac{1}{2}A^{2}+AB+\dfrac{1}{2}B^{2}-\dfrac{1}{2}\left(  A+B+\cdots
\right)  ^{2}%
\]
by%
\[
A+B+\dfrac{1}{2}A^{2}+AB+\dfrac{1}{2}B^{2}-\dfrac{1}{2}\left(  A+B\right)
^{2}+\cdots
\]


\item \textbf{page 8, the formula after (1.1):} The left hand side,%
\[
\dfrac{1}{12}\left(  A^{2}B+AB^{2}+B^{2}A+BA^{2}-2ABA-BAB]\right)
\]
has an $]$ bracket too much.

\item Something that is generally confusing me in this chapter (at least its
geometric parts) is the unclarity about whether we are considering linear Lie
groups or general Lie groups. When you write ``One of the important
consequences of the mere existence of this formula is the following'' on page
8, you suddenly switch from $\operatorname*{GL}\nolimits_{n}$ to general Lie
groups; it would be helpful if you would tell when you do so.

\item \textbf{page 8, third paragraph:} ``closed'' should be ``close'' in ``two
elements of $G$ sufficiently closed to the identity''.

\item \textbf{page 8, third paragraph:} There is a ``the'' too much in ``the the
product of $a=\exp\xi$ and $b=\exp\eta$''.

\item \textbf{page 9, remark 1:} In the last formula of remark 1, I would add
a bracket to the left hand side ($\log\left(  \exp A\right)  \left(  \exp
B\right)  $).

\item \textbf{page 9, remark 2:} You write: ``On the right hand side,
exponentiation takes place in the algebra of endomorphisms of the ring in
question.''. You probably mean ``of the Lie algebra in question''.

\item \textbf{page 11, middle of the page:} Why do we have%
\[
\dfrac{d}{dt}\operatorname*{Ad}\left(  \exp tA\right)  X=\left(  \exp
tA\right)  AX\left(  \exp-tA\right)  +\left(  \exp tA\right)  X\left(
-A\right)  \left(  \exp-tA\right)
\]
for a \textit{general} (non-matrix) Lie algebra? I can't apply the product
rule in this case as things are happening in different tangent spaces...

\item \textbf{page 11, middle of the page:} In $\dfrac{d}{dt}%
\operatorname*{Ad}\exp tA$ you could add a bracket: $\dfrac{d}{dt}%
\operatorname*{Ad}\left(  \exp tA\right)  $. (But it's clear anyway).

\item \textbf{page 11, middle of the page:} In $\exp t\operatorname*{ad}A$ you
could add a bracket, lest it looks like $\left(  \exp t\right)  \left(
\operatorname*{ad}A\right)  $.

\item \textbf{page 12, example Sp(n):} Here you write:%
\[
dAJa^{\dag}+AJdA^{\dag}=0.
\]
The $a$ should be an $A$ here.

\item \textbf{page 12, example Gl(n,C):} Here you write:%
\[
dAJ=JdA=0.
\]
Do you really mean this, or should the first equality sign be a minus sign actually?

\item \textbf{page 19, before the long formula:} ``On then other hand'' has an
obvious typo.

\item \textbf{page 20, \S 1.8.1, Uniqueness:} You write: ``By the universal
property $t=\ell_{t}^{\prime}\circ t^{\prime},$ $t^{\prime}=\ell_{t}^{\prime
}\circ t$ so $t=\left(  \ell_{t}^{\prime}\circ\ell_{t^{\prime}}\right)  \circ
t$, but also $t=t\circ\operatorname*{id}$.'' Two errors here: first,
$t^{\prime}=\ell_{t}^{\prime}\circ t$ should be $t^{\prime}=\ell_{t^{\prime}%
}\circ t$. Second, $t=t\circ\operatorname*{id}$ should be
$t=\operatorname*{id}\circ t$.

\item \textbf{page 20, \S 1.8.1, Existence:} You write: ``Let $M$ be the free
vector space on the symbols $x_{1},\ldots,x_{m},\ x_i \in E_i$''. I'd rather say ``on the symbols
$\left(  x_{1},\ldots,x_{m}\right)  $ with $x_i \in E_i$'', as otherwise it sounds like you are
defining the direct
sum of the $E_{i}$.

\item \textbf{page 20, last formula on this page:} There is one comma too much
in ``$\left(  x_{1},\ldots,,ax_{i},\ldots,x_{m}\right)  $''.

\item \textbf{page 21, first line of \S 1.8.2:} Replace ``they are they are'' by
``then they are''.

\item \textbf{page 21, \S 1.8.2:} You hardly mean ``isomorphism'' in ``We have an
isomorphism of $A$ into $A\otimes B$''.

\item \textbf{page 21, first line of \S 1.8.3:} Replace ``The \textbf{tensor
algebra} of a vector space'' by ``The \textbf{tensor algebra} of $V$''.

\item \textbf{page 21, second line of \S 1.8.3:} ``the universal problem for
maps'' should be ``the universal problem for linear maps''.

\item \textbf{page 22, second line from above:} ``of the last subsection''
should be ``of Subsection 1.8.1''.

\item \textbf{page 22, second line of \S 1.8.4:} ``generated the elements''
should be ``generated by the elements''. Also, you might want to add ``for
$x,y\in L$'' after $\left[  x,y\right]  -x\otimes y+y\otimes x$, lest it looks
like ``for $x,y\in TL$'' or something like that.

\item \textbf{page 22, last line of \S 1.8.4:} You write: ``must vanish on $I$
if it is to be a Lie algebra homomorphism''. The ``it''\ here refers to $\alpha$,
not $\psi$; probably better just to write $\alpha$ instead.

\item \textbf{page 23, \S 1.8.7:} In the second formula of \S 1.8.7, you
write:%
\[
xy\otimes1+x\otimes y+y\otimes x++1\otimes xy,
\]
There is a $+$ sign too much in here.

\item \textbf{page 24, \S 1.8.7:} You write: ``is the identity (on $1$ and on)
$L$ and hence is the identity''. The closing bracket is misplaced here; it
should be after the first ``and''.

\item \textbf{page 24, \S 1.8.7:} Replace $D$ by $C$ in ``a map $\varepsilon
:D\rightarrow k$ (called a \textbf{co-unit})''.

\item \textbf{page 24, \S 1.8.7:} I was very surprised to see that you are
using the word ``co-algebra'' to mean
``not necessarily co-associative co-algebra''. You might
want to point out in a footnote that you are using this nonstandard notation.

\item \textbf{page 24, \S 1.8.7, middle of the page:} You write: ``Hence the
comultiplication is is coassociative.'' Redundant ``is'' in this sentence.

\item \textbf{page 24, \S 1.9:} One comma too much in ``For example,,''.

\item \textbf{page 26, middle of the page:} There is one closing bracket too
much in ``Recall that if $M=\left(  i_{1},\ldots,i_{n}\right)  )$ we set
$\ell\left(  M\right)  =n$ and call it the length of $M$''.

\item \textbf{page 26, proof of Lemma 1:} You write: ``We will inductively
define a map''. I think ``a $k$-bilinear map'' would be slightly better here, as
you define it on products of bases.

\item \textbf{page 27, first paragraph:} You write: ``Notice that the first of
these two cases is consistent with (and forced on us) by (1.20)''. Again,
brackets are misplaced: It should be ``[...] (and forced on us by) (1.20)''.

\item \textbf{page 27, middle of the page:} You write: ``Furthermore (*) holds
with $j$ and $N$ replaced by $M$''. Should be ``replaced by $i$ and $M$`''.

\item \textbf{page 27, last paragraph:} I don't see what is happening beginning
with ``So we know that (1.21) holds for $x=x_{i},y=x_{k}$ and $v=z_{\left(
jP\right)  }$ (if $j\leq P$) or $v=z_{Q}$ (otherwise).'' How do you know this?
I think you need a stronger inductive assumption for this.\newline Also, it
seems to me that you don't just use that $I$ is totally ordered, but also that
$I$ is well-ordered, as you do transfinite induction over $i$ (assuming that
something is already done for all $j<i$ and then doing it for $i$). I may be
seriously off track here.

\item \textbf{page 28, proof of Lemma 1:} There are missing $]$ brackets in
here (three of them; each one in $\left[  x_{j},[x_{i},x_{k}\right]  $).

\item \textbf{page 28, \S 1.10:} Maybe point out that you are in
characteristic $0$ (otherwise, the claim that the elements of $L$ are the only
primitives in $U\left(  L\right)  $ is invalid).

\item \textbf{page 29, \S 1.11.1:} You write: ``We have a ``multiplication'' map
$M_{X}\times M_{X}$ given by the inclusion''. The ``$M_{X}\times M_{X}$''
should be ``$M_{X}\times M_{X}\rightarrow M_{X}$'' here.

\item \textbf{page 29, \S 1.11.1:} A reminder that algebras are not
necessarily assumed associative wouldn't be out of place here.

\item \textbf{page 30, \S 1.11.2, middle of the page:} ``each of the $a_{n}$
also belong to $I$'' should be ``[...] belongs to $I$''.

\item \textbf{page 31, \S 1.11.3, third paragraph:} In ``thus give rise to a Lie
algebra homomorphism'', replace ``give'' by ``gives''.

\item \textbf{page 31, \S 1.11.3, third paragraph:} ``both compositions $\Phi
\circ\Psi$ and $\Psi\circ\Phi$ are the identity on $X$'' should be ``Both [...]''.

\item \textbf{page 31:} There is a point too much in (1.24).

\item \textbf{page 32, \S 1.12:} Why is $\mathbf{L}_{X}$ the set of all
$\alpha\in m$ satisfying $\Delta\alpha=\alpha\otimes1+1\otimes\alpha$ ? (It's
not hard, but you might mention it as an exercise - it is tacitly used in \S 1.21.1.)

\item \textbf{page 33, fourth line from above (not counting the formulas):}
Please add a whitespace after ``general'' in ``and in general$L_{X}^{n}$ is
spanned by elements of the form''.

\item \textbf{page 34, fifth line of formulas:} Add a $\sum\limits_{m}$ sign
in front of \newline$\sum\limits_{p_{i}+q_{i}\geq1}\dfrac{\left(  -1\right)
^{m+1}}{m}\dfrac{x^{p_{1}}y^{q_{1}}x^{p_{2}}y^{q_{2}}\cdots x^{p_{m}}y^{q_{m}%
}}{p_{1}!q_{1}!\cdots p_{m}!q_{m}!}$.

\item \textbf{page 34, definition of }$z_{p,q}^{\prime}$\textbf{:} There is a
mistake in here (which comes directly from Serre, I think): The condition
$p_{m}\geq1$ is wrong; for example it leads to a wrong formula for $n=1$.

\item I would really like to see a final paragraph about the relation between
the algebraic BCH and the geometric BCH. The formulas both compute
$\log\left(  \left(  \exp A\right)  \left(  \exp B\right)  \right)  $, but the
$A$ and $B$ are different, the meanings of $\log$ and $\exp$ are different,
and the right hand sides are different, too. This is probably obvious for a
geometer, but how can one derive the geometric BCH from the algebraic one?
\end{itemize}


\end{document}