Princeton U Press, Jan 2003

Reviewer 2:

P. Cvitanovic notes in red font,
all edits incorporated
13 Oct 2007

Some referees might want to Cvitanovic to `tone down' the idiosyncratic aspects of this book, but I think that would be a big mistake.

My only complaint along these lines is that the bland title {\sl Group Theory} scarcely conveys the real contents of the book, and the quirky subtitle {\sl Lie's, Tracks, and Exceptional Groups} helps only a little bit to remedy this --- for example, the word `Lie's' is not exactly English, and nobody will know what `tracks' are until after they read the book. But this is a fairly minor point.

My main worry is that while the chapters on exceptional Lie groups and the magic triangle (Chaps.\ 16-22) form the real heart of the book, they are still not complete. In the version I received Chapter 22 is missing, and Chapters 16-21 have gaps and many more typos and other errors than the rest of the book. In short, this is very much a book {\it in the process of being written}. It still needs polishing and careful correction before it is suitable for publication.There are a lot of calculations that I did not have the energy to verify. Some expert needs to do this after the book is completely finished.

Here are some mistakes I caught:

** Edits still outstanding**

(PC:
all done)

** Edits incorporated**

(PC: internal joke, now removed)

p. 3 -- Here the word ``octonion" is misspelled ``octonian" in two places. While this misspelling is fairly common in the literature, the correct spelling is definitely ``octonion'', since this correctly parallels the word ``quaternion''.

(PC: corrected throughout)

p. 4 -- In the caption to Figure 1.1 he says the Freudenthal magic square is marked by the dotted line. While there are some dotted lines in this figure, they don't actually mark the magic square.

(PC: done)

p. 5 -- ``i.e." is misspelled ``{\sl ie.}"

(PC: corrected throughout)

p. 6 -- In equation (2.2) he introduces overlined symbols without explanation. If these stand for conjugate reps, he should have introduced this notation when discussing antiquarks in the second equation on page 5.

(PC: done)

p. 7 -- The neologism ``clebsches" should be italicized since it's being defined here.

(PC:
done)

p. 17 -- His definition of the conjugate of a complex vector
space is wrong, since ``complex conjugation of elements $x \in V$"
makes no sense when $V$ is an abstract vector space. (We can
conjugate components with respect to a basis, but not elements
of an abstract vector space!) The correct definition says that
the conjugate space $\overline V$ has the same underlying set as
$V$: for each element $v \in V$ there is a corresponding element
$\overline v$ in $\overline V$. The difference is that $\overline
V$ is made into a complex vector space in a different way:
addition is the same, but multiplying an element
$\overline{v} \in \overline{V}$ by a complex number $c$ gives
$\overline{\overline{c} v}$. This mistake will come to haunt
him on page 211.
(PC the Haunted: after consultations
with my lawyers, this has been replaced by the more palatable
"... dual space $\overline V$ is the set
of all linear forms on V over the field F...."
)

p. 32 -- The idea of ``infinitesimal transformations" as matrices with ``small elements" is sloppy, harking back to the days before calculus was made rigorous through the concept of limit. Since this is a mathematics textbook rather than a physics one, it would be good to at least mention that a precise treatment exists.

(PC:
why me? OK, OK, I put some "rigor" there now...)

p. 39-40 -- The word ``birdtracks'' is misspelled ``birtracks'' at least three times

(PC: 6 times!
done)

``succeeded'' is spelled ``suceeded''
(PC:
done)

``down'' is spelled ``donw''.
(PC:
done)

p. 44 -- The last sentence on this page also breaks off in the middle: ``Fortunately, and $3n-j$ symbol which contains as a sub-diagram a loop, with, let us say, seven vertices''....

(PC:
done)

p. 47 -- The last sentence on this page breaks off in the middle: ``If $G_1$ does not exist (the invariant relations are so stringent that there is no space on which they can be realized).''

(PC:
text completed, move to p. 41 or so)

p. 50 -- ``Factor $1/p!$'' should read ``A factor of $1/p!$''.

(PC:
done)

p. 130 -- The factor of $(a-1)$ in equation (11.5) should be $(p-1)$.

(PC:
done)

p. 145 -- The sentence beginning ``Matrix $A_a^b$'' should read ``The matrix $A_a^b$...''.

(PC: done)

p. 152 -- The author says to compare ``table ?? and table 10.1''.

(PC:
done)

p. 158 -- The author uses the symbol $\overline{SO(3)}$; the usual term for the universal cover of $SO(3)$ is $\widetilde{SO}(3)$ or $Spin(3)$. This is particularly important because the author uses $\overline{SO(3)}$ to mean something completely different in Chapter 13.

(PC:
done)

p. 159 - There is an equation on this page which is too long and shoots off into the margin; also, this equation contains a double-headed arrow which should probably be a minus sign.

(PC:
rewritten)

I think that without much extra work the author could definitively answer his question ``what are spinsters?'' As he notes, just as the Clifford algebra has its spinor representation(s), the Heisenberg algebra has its Fock representation, which is infinite-dimensional. Moreover, just as spinors form a representation of the Spin group (i.e.\ the double cover of the rotation group), Fock space forms a representation of the metaplectic group (i.e. the double cover of the symplectic group). The whole Heisenberg/Fock/metaplectic story works almost like the Clifford/spinor/Spin story except for a bunch of minus signs. Thus, it would be utterly shocking if ``spinsters'' were anything other than the Fock representation of the metaplectic group. It should be easy to check the necessary diagrammatic equations to prove this. (PC: incorporated 9 Oct 2005)

p. 165 -- There is an unnecessary left parenthesis in equation (15.23).

(PC:
done)

p. 166 -- The author writes that the invariance group of $f^{abc}$ is $SU(3)$; this is only true if we restrict attention to unitary transformations; in general the whole group $SL(3)$ preserves this tensor.

(PC:
done)

p. 171 -- In the sentence before equation (16.9), ``takes form'' should read ``takes the form''(PC: done). In the equation, I believe the plus sign should be a minus sign. Also, the symbol $A"$ should read ${A'}' $, both in this equation and the text following.

(PC: fixed signs. ${A'}' $ looks horrible, no thanks.
done)

p. 169 -- I think that the author is implicitly assuming the equation $g^{ai}f_{ibc} = f_{abi}g^{ic}$ in all the diagrammatic calculations in Chapter 16. This equation has a nice pictorial interpretation that allows one to wiggle diagrams around without changing the tensor they represent. Moreover, this equation holds for the Lie bracket and Killing form in a semisimple Lie algebra, where it's equivalent to $ g([X,Y],Z) = g(X,[Y,Z]) . $ For the Lie algebra of $SO(3)$, it says simply $ (X \times Y) \cdot Z = X \cdot (Y \times Z) .$ So, I think this equation needs to be added to the definition of the $G_2$ family.

(PC: here I disagree - I have disposed of the symmetric
2-index invariant tensor
$g^{ab}$ in Chapter 10 "Orthogonal groups", here I am looking
at subgroups of SO(n). $g^{ab}$ is not a Killing form, this
is the defining, not the adjoint representation. Later on
it does become the Killing form for the $E_8$ family, but that is
a result of calculation, not an assumption.
)

p. 172 -- ``Octonion'' is again misspelled, as is ``Gordan'' in ``Clebsch-Gordan''.

(PC: 6 times!
done)

p. 180 -- A purely diagrammatic proof of Hurwitz's theorem has also been given by Dominic Boos, and this should be cited here! See

Dominik Boos, Ein tensorkategorieller Zugang zum Satz von Hurwitz (A tensor-categorical approach to Hurwitz's theorem), Diplomarbeit ETH Zurich, March 1998, available at http://www.math.ohio-state.edu/$\sim$rost/tensors.html and also Markus Rost, On the dimension of a composition algebra, Documenta Mathematica 1 (1996), 209-214, also available at http://www.mathematik.uni-bielefeld.de/DMV-J/vol-01/10.html

Like the author's proof, Boos' proof makes use of the alternativity relation (equation (16.59)). More precisely, he defines a ``vector product algebra'' to be a vector space with a dot product and cross product satisfying properties that generalize those of $R^3$ with its usual dot product and cross product, and gives a purely diagrammatic argument that the dimension of such a thing is 0,1,3, or 7.

(PC: I did it in 1975,
and it was stated in 1976 paper, derived in detail in 1977 preprint,
included in the 1984 Nordita book, on the web since 1996.
I'am now referring to Rost and Boos, in hope that
a different write-up helps some reader)

p. 181 -- Again I think the author is implicitly assuming throughout this chapter that his symmetric quadratic invariant and antisymmetric cubic invariant satisfy the equation $g^{ai}f_{ibc} = f_{abi}g^{ic}$. If so, this should be pointed out explicitly.

(PC: here I disagree, as in p. 169 - I have disposed of the symmetric
2-index invariant tensor
$g^{ab}$ in Chapter 10 "Orthogonal groups", here I am looking
at subgroups of SO(n).
As it stands, $g^{ai}f_{ibc} = f_{abi}g^{ic}$ has mismatched up
and down indices, but in the $SO(n)$ normalization we are
using, $g^{ab}$ is the identity,
anyway, so the relation is trivially
satisfied.
)

p. 182 -- I can't tell if ``the $E_sting$ family Lie algebras'' is a typo for ``the existing families of Lie algebra'' or ``the $E_8$ family of Lie algebras''.

(PC:
fixed)

In equation (17.8) the operator $P_A$ has not been defined; the author probably means $C_A P_\Box$.

(PC:
fixed)

p. 187 -- The diagram in equation (17.36) still needs to be drawn.

(PC:
fixed)

p. 188 -- There is a reference to (??) above equation (17.43).

(PC:
different derivation now, not applicable)

p. 190 - There is a reference to (??) in the caption of table 17.2 Equations (17.55) and (17.56) are incomplete.

(PC:
different derivation now, not applicable)

p. 211 -- the author's mention of the `complex conjugate' vector space $\overline A$ is meaningless here, since $A$ is not a complex vector space. And even if by $\overline A$ he means the {\it dual} vector space, equations (18.75) and (18.76) do not make sense.

(PC:
you are right, removed now)

p. 211 -- Again ``octonion'' is misspelled.

(PC:
done)

p. 211 -- Definition (18.75) and (18.76) don't make sense, for too many reasons to easily sort out. For example, is equation (18.76) a definition of the cross product or the letter $z$? Neither option is viable. Is $\overline x$ an element of $A$ or the dual vector space of $A$? Neither option is viable.

If the algebra $A$ being discussed here is really the Jordan algebra of $3 \times 3$ hermitian octonionic matrices, correct definitions are as follows: define $(x,y) = {\rm tr}(xy)$, set $\langle x,x,x \rangle = {\rm det}(x)$ and then extend $\langle \cdot, \cdot, \cdot \rangle$ to be a symmetric trilinear form, and then let $x \times y$ be the unique element of $A$ such that $(x \times y, z) = 3(x,y,z)$. Of course, for all this to make sense, we need to define the trace and (more subtly) the determinant of $3 \times 3$ hermitian octonionic matrices. A correct treatment along these lines can be found in various places, perhaps including the work by Springer on which the author claims to be basing his treatment.

(PC: you are right.
Now I cite Springer word for word)

p. 212 - There are a lot of references to (??) on this page, and one on the previous page.

(PC:
different derivation now, not applicable)

p. 215 -- Again ``octonion'' is misspelled.

(PC: done)

p. 216 -- The first sentence should start with `the': ``The $V \otimes A$ space....''

(PC: done)

p. 220 -- The first sentence trails off in ``we obtain''. There is a reference to a nonexistent equation called (?!). $F_4(28)$ should be $F_4(26)$. Table 19.1 apparently needs editing.

(PC:
fixed)

p. 219 -- The author speaks of ``the exceptional simple Jordan algebra of traceless Hermitian $3 \times 3$ matrices with octonions matrix elements''. The {\it traceless} hermitian octonion matrices do not form a Jordan algebra, since they aren't closed under the Jordan product and don't include the identity. $F_4$ acts as automorphisms of the exceptional Jordan algebra, and as a rep of $F_4$ this algebra splits as the direct sum of a 1-dimensional trivial rep and a 26-dimensional irrep given by the traceless matrices.

(PC:
you are right - now fixed)

p. 221 -- The second sentence should say ``...an $n \to -n$ substitution...''

(PC:
fixed)

There is a reference to chapter ??.

(PC:
fixed)

p. 227 -- After equation (20.36), $SO(7)$ should probably be $SO(10)$.

(PC: done)

p. 228 -- There's a typo in ``fermionic''.

(PC: done)

p. 229 -- The author writes ``The Dynkin indices listed in tables ?? and ??gree with....''

(PC:
fixed)

p. 232 -- There's a reference to chapter ??.

(PC:
fixed)

p. 235 -- The sentence beginning ``Role of the'' should start with a `The'.

(PC: done)

p. 236 -- There should be another `the' in ``In the dimension of the associated reps, eigenvalue....''

(PC: done)

The author writes ``... because it contains Freudenthal's Magic Square [72], marked by the dotted line in table 21.1.'' The table being referred to is really table 21.2, and there is no dotted line marking out the magic square in this table (PC: frame drawn). Also, the magic square was actually first noticed by Rosenfeld:

Boris A. Rosenfeld, Geometrical interpretation of the compact simple Lie groups of the class (Russian), Dokl. Akad. Nauk. SSSR (1956) 106, 600-603.

and then independently made rigorous by Freudenthal:

Hans Freudenthal, Beziehungen der $E_7$ und $E_8$ zur Oktavenebene, I, II, Indag. Math. 16 (1954), 218-230, 363-368. III, IV, Indag. Math. 17 (1955), 151-157, 277-285. V -- IX, Indag. Math. 21 (1959), 165-201, 447-474. X, XI, Indag. Math. 25 (1963) 457-487.

and Tits:

Jacques Tits, Alg\'ebres alternatives, alg\'ebres de Jordan et alg\'ebres de Lie exceptionnelles, Indag. Math. 28 (1966) 223-237.

so it should probably just be called the Magic Square.

(PC:
Incorporated. I already had Rosenfeld, Freudenthal, Tits references, but this
clarifies who did what a bit. Seems that Ned. Akad. Weternsch. Proc. =
Indignations of Mathematics; go figure.)

The author misspells `Freudenthal Magic Square' (PC: fixed)

p. 238 -- The author speaks of a `division algebra of dimension 6', but there is no such thing! It's a well-known hard theorem that finite-dimensional real division algebras occur only in dimensions 1, 2, 4, and 8. (I would like to know more about the sextonions, but I already know they're not a division algebra.)

(PC: you are right - removed now)

p. 242 -- The author asks why the magic triangle is symmetric across the diagonal. I don't know: this is a fascinating puzzle! But it should be emphasized that while the Freudenthal-Tits construction of the magic square leaves this symmetry mysterious, Vinberg's construction makes it obvious:

E. B. Vinberg, A construction of exceptional simple Lie groups (Russian), Tr. Semin. Vektorn. Tensorn. Anal. 13 (1966), 7-9.

A more accessible reference is:

A. L. Onishchik and E. B. Vinberg, eds., Lie Groups and Lie Algebras III, Springer, Berlin, 1991, pp. 167-178.

A more recent construction of the magic square due to Barton and Sudbery also has manifest symmetry:

Chris H. Barton and Anthony Sudbery, Magic squares of Lie algebras, preprint available as math.RA/0001083.

A tour of all these constructions, explaining how they are related, can be found here:

John Baez, The octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205.

These references should be cited, for the benefit of the reader who wants to ponder the mysterious symmetry of the magic triangle. (PC 1 Aug 2007: all of this now promoted to a higher level of limbo, into the \Preliminary{ parts of the webbook manuscript} )

p. 263 -- Reference [100] is missing.

(PC:
fixed)

p. 267 -- For some reason the references from [1] to [165] are listed alphabetically by author, and then [166] to [204] go alphabetically by author starting at the beginning of the alphabet again. Now I see that some of the references I suggested including do in fact appear --- but without citations at the appropriate places in the text.

(PC:
fixed)

** Edits of the web epilogue
(not in the book, so not entered yet)**

p. 242 -- in the second to last sentence, ``outstriping'' should be ``outstripping''.

Despite these somewhat grumpy corrections, I must emphasize that I found reading the book quite enjoyable.