Date: 25 Feb 2003

From: hpfeiffer <hpfeiffer [ at ] Oct 2005
moved to A Einstein Institute, pfeiffer [ at ]

Hendryk Pfeiffer: I refer to the version of December 2, 2002


P. Cvitanovic July 2003 notes in red font
or: How I spent my summer holidays in Ciovo, Croatia

Edits still outstanding

Chap 2:


mix up Lie group and Lie algebra, write things

such as `SO(6) is isomorphic to SU(4)'.

Actually SU(4)=Spin(6) is a double cover of SO(6), therefore locally

isomorphic to SO(6) and both have got the same Lie algebra. Even locally,

you can tell the difference as their (real) Lie algebras are different.

However, the complexifications of these Lie algebras are the same. This is

what you see in the Cartan-Dynkin classification.

At first sight I would say that you are safe if you define Lie

algebras over the field of complex numbers, then say that you study

their complex representations and do all the bird track business in

this setting.

Otherwise you get into trouble at several occasions when you DEFINE a

group from the invariants of its representations. Firstly, specifying

the Kronecker delta and the epsilon, if you work over the field of

complex numbers, you get the complex Lie algebra sl_2 as the

invariance algebra. Then you can pass to su(2) by chosing a particular

real form or alternatively to su(1,1) etc. Secondly, if you want to

talk about groups rather than algebras, you first have to choose a

real form, say su(2), then exponentiate and get a group, say SU(2),

but other than this connected and simply connected group, you can have

quotients such as SO(3) or even bigger groups whose component of the

unit this is, say O(3). You cannot tell the difference if you look

only at the reps of the Lie algebra. All the SO(6) vs SU(4) questions are

of this sort.

There is a second related issue. If you define quarks and antiquarks

you have a complex vector space with a hermitean scalar product in

mind, ie raising an index would be an anti-linear operation (complex

conjugate representation of SU(3)). However, if you study reps of

SO(17), a real vector space with a symmetric scalar product seems

sufficient. Raising an index is then a linear operation,

complexification unnecessary. Why this difference?

The answer is that the `true' structure which gives indices upstairs,

is the dual representation which makes sense independently of the

chosen real form. If for SU(3), you then choose the reps to be

unitary, then the dual rep is the complex conjugate one, etc. For

SO(3,1), for example, you have finite-dimensional reps, but they are

not unitary, and the analogy is gone. But the dual reps are always

there and help you avoid this trouble.

Chap 3:


* It makes a difference whether a group is unitary or whether a

  particular representation is unitary

(PC: mhm... honestly, this is not a book about groups and their representations, but only about Lie algebras... Will try to re-emphasize this n times.)

* below (3.42), `minimal characteristic' -> `minimal'.

(PC: I need to distinguish the characteristic polynomial
det(M-lambda 1)=0,
with in general degenerate roots
and the minimal characteristic polynomial

Chap 4:


* below (4.1), I am not sure whether reality is the right

  condition. See my comment on the dual rep above.

Chap 5:


* I have a general comment on the notation in Sect 5. Assume in (5.1)

  there are degenracies, for example, as adjoint tensor adjoint of

  SU(3) contains multiple direct summands of adjoint irreps in its

  decomposition. In the remainder of Sect 5 you need that lambda is

  irreducible, ie. the triangle symbol needs to know on which of the

  two adjoints it projects. One has to be very careful when one

  replaces the triangle by a dot in order not to suggest a symmetry

  which is not there. This notation originates from SU(2) where this

  is no problem as the intertwiner associated with the trivalent

  vertex is unique up to normalization. In general, this breaks down,

  and I have already seen wrong derivations based on negligence with

  the multiplicities.

Chap 7:


* below (7.12), the fact that all Casimirs can be expressed in terms

  of the defining rep is a consequence of the fact that this

  representation generates the representation ring, ie is

  fundamental. The question is what algebra you are actually looking at.

(PC: for rank $r$ Lie algebra there are $r$ fundamental representations, and my "defining" one is the lowest dimensional 0f these. But what do you think I should write here?)

Edits incorporated

Chap 1


* Why do you call the traditional way `canonical'? Usually, canonical

  means `free of choices' or `independent of basis', and actually it

  is the diagrammatical method that is free of choices whereas old

  fashioned people would write explicit matrices.

* p. 3, par. 3, the last sentence reads "...the defining

  representations... are groups..."

(PC: sorry, I do not see what to correct here)

Chap 2:


There is a general issue here. Compared with the standards of pure

maths, you are very sloppy in this Chapter. I suggest that you check

this chapter very carefully, possibly together with someone from puremaths,

(PC: were it only that easy. I have one good, card carrying mathematician friend. I'll try trading some babysitting for proofreading with her. But this is what my wife Sara A Solla has to say about the suggestion: "Two physicists balooning get lost in a storm. When it finally clears up they find themselves in a totally unfamiliar landscape. They see a man on the ground, and they yell down to him: "wheeee-re aaaaare we?". He looks at them. They yell again: "wheeee-re aaaaare we?" He looks at them. Finally, as the wind is about to blow them away from the hearing range, he yells back: "Youuuuu aaaare in a balooooooon". One physicist says to the other: "Just our luck to run into a mathematician!". How do you know he is a mathematician?" asks the second physicist. "First, it takes him a long time to answer. Second, the answer is absolutely correct. Third, the answer is totally useless". If you have a "puremaths" friend who is not the classical give-him-a-round-around ma-the-ma-tician, you'll do me a great favor if you asked him to try reading this chapter.)

because potential reviewers might get very upset if you define

`infinitesimal groups',

(PC: the notion of `infinitesimal groups' is hair-raising, did I really use this expression? Cannot find it anywhere... I put some "rigor" now into `infinitesimal transformations'.)

* Sect 2.1, the relation bewteen G^a_b and G_b^a is not explained

(PC: fixed)

* equation below (2.6) needs a T on the r.h.s.

(PC: caught that one already)

* p. 12, you use the fact that defining rep tensor its dual contains

  the adjoint as one of its irreducible components. Maybe you can

  stress this fact.

(PC: done)

* p. 13, second equation, there should be a straight line whose

  prefactor is (n+B+C).

(PC: done)

* p. 13, v) As I recall the terminology, P_aP_b=\delta_ab P_a is

  called `transversal', not `orthogonal'. A projector is `orthogonal'

  if it is symmetric with respect to a given scalar product.

(PC: dunno... - can you give me a reference?)

Chap 3:


   the set of integers Z is a cyclic group, but not covered by your


(PC: as I do not use this, I removed the reference to cyclic groups)

   neither quaternions (skew-field) nor octonians (non-associative

  algebra) form a field

(PC: removed the reference to ``fields'')

   sect 3.1.3, you should say that you need a basis for your vector

  space in order to define the structure constants. The formulas are

  correct, but over what range do your indices go?

(PC: done)

   definition of a Lie algebra is missing, you just give an example using the commutator in an associative algebra.

(PC: removed Lie algebra reference)

You are also using the \cdot in two different meaning in the same definition.

(PC: I do not see where?)

   I am not quite happy with the notion of `defining rep'. I could use

  any representation here. Why is one of them special?

(PC: I prefaced the definition with a plea to the reader for much patience...)

* above (3.36) replace T\in by T\subset

(PC: done)

* sect 3.2.1, the algebra is certainly not finite, but it is


(PC: done)

* 'minimal basis' -> 'basis'

(PC: done)

* below (3.51), the notion of "c-number" was not introduced, but is

  actually not required either.

(PC: done)

* (3.53), which projector belongs to which invariant matrix?

(PC: done)

Chap 4:


* (4.38) are the signs correct?

(PC: they were not, thanks - fixed now)

Chap 5:


* (5.12) there should be two sums over \lambda_1 and \lambda_2

(PC: fixed now)

Chap 7:


* (7.10) typo, see (6.51)

(PC: fixed)

* below that, replace [nxn] by n

(PC: fixed)

* before headline of 7.3, are you sure about the Betti numbers?

(PC: I am rarely sure of anything, but this is what Chevalley calls them... The definition is global - cohomologies, the Casimirs are local, would love to understand the connection.)

* (7.40)-(7.43) why do you call them Casimirs?

(PC: changed casimir -> invariant, fixed)

* below (7.49), A and B are not yet in table 7.5

(PC: wrong table reference, fixed)

* I think the result that the exceptionals do not have a primitive

  quartic Casimir, was found by Okubo (PC: I was and am citing him for that) .

A very clean proof is by

  K. Meyberg, Spurformeln in einfachen Lie-algebren, Abhandlungen des

  Mathematischen Seminars der Universitat Hamburg 54 (1984) 177--189

  Meyberg actually uses the structure of the decomposition of adjoint

  tensor adjoint as the input. His calculation shows that the absence

  of a primitive quartic Casimir IS directly related to the Deligne

  conjecture. Your (7.53) should agree with Meyberg.

  By the way, there is another Meyberg article in an obscure journal

  which is definitely worth looking at (PC: I cite it now, sight unseen...) ,

  K. Meyberg, Trace formulas in various algebras and L-projections,

  Nova Journal of Algebra and Geometry 2 No 2 (1993) 107-135

Chap 8:


* hep-th/0108095 has a further applications of group integrals in

  lattice gauge theory and derives a closed form for the strong

  coupling expansion

Chap 9:


* Sect 9.3-9.4, it is not always clear what refers to U(n) and what to


* below (9.23), below and to the left -> below and to the right

(PC: fixed)

* p. 92, line 3, arrangement -> arrangements

(PC: fixed)

Chap 10:


* first line, leave invariant a non-degenerate symmetric bilinear form

* why complex conjugation? See my comment above. For SO(N), I just

  need real vector spaces.

* below (10.44), when you mention Young diagrams, this refers first to

  SU(n) - I was not aware of this and looked at table 10.3 which gave

  10 independent components of R in d=4 (which is wrong). On the next

  page, I realized that the (2,2) diagram was meant to belog to SU(4)

  rather than SO(4).

Chap 15:


* below (15.4), this is actually SU(2) spin rather than SO(3) angular

  momentum as there are non-integer spin irreps involved here

(PC: fixed)

* (15.23) typo

(PC: fixed)

Chap 16:


* below (16.8), "To spell it out..." This is correct, but you have not

  proved it. You have shown only that E \neq F' and A'\neq 0 implies

  n=3 and ff=\delta\delta, but not the converse implication.

* p.172, I suggest to replace "Contracting with \delta" by

  "Contracting the free ends of the top line with \delta". It took me

  a while to see what you did.

(PC: fixed)

* below (16.11), the word 'follows' confused me. It follows only if

  you assume the general form of the decomposition and the

  explanations below (16.12).

(PC: fixed)

* (16.14) last tensor, the dots for the f-tensor are missing

(PC: fixed)

Chap 17:


* p. 180, line 2, typo

(PC: cannot see it...)

* (17.8), why is this =0?

(PC: fixed)

* p. 189, the factorization into linear factors breaks down for higher

  tensor powers. Landsberg and Manivel have got a construction which

  suggests that the correct notion are Casimir eigenspaces (which can

  contain more than one irreducible component) which yields higher

  order polynomials that do not factorize. The same is observed in


* p. 195, how do you match the list of n=3,6,9,15,27 with the algebras

  a2,a2+a2,a5,e6? The case of e6 is obvious. But you have then to

  start with a 6-dim irrep of a2. Is it that? It is then probably

  quite difficult to show that it was really a2 (yes, I see 18.9

  now). However, the logical structure of these arguments is not quite

  obvious. Can you make more transparent what you assume and what you

  acually prove?

Chap 19:


* below (19.19) typo

* before (19.47) typo

(PC: fixed)

* Sect 19.4, "Jordan algebra of traceless Hermitean matrices",

  something is odd here.

Chap 20:


* Sect 20.3.2, it says "the 7-dim rep ... is a subgroup ..."

(PC: do not know what to do about this - fixed equation reference closeby)

Chap 21:


* Table 21.2, there should be a dotted line marking the

  Freudenthal-Tits magic square

App A:


* below (A.3) "As P_r is..." -> "As P_r projects onto ..."

(PC: fixed)

These chapters need more careful proof reading of the

language. It happens quite often that the definite article `the' is


(PC: welcome to slavic (non)use of articles - very efficient, saves lots of print)

Also, you should have a look at

  C. H. Barton and A. Sudbery, Magic squares of Lie algebras, math/0001083.

for the symmetries of the magic triangle.