Date: 25 Feb 2003
From: hpfeiffer <hpfeiffer [ at ] perimeterinstitute.ca
http://www.damtp.cam.ac.uk/user/hp224/ Oct 2005
moved to A Einstein Institute, pfeiffer [ at ] aei.mpg.de
Hendryk Pfeiffer: I refer to the version of December 2, 2002
Edits still outstanding
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
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.
* It makes a difference whether a group is unitary or whether a
particular representation is unitary
* below (3.42), `minimal characteristic' -> `minimal'.
* below (4.1), I am not sure whether reality is the right
condition. See my comment on the dual rep above.
* 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
* 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.
* 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..."
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,
because potential reviewers might get very upset if you define
* Sect 2.1, the relation bewteen G^a_b and G_b^a is not explained
* equation below (2.6) needs a T on the r.h.s.
* 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.
* p. 13, second equation, there should be a straight line whose
prefactor is (n+B+C).
* 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.
the set of integers Z is a cyclic group, but not covered by your
neither quaternions (skew-field) nor octonians (non-associative
algebra) form a field
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?
definition of a Lie algebra is missing, you just give an example using the commutator in an associative algebra.
You are also using the \cdot in two different meaning in the same definition.
I am not quite happy with the notion of `defining rep'. I could use
any representation here. Why is one of them special?
* above (3.36) replace T\in by T\subset
* sect 3.2.1, the algebra is certainly not finite, but it is
* 'minimal basis' -> 'basis'
* below (3.51), the notion of "c-number" was not introduced, but is
actually not required either.
* (3.53), which projector belongs to which invariant matrix?
* (4.38) are the signs correct?
* (5.12) there should be two sums over \lambda_1 and \lambda_2
* (7.10) typo, see (6.51)
* below that, replace [nxn] by n
* before headline of 7.3, are you sure about the Betti numbers?
* (7.40)-(7.43) why do you call them Casimirs?
* below (7.49), A and B are not yet in table 7.5
* 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
* hep-th/0108095 has a further applications of group integrals in
lattice gauge theory and derives a closed form for the strong
* 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
* p. 92, line 3, arrangement -> arrangements
* 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).
* below (15.4), this is actually SU(2) spin rather than SO(3) angular
momentum as there are non-integer spin irreps involved here
* (15.23) typo
* 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.
* 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).
* (16.14) last tensor, the dots for the f-tensor are missing
* p. 180, line 2, typo
* (17.8), why is this =0?
* 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
* below (19.19) typo
* before (19.47) typo
* Sect 19.4, "Jordan algebra of traceless Hermitean matrices",
something is odd here.
* Sect 20.3.2, it says "the 7-dim rep ... is a subgroup ..."
* Table 21.2, there should be a dotted line marking the
Freudenthal-Tits magic square
* below (A.3) "As P_r is..." -> "As P_r projects onto ..."
These chapters need more careful proof reading of the
language. It happens quite often that the definite article `the' is
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.