Report for Princeton University press on Group Theory by P Cvitanovic

Princeton U Press, Jan 2003

Reviewer 1:

• I did not spend much time on Chs. 4, 8, 11 and some other sections:10.5, 16.6... .
• I must admit to finding Ch. 17, especially Sec. 17.2, and 18.7 hard going. Perhaps this is due to the built up in the elements of bird-track diagrams that one has to carry along in one's head.

• I think that PUP might require that the epilogue be omitted (PC: included here to answer questions that arise in conversations and seminars - now moved to the book homepage) , and the `acknowledgement' to B. Julia be changed (PC: internal joke, now removed).

• (2)}. The ms, as sent to me, does not have most of the listed references flagged in the text at the point where they become relevant. (PC 12 Jul 2007: I decided to err on side of citing too many - almost all of the papers cited I have looked over in 1974-2007 time window. Cross-referencing all of them? I give up. The PDF version of the book back-links those actually cited. )

(1). It is important that Sec. 2.2 be crystal clear. In fact some of the bird-track lines on P9 of the ms seem to be of the wrong thickness. (PC 12 Jul 2007: 2 `identity' birdtracks were thin instead of thick. Fixed now )

(2). The ms, as sent to me, does not have most of the listed references flagged in the text at the point where they become relevant. No doubt that will be taken care of for a final draft. (PC 12 Jul 2007: I decided to err on side of citing too many - almost all of the papers cited I have looked over in 1974-2007 time window. Cross-referencing all of them? I give up. The PDF version of the book back-links those actually cited. )

(3). There are places where the text just appeared to peter out, e.g. Sec. 16.7. (PC 12 Jul 2007: G_2 reps are too boring, and many references do them well - will most likely omit this section from the book, or really write it. )

(4). Fischler notation on P127. Is this of importance (in which perhaps some further discussion) or an inessential distraction? (PC 12 Jul 2007: mostly a pointer to further reading, if a reader needs to get SO(n) reps right. Though there are probably other refs to point to, too. )

(5). Ch. 14 has wrong notation in the odd page headings. (PC 12 Jul 2007: PUP macros capitalize SP(N). Thanks, changed title to ``Spinors' symplectic sisters". )

(6). The ms says that the sp(n) analogue of the so(n) spinor representations needs to be invented. It is in fact well-known that the metaplectic representations fulfil the role and at a fairly deep level; see the paper [3] by G. Segal. Sec. 4 of [4] employs an operator view: fermionic vis-á-vis bosonic creation operator. (PC: incorporated 9 Oct 2005)

(7). There is one very important family of tensors, to which bird tracks methods could easily be applied, which receives no mention in the ms. These are the totally antisymmetric tensors that correspond to the cohomology cocycles of any Lie algebra. Their relation to the symmetric tensors and Casimir operators that feature centrally in the PC ms is outlined in reference 48 of the ms. Their properties, for the su(n) family, are systematically developed in [4] below. Surely they deserve a mention, they are of much greater use in applications (e.g. supercharges more than cubic in fermionic operators) than the Levi-Civita tensors.

(8). I was astonished to see no mention of the very fine book: F.  Gürsey and C.-H.  Tze, Division, Jordan and related algebras in theoretical physics, World Sci., Singapore, 1996. There is much material relevant to the PC ms in this book. (PC: I was astonished to hear such book exists. Cited now, but still need to read incorporated 12 Jul 2007)

Also Okubo has a book of relevance, does he not? (PC 12 Jul 2007: thanks, will do)

The book by L.  Frappat, A. Sciarrino and P.  Sorba, A dictionary of Lie algebras and superalgebras, Academic Press, 2001, deserves mention, perhaps alongside the Physics Report of Slansky in the ms. (PC: incorporated 12 Jul 2007)

(9). I think the author should think carefully about moderating the impression given at the foot of P236 that the magic triangle emerged from his own work. See e.g the paper [5] below. (PC 12 Jul 2007: sorry, I cannot find earlier work that you seem to imply. I constructed it in 1977 from my own invariance groups approach (not octonion matrices) and published in 1981. Who has done it earlier? )

(10). The fact that the decomposition of ad^Ä2 into irreducibles occurs uniformly over the family of groups in the last line of the magic triangle, which includes the exceptionals, does not, as is stated on P191, originate with Vögel, Deligne, etc., but in fact with Meyburg [6]. Fairplay demands that this be acknowledged despite the fact that the paper appears in a possibly inaccessible journal. I have a copy of the paper; it covers the matter comprehensively. (PC: If this is the earlier work that you refer to, that is 3 years after my 1981 paper. Thanks, now incorporated 12 Jul 2007)

(10). Paul Howe not Poul. Also Ref 10 of the ms: the fourth author is B. W.  Lee; the journal (Rev. Mod. Phys.) made a correction later in volume 34 to indicate this. (PC: incorporated 12 Jul 2007)

(11). Many people will see sextonians in the ms for the first time and want to take the idea on board, but in fact Sec. 16.4 is very condensed, and a reader dipping into the ms at the section will find it difficult. In fact the text does not really say what a sextonian is, it just gives their algebra bird-track wise. The section is nevertheless accessible with some effort. (PC 16 Jul 2007: will probably omit sextoinians altogether)

Some References

(PC 16 Jul 2007: all suggested refs now promoted to a higher level of limbo, into the \Preliminary{ parts of the webbook manuscript} )

[1]. J. A.  de Azcarraga and A. J.  Macfarlane, Optimally defined Casimir operators..., math-ph/0006013, JMP 42 419-437 (2000). (PC: incorporated, but not referred to yet 12 Jul 2007)

[2]. A. J.  Macfarlane and H.  Pfeiffer, Representations of the exceptional and other Lie algebras with integral values of the Casimir operator, math-ph/0208014, J. Phys A, in press.

[3]. G.  Segal, Commun. Math. Phys., 80 301-... (1981).

[4]. A. J.  Macfarlane, H.  Pfeiffer and F.  Wagner, Symplectic and orthogonal Lie algebra technology for ... oscillator models of integral systems, math-ph/0007040, Internat. J. Mod. Phys. A16 1199-1225 (2001).

[5]. J. A.  de Azcarraga and A. J.  Macfarlane, Compilation of identities for the antisymmetric tensors of the higher coccyles of su(n), math-ph/0006026, Internat. J. Mod. Phys. A16 1377-1409 (2001).

[6]. J. R.  Faulkner and J. C.  Ferrar, Exceptional Lie algebras and related algebraic and geometric structures, Bull. London Math. Soc. 9 1-35 (1977).

[7]. K.  Meyburg, Spurformeln in einfachen Lie-algebren, Abh. Math. Sem. Univ. Hamburg, 54 177-189 (1984). (I now see this is reference 195 in the ms.)

I would suggest that my papers [1,2,4,5] deserve citation by PC and also [8].

[8]. A. J.  Macfarlane, Lie algebra and invariant tensor technology for g₂, math-ph/0103021, Internat. J. Mod. Phys. A16 3067-3097 (2001). (PC: referred to in G₂ chapter 6 Oct 2007)