Date: 2 Dec 2002
Limbo, limbo and catholicism.
As far as the literature is concerned, here are some ideas that come to my
G Kuperberg, Spiders for rank 2 Lie algebras, Comm Math Phys 180 (1996) 109
P Vogel, the material on Vassiliev Theory on his web site
The question is whether one can construct the representation theory of simple Lie algebras using only diagrams. The answer is `yes' at least for rank 2. It is possible to construct the entire representation category from diagrams, both of the Lie algebra and of the q-deformed envelope.
People are now studying whether one can do this for all simple Lie algebras or for an even wider class. I am not an expert on the following, but as I understand it, this is exciting because of the Vassiliev invariants: There are more invariants than those coming from Lie algebras. So there is the conjecture that there exists a more general structure, some `universal Lie algebra' which is defined in terms of diagrams for its representations and which can be specialized to a large class of Lie algebras, but which also covers the more general cases.
This is probably the most advanced application of `bird track' technology in pure maths. I think most people involved there did not know your Nordita book and just reinvented a lot of it.
Bruce Westbury explained most of this to me. Probably he, Vogel or Kuperberg are the people to contact for more details.
There is also some connection to the extended Freudenthal magic square (I have seen that you mention Barton-Sudbery). The algebras in one row of it form a family in a new sense. The diagrams of their representation category are the same and it is only a single rational coefficient which differs and which is responsible for the specialization to one particular algebra.
The new series behavior applies in particular to the exceptional algebras and motivates Deligne's conjecture. I have seen that you refer to the paper by Cohen and DeMan. We have recently extended some aspects to higher order tensor products in
I think the first paper on the use of diagrams for SU(N) irreps was in M Creutz, On invariant integration over SU(N), J Math Phys 19 No 10 (1978) 2043
I am not sure how relevant the following is: There are many results on representation theory formulated in tensor notation. They could be translated into diagrams, but here they are presented with many indices:
AJ Macfarlane, Lie algebra and invariant tensor technology for g_2, Int J Mod Phys A 16 (2001) 3067
AJ Macfarlane, H Pfeiffer, On characteristic equations, trace identities and Casimir operators of simple Lie algebras, J. Math. Phys. 41 No. 5 (2000) 3192-3225, Erratum: Vol. 42 No. 2 (2001) 977.
AJ Mountain, Invariant tensors and Casimir operators for simple compact Lie groups, J Math Phys 39 No 10 (1998) 5601
and on some problems with the naive version of this tensor calculus which implies the same trouble for the diagrammatical calculus:
M Forger, Invariant polynomials and Molien functions, J Math Phys 39 (1998) 1107
Finally, there is a different flavour of diagrams for quantum groups and knot invariants when the over- and undercrossings and the twists in the framing are relevant.
On the one hand, there are the diagrams of
N Reshetikhin,VG Turaev, Ribbon graphs and their invariants derived from quantum groups, Comm Math Phys 127 (1990) 1
VG Turaev, Quantum invariants of knots and 3-manifolds Walter deGruyter, Berlin, 1994
and at first sight it seems they are far off the bird track idea. However, they can be specialized to representations of compact Lie groups, and one can prove results about products of 6j-symbols from them:
gr-qc/0211106 and the references therein
On the other hand, there is a second type of diagrams using the ideas of skein theory in order to accomplish the same, for example the `Chain Mail'. I do not have very good references on this, but maybe the following:
DV Boulatov, Quantum deformation of lattice gauge theory, Comm Math Phys 186 (1997) 295
JE Roberts, Skein Theory and Turaev-Viro invariants, Topology 34 (1995) 771
And finally the comment that the spin networks of Penrose which are now heavily used in loop quantum gravity and in spin foam quantum gravity (the area I am now specializing in), are just bird track diagrams - again the communities are rather disjoint, and I am not sure whether the gravity people have read your book.
e-mail: hpfeiffer [snail] perimeterinstitute.ca