Tracks, Lie’s, and Exceptional Magic

1 Introduction Sometimes a solution to a mathematical problem is so beautiful that it can impede further progress for a whole century. So is the case with the Killing-Cartan classification of semi-simple Lie algebras [Killing 1888; Cartan 1952]. It is elegant, it is beautiful, and it says that the 3 classical families and 5 exceptional algebras are all there is, but what does that mean? The construction of all Lie algebras outlined here (for a more detailed presentation , consult [Cvitanovi´c 2004]) is an attempt to answer to this question. It is not a satisfactory answer – as a classification of semi-simple Lie groups it is incomplete – but it does offer a different perspective on the exceptional Lie algebras. The question that started the whole odyssey is: What is the group theoretic weight for Quantum Chromodynamic diagram = ? (1.1) A quantum-field theorist cares about such diagrams because they arise in calculations related to questions such as asymptotic freedom. The answer turns out to require quite a bit of group theory, and the result is better understood as the answer to a different question: Suppose someone came into your office and asked " On planet Z, mesons consist of quarks and antiquarks, but baryons contain 3 quarks in a symmetric color combination. What is the color group? " If you find the particle physics jargon distracting, here is another way to posing the same question: " Classical Lie groups preserve bilinear vector norms. What Lie groups preserve trilinear, quadrilinear, and higher order invariants? " The answer easily fills a book [Cvitanovi´c 2004]. It relies on a new notation: invariant tensors ↔ " Feynman " diagrams, and a new computational method, diagrammatic from start to finish. It leads to surprising new relations: all exceptional Lie groups emerge together, in one family, and groups such as E 7 and SO(4) are related to each other as " negative dimensional " partners. Here we offer a telegraphic version of the " invariance groups " program. We start with a review of basic group-theoretic notions, in a somewhat unorthodox notation suited to the purpose at hand. A reader might want to skip directly to the interesting part, starting with sect. 3. The big item on the " to do " list: prove that the resulting classification (primitive invariants → all semi-simple Lie algebras) is exhaustive, and prove