Group Theory: Birdtracks, Lie’s, and Exceptional Groups by Predrag Cvitanović

By Predrag Cvitanović

If classical Lie teams defend bilinear vector norms, what Lie teams look after trilinear, quadrilinear, and better order invariants? Answering this question from a clean and unique viewpoint, Predrag Cvitanovic takes the reader at the outstanding, four-thousand-diagram trip in the course of the idea of Lie teams. This ebook is the 1st to systematically advance, clarify, and follow diagrammatic projection operators to build all semi-simple Lie algebras, either classical and exceptional.

The invariant tensors are provided in a a little unconventional, yet in recent times standard, "birdtracks" notation encouraged by means of the Feynman diagrams of quantum box thought. significantly, invariant tensor diagrams exchange algebraic reasoning in accomplishing all group-theoretic computations. The diagrammatic strategy is especially powerful in comparing complex coefficients and crew weights, and revealing symmetries hidden by means of traditional algebraic or index notations. The ebook covers so much issues wanted in functions from this new viewpoint: diversifications, younger projection operators, spinorial representations, Casimir operators, and Dynkin indices. past this well-traveled territory, extra unique vistas open up, resembling "negative dimensional" family among a number of teams and their representations. the main interesting results of classifying primitive invariants is the emergence of all extraordinary Lie teams in one family members, and the attendant development of outstanding and classical Lie teams, the so-called Magic Triangle. Written in a full of life and personable sort, the booklet is aimed toward researchers and graduate scholars in theoretical physics and mathematics.

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49), the matrix G has a block diagonal form:   G1 0 0 CGC † =  0 G2 0  , G= C i Gi Ci . i . 60) The rep Gi acts only on the di -dimensional subspace Vi consisting of vectors Pi q, q ∈ V˜ . In this way an invariant [d×d] hermitian matrix M with r distinct eigenvalues induces a decomposition of a d-dimensional vector space V˜ into a direct sum of di dimensional vector subspaces Vi : M V˜ → V1 ⊕ V2 ⊕ . . ⊕ Vr . 61) For a discussion of recursive reduction, consult appendix A. The theory of class algebras [155, 156, 157] offers a more elegant and systematic way of constructing the maximal set of commuting invariant matrices Mi than the sketch offered in this section.

18) A specific choice of clebsches is quite arbitrary. All relevant properties of projection operators (orthogonality, completeness, dimensionality) are independent of the explicit form of the diagonalization transformation C. Any set of Cλ is acceptable as long as it satisfies the orthogonality and completeness conditions. a , α = δβα δλµ 2 a1 λ µ ... = λ µ . 19) Here 1 is the [dλ × dλ ] unit matrix, and Cλ ’s are multiplied as [dλ × d] rectangular matrices. c = δca11 δca22 . . δbqq 2 c1 λ λ ...

2. , it transposes a diagram into its mirror image. bp = X Y . 1, April 8, 2011 29 DIAGRAMMATIC NOTATION In sections. 32) matrices M : V p ⊗ V¯ q → V p ⊗ V¯ q in the multi-index notation ... d1 dp c1 cq ... 8) ... M ... b1 bp a1 aq in such a way that the matrix multiplication ... MN ... = ... ... N ... ... 9) and the trace of a matrix ... 10) ... M can be drawn in the plane. Notation in which all internal lines are maximally crossed at each multiplication [319] is equally correct, but less pleasing to the eye.

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