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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2019

    am splitting this off from n-Lie algebra (which should better be renamed to Filippov algebra)

    – am being interrupted now – not done yet…

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2019

    have added detailed statement of the definition of M2-brane 3-algebras, of the way they are induced from metric Lie representations, and the theorem that this construction is a bijection.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 6th 2019
    • (edited Dec 6th 2019)

    added graphics showing the string diagram which represents the 3-bracket induced from a metric Lie representation (here)

    diff, v4, current

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2019

    The book at that link contains a treatment of Lie groups using ‘bird track’ notation.

    If classical Lie groups preserve bilinear vector norms, what Lie groups preserve trilinear, quadrilinear, and higher order invariants? Answering this question from a fresh and original perspective, Predrag Cvitanovic takes the reader on the amazing, four-thousand-diagram journey through the theory of Lie groups. This book is the first to systematically develop, explain, and apply diagrammatic projection operators to construct all semi-simple Lie algebras, both classical and exceptional.

    The invariant tensors are presented in a somewhat unconventional, but in recent years widely used, “birdtracks” notation inspired by the Feynman diagrams of quantum field theory. Notably, invariant tensor diagrams replace algebraic reasoning in carrying out all group-theoretic computations. The diagrammatic approach is particularly effective in evaluating complicated coefficients and group weights, and revealing symmetries hidden by conventional algebraic or index notations. The book covers most topics needed in applications from this new perspective: permutations, Young projection operators, spinorial representations, Casimir operators, and Dynkin indices. Beyond this well-traveled territory, more exotic vistas open up, such as “negative dimensional” relations between various groups and their representations. The most intriguing result of classifying primitive invariants is the emergence of all exceptional Lie groups in a single family, and the attendant pattern of exceptional and classical Lie groups, the so-called Magic Triangle. Written in a lively and personable style, the book is aimed at researchers and graduate students in theoretical physics and mathematics.

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 7th 2019

    Sorry meant to add this to the other thread. Bad connection on a phone.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2019

    Could you give a page number.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2019

    Got it: the notation is introduced on pages 9-10

    (this “webbook” doesn’t work well on my machine)

    So that is just standard Penrose/string diagram notation, as far as I can see!

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeDec 7th 2019

    replaced the string diagram graphics with one that also shows the Faulkner construction (here)

    diff, v6, current

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