Is there a definition of "quantum group"?

Discussion in 'Math Research' started by J.S. Milne, Sep 10, 2009.

  1. J.S. Milne

    J.S. Milne Guest

    I mean a definition as we have had, for example, for "group" since
    about 1882. In scanning the copious literature I have been unable to
    find one. For example, the books on quantum groups by Guichardet,
    Kassel, and Chari and Pressley don't list "quantum group" in the
    index. That by Shnider and Sternberg does, but the page indexed
    doesn't define "quantum group" (or anything else).

    Quantum groups (unlike algebraic groups) have their own entry in the
    Princeton Companion, but I couldn't find a definition there either,
    and nor in "What is a Quantum Group?", Notices AMS 2006.

    Am I correct in presuming that the state of the theory of quantum
    groups is similar to that of groups about 1870: lots of interesting
    examples, but no definition?
    J.S. Milne
     
    J.S. Milne, Sep 10, 2009
    #1
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  2. J.S. Milne

    tchow Guest

    I don't work in that area, but my impression is that the answer is yes,
    there is no single definition of a quantum group. Similarly vague terms
    include "large cardinal" and "sieve."
     
    tchow, Sep 14, 2009
    #2
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  3. I've been told that this means a Hopf algebra that is neither
    commutative not cocommutative. More logical would be to say that this
    means any Hopf algebra.
     
    Julia Kuznetsova, Sep 15, 2009
    #3
  4. J.S. Milne

    JoeShipman Guest

    "Large Cardinal" is defined as a cardinal number K such that there
    exists an inaccessible cardinal I <= K. "Large Cardinal Axiom" is a
    somewhat vaguer concept but a reasonable working definition is "a
    proposition of set theory that implies the existence or consistency of
    an inaccessible cardinal".
     
    JoeShipman, Sep 15, 2009
    #4
  5. Well, Chari and Pressley say that the category of quantum groups is
    opposite to the category of (not nec. commutative) Hopf algebras. But
    other people like to have an R-matrix in their quantum groups.

    So, yes, the field might still be in so much development that it is too
    early to decide on the most useful definition.
     
    Maarten Bergvelt, Sep 15, 2009
    #5
  6. J.S. Milne

    J.S. Milne Guest

    If "quantum group" = "Hopf algebra" there would be no need for a new
    name. In fact, all quantum groups seem to be Hopf algebras with
    additional structure, but that is about as helpful as saying that a
    group is a set with additional structure. It puzzles me that there is
    such a large literature on quantum groups but no definition (that I
    can find) or even a clear discussion of what is a quantum group and
    what is not. This, after all, is 2009, not 1870 (we are all post
    Bourbakists).
     
    J.S. Milne, Sep 15, 2009
    #6
  7. J.S. Milne

    tchow Guest

    In 2009 there are other examples of words that are not defined precisely.
    Take "sieve" for example. We have the Brun sieve, the Selberg sieve,
    the large sieve...etc. But what is a sieve? You will search the
    number-theoretic literature in vain for a precise definition that allows
    you to say, "This is a sieve and this is not." The reason is that there
    is no compelling motivation for such a precise definition. You could
    come up with one, but you would be coming up with a definition just for
    the sake of coming up with a definition.

    In the theory of quantum groups, we have quasitriangular Hopf algebras,
    Yangians, etc., which have precise definitions. Apparently, the workers
    in the field feel that this suffices, and are content to leave the
    term "quantum group" at the vague level of "deformation of a surrogate
    (e.g., a universal enveloping algebra) for a Lie-theoretic object
    (e.g., a Lie algebra or compact Lie group)." If the specialists don't
    see a reason for a more precise definition, then as an outsider I would
    hesitate to demand it.
     
    tchow, Sep 16, 2009
    #7
  8. I would say that, in most general sense, qg is a Hopf algebra which
    may be deformed to a commutative or a cocommutative one. But I also
    agree with other posters that there are quite limited benefits coming
    of those which do not allow some kind of braiding (R-matrix or
    somesuch).

    On the other hand, the question is quite moot anyway: people do not
    study "quantum groups"; they usually consider some particular
    flavors...

    Ilya
     
    Ilya Zakharevich, Sep 22, 2009
    #8
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