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This Week's Finds in Mathematical Physics (Week 249)

  1. Apr 10, 2007 #1
    Also available as http://math.ucr.edu/home/baez/week249.html

    April 8, 2007
    This Week's Finds in Mathematical Physics (Week 249)
    John Baez

    As you may recall, I'm telling a long story about symmetry, geometry,
    and algebra. Some of this tale is new work done by James Dolan, Todd
    Trimble and myself. But a lot of it is old work by famous people which
    deserves a modern explanation.

    A great example is Felix Klein's "Erlangen program" - a plan for
    reducing many sorts of geometry to group theory. Many people tip
    their hat to the Erlanger program, but few seem to deeply understand
    it, and even fewer seem to have read what Klein actually wrote about it!

    The problem goes back a long ways. In 1871, while at Goettingen,
    Klein worked on non-Euclidean geometry, and showed that hyperbolic
    geometry was consistent if and only if Euclidean geometry was.
    In the process, he must have thought hard about the role of symmetry
    groups in geometry. When he was appointed professor at Erlangen
    in 1872, he wrote a lecture outlining his "Erlanger Programm" for
    reducing geometry to group theory.

    But, he didn't actually give this lecture as his inaugural speech!
    He spoke about something else.

    So, nobody ever heard him announce the Erlangen program. And, until
    recently, the lecture he wrote was a bit hard to find. Luckily, now
    you can get it online:

    1) Felix Klein, Vergleichende Betrachtungen ueber neuere geometrische
    Forsuchungen, Verlag von Andreas Deichert, Erlangen, 1872. Also
    available at the University of Michigan Historical Mathematics Collection,

    Even better, Johan Ernst Mebius has recently prepared an HTML version,
    with links to the above version:

    2) Johan Ernst Mebius, Felix Klein's Erlanger Programm,

    But what if you have the misfortune of only reading English, not
    German? Until now the only translation was quite hard to obtain:

    3) Felix Klein, A comparative review of recent researches in geometry,
    trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249.

    In case you're wondering, the "New York Mathematical Society" no longer
    exists! It was founded in 1888, but in 1894 it went national and became
    the American Mathematical Society.

    Luckily, Chris Hillman got ahold of this old journal and scanned it in!
    So now, you can read Klein's paper in English here:

    4) The Erlangen program, http://math.ucr.edu/home/baez/erlangen/

    English-speakers can read more about the Erlangen program here:

    5) Felix Klein, Elementary Mathematics from an Advanced Standpoint:
    Geometry, part 3: Systematic discussion of geometry and its foundations,
    Dover, New York, 1939.

    Luckily Dover keeps its books in print!

    For more on the Erlangen program, try these:

    6) Garrett Birkhoff and M. K. Bennett, Felix Klein and his
    "Erlanger Programm", in History and Philosophy of Modern Mathematics,
    eds. W. Aspray and P. Kitcher, Minnesota Stud. Philos. Sci. XI,
    University of Minnesota Press, Minneapolis, 1988, pp. 145-176.

    7) Hans A. Kastrup, The contributions of Emmy Noether, Felix Klein and
    Sophus Lie to the modern concept of symmetries in physical systems,
    in Symmetries in Physics (1600-1980), ed. M. G. Doncel, World Scientific,
    Singapore, 1987, pp. 113-163.

    8) I. M. Yaglom, Felix Klein and Sophus Lie: Evolution of the
    Idea of Symmetry in the Nineteenth Century, trans. S. Sossinsky,
    Birkhauser, Boston, 1988.

    For more about Klein, try "week213" and this little biography:

    9) MacTutor History of Mathematics Archive, Felix Klein,

    But what does the Erlangen program actually amount to, in the language
    of modern mathematics? This will take a while to explain, so the best
    thing is to dive right in.

    Last week in the Tale of Groupoidification I tried to explain two slogans:



    They're a bit vague; they're mainly designed to give you enough intuition
    to follow the next phase of the Tale, which is all about how:



    But before the next phase, I need to say a bit about how groupoids
    and spans of groupoids fit into Klein's Erlangen program.

    Groupoids are a modern way to think about symmetries. A more
    traditional approach would use a group acting as symmetries of
    some set. And the most traditional approach of all, going back to
    Galois and Klein, uses a group acting *transitively* on a set.

    So, let me explain the traditional approach, and then relate it
    to the modern one.

    I hope you know what it means for a group G to "act" on a set X.
    It means that for any element x of X and any guy g in G, we get a
    new element gx in X. We demand that

    1x = x


    g(hx) = (gh)x.

    More precisely, this is a "left action" of G on X, since we write
    the group elements to the left of x. We can also define right actions,
    and someday we may need those too.

    We say an action of a group G on a set X is "transitive" if given
    any two elements of X, there's some guy in G mapping the first
    element to the second. In this case, we have an isomorphism of sets

    X = G/H

    for some subgroup H of G.

    For example, suppose we're studying a kind of geometry where the symmetry
    group is G. Then X could be the set of figures of some sort: points, or
    lines, or something fancier. If G acts transitively on X, then all
    figures of this sort "look alike": you can get from any one to any other
    using a symmetry. This is often the case in geometry... but not always.

    Suppose G acts transitively on X. Pick any figure x of type X and let
    H be its "stabilizer": the subgroup consisting of all guys in G that map
    x to itself. Then we get a one-to-one and onto map

    f: X -> G/H

    sending each figure gx in X to the equivalence class [g] in G/H.

    If you haven't seen this fact before, you should definitely prove it -
    it's one of the big ways people use symmetry!

    Here's one kind of thing people do with this fact. The 3d rotation
    group G = SO(3) acts on the sphere X = S^2, and the stabilizer of
    the north pole is the 2d rotation group H = SO(2), so the sphere is
    isomorphic to G/H = SO(3)/SO(2). The same sort of result holds in any
    dimension, and we can use it to derive facts about spheres from facts
    about rotation groups, and vice versa.

    A grander use of this fact is to set up a correspondence between
    sets on which G acts transitively and subgroups of G. This is
    one of the principles lurking behind Galois theory.

    Galois applied this principle to number theory - see "week201" for
    details. But, it really has nothing particular to do with number theory!
    In his Erlangen program, Klein applied it to geometry.

    Klein's goal was to systematize a bunch of different kinds of non-Euclidean
    geometry. Each kind of geometry he was interested in had a different
    group of symmetries. For example:

    n-dimensional spherical geometry has the rotation group SO(n+1) as
    symmetries. (Or, if you want to include reflections, the bigger
    group O(n+1).)

    n-dimensional Euclidean geometry has the Euclidean group ISO(n) as
    symmetries. (This group is built from rotations in SO(n) together
    with translations in R^n.)

    n-dimensional hyperbolic geometry has the group SO(n,1) as symmetries.
    (This group also shows up in special relativity under the name of
    the "Lorentz group": it acts on the "mass hyperboloid", and that's how
    hyperbolic geometry shows up in special relativity.)

    n-dimensional projective geometry has the group SL(n+1) as symmetries.
    (This group consists of nxn matrices with determinant 1. Scalar multiples
    of the identity act trivially on projective space, so you can mod out by
    those and use the "projective special linear group" PSL(n+1).)

    The details here don't matter much yet; the point is that there are lots
    of interesting kinds of geometry, with interesting symmetry groups!

    Klein realized that in any kind of geometry like this, a "figure"
    corresponds to a subgroup of G: namely the stabilizer group H of
    this figure. Here a "figure" could be a point, a line, a plane,
    or something much fancier. Regardless of the details, the set
    of all figures of the same type is G/H, and G acts transitively
    on this set.

    The really cool part is that we can use Klein's idea to *define* a
    geometry for any group G. To do this, we just say that *every* subgroup
    H of G stabilizes some figure. So, we work out all the subgroups of G.
    Then, we work out all the incidence relations - relations like "a point
    lies on a line". To do this, we take two sets of figures, say

    X = G/H


    Y = G/K

    and find all the invariant relations between them: that is, subsets of
    X x Y preserved all the symmetries. I'll say more about how to do this
    next time - we can use something called "double cosets". In nice cases,
    like when G is a simple Lie group and H and K are so-called "parabolic"
    subgroups, these let us express all the invariant relations in terms of
    finitely many "atomic" ones! So, we can really carry out Klein's program
    of thoroughly understanding geometry starting from groups - at least in
    nice cases.

    In short, group actions - especially transitive ones - are a traditional
    and very powerful way of using symmetry to tackle lots of problems.

    So, to bridge the gap between the traditional and the new, I should
    explain how group actions give groupoids. I'll show you that:


    It's not very hard to get a groupoid from a group action. Say we have
    a group G acting on a set X. Then the objects of our groupoid are
    just elements of X, and a morphism

    g: x -> y

    is just a group element g with

    gx = y.

    Composing morphisms works the obvious way - it's basically just
    multiplication in the group G.

    Some people call this groupoid an "action groupoid". I often call
    it the "weak quotient" X//G, since it's like the ordinary quotient
    X/G, but instead of declaring that x and y are *equal* when we have
    a group element g sending x to y, we instead declare they're
    *isomorphic* via a specified isomorphism g: x -> y.

    But for now, let's call X//G the "action groupoid".

    So, group actions give action groupoids. But, these groupoids come
    with extra stuff!

    First of all, the action groupoid X//G always comes equipped with a


    sending any object of X//G to the one object of G, and any morphism
    g: x -> y to the corresponding element of G. Remember, a group is
    a groupoid with one object: this is the 21st century!

    Second of all, this functor p is always "faithful": given two morphisms
    from x to y, if p maps them to the same morphism, then they were equal.

    And that's all! Any groupoid with a faithful functor to G is
    equivalent to the action groupoid X//G for some action of G on some
    set X. This takes a bit of proving... let's not do it now.

    So: in my slogan


    the "certain extra stuff" was precisely a faithful functor to G.

    What if we have a *transitive* group action? Then something nice

    First of all, saying that G acts transitively on X is the same as
    saying there's a morphism between any two objects of X//G. In
    other words, all objects of X//G are isomorphic. Or in other words,
    there's just one isomorphism class of objects.

    Just as a groupoid with one object is a group, a groupoid with one
    *isomorphism class* of objects is *equivalent* to a group. Here
    I'm using the usual notion of "equivalence" of categories, as
    explained back in "week76".

    So, G acts transitively on X on precisely when X//G is equivalent
    to a group!

    And what group? Well, what could it possibly be? It's just the
    stabilizer of some element of X! So, in the case of a transitive
    group action, our functor


    is secretly equivalent to the inclusion


    of the stabilizer group of this element.

    So, we see how Klein's old idea of geometrical "figures" as subgroups of G
    is being generalized. We can start with any groupoid Y of "figures" and
    "symmetries between figures", and play with that. It becomes an action
    groupoid if we equip it with a faithful functor to some group G:


    Then the action is transitive if all the objects of Y are isomorphic.
    In that case, our functor is equivalent to an inclusion


    and we're back down to Klein's approach to geometry. But, it's actually
    good to generalize what Klein did, and think about arbitrary "groupoids
    over G" - that is, groupoids equipped with functors to G.

    So, when we blend our ideas on spans of groupoids with Klein's ideas,
    we'll want to use spans of groupoids "over G" - that is, commutative
    diamonds of groupoids and functors, like this:

    / \
    / \
    / \
    / \
    v v
    X Y
    \ /
    \ /
    \ /
    \ /
    v v

    There's much more to say about this, but not today!

    I'll say one last thing before quitting. It's a bit more technical,
    but I feel an urge to see it in print.

    People often talk about "the" stabilizer group of a transitive action
    of some group G on some set X. This is a bit dangerous, since every
    element of X has its own stabilizer, and they're not necessarily all equal!

    However, they're all *conjugate*: if the stabilizer of x is H,
    then the stabilizer of gx is gHg^{-1}.

    So, when I say above that


    is equivalent to


    I could equally well have said it's equivalent to


    where the inclusion i' is the inclusion i conjugated by g. If you
    know some category theory, you'll see that i and i' are naturally
    isomorphic: a natural isomorphism between functors between groups
    is just a "conjugation". Picking the specific inclusion i requires
    picking a specific element x of X.

    Of course, I'll try to write later issues in a way that doesn't force
    you to have understood all these nuances!

    Previous issues of "This Week's Finds" and other expository articles on
    mathematics and physics, as well as some of my research papers, can be
    obtained at


    For a table of contents of all the issues of This Week's Finds, try


    A simple jumping-off point to the old issues is available at


    If you just want the latest issue, go to

  2. jcsd
  3. Apr 11, 2007 #2
    Hi John,

    Long time no see or hear. I thought I'd add a few comments on the
    Erlanger program
    which are certain to be ill-informed.

    One way I look at the question is this: take a category C and an
    object X in it.
    Consider natural ways to attach to X a subcategory G(X) of C. One can
    when it's possible to recover X from G(X). In Klein's situation, at
    least as far as
    my ignorant understanding goes, one takes G(X) to be the category
    one object and automorphism group Aut(X). In this form, the
    of X is impossible in most situations. For example, if C consists of
    compact two-manifolds, most of them have trivial automorphism group.
    So Klein was thinking of a quite `homogeneous' setting in his notion
    of geometry. I guess one can make this rather precise
    in the study of homogeneous spaces or hyperbolic manifolds.

    But one other interesting case looks at a different
    category associated to X, namely Cov(X), the category of covering
    spaces. Here, if C is the category of the spectrum of algebraic number
    fields, for example,
    and Cov(X) is interpreted as etale coverings in the sense of algebraic
    then it is a theorem of Neukirch and Uchida that you can indeed
    recover X
    from Cov(X). The same works when you let the X run over the category
    hyperbolic algebraic curves over number fields. This was part of the
    `anabelian program' discussed by Grothendieck in the 80's.
    >From this point of view, the anabelian program has quite a bit of

    the Erlanger program in it. I should admit, however, that the main
    experts in anabelian geometry feel quite luke-warm about this
    way of looking at things.

    In any case, the X -> G(X) -> X problem could be a not-entirely-
    uninteresting question in many other situations.

    BTW, I believe (but I' m not sure) the recovery of X from Cov(X) works
    in the category of hyperbolic manifolds (in the usual sense, no
    arithmetic here) of dimension at least three.

    One other theorem I know about that doesn't require arithmetic
    to discuss is when C is the category of compact smooth manifolds.
    Here let G(X) be the the group of diffeomorhisms of X. Then one can
    X from G(X).

    I had thought that the `Yoneda Lemma' could be thought of as a trivial
    of such recovery in a very general setting, but then realized while
    writing this post is that it's not
    so simple to formulate it in that way.

    I hope things are well.

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