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Let $\operatorname{Klein}$ denote the category of principal homogeneous bundles. An object in this category is a tuple $\mathbf Q = (Q, P; G, H; q, a, \tilde a)$, where:

  • $G$ is a Lie group, and $H$ is a closed subgroup;

  • $P$ is a $G$-torsor and the manifold $Q$ is diffeomorphic to the quotient $G/H$;

  • The (left) actions $\tilde a : G \to \operatorname{Aut}(P)$ and $a : G \to \operatorname{Aut}(Q)$ are morphisms of topological groups; and

  • $q :P \to Q$ is a principal $H$-bundle.

A morphism $\Phi : \mathbf Q \to \mathbf Q'$ is described by a tuple $\Phi = (\varphi, \tilde \varphi, f)$, where $\varphi : Q \to Q'$ and $\tilde \varphi : P \to P'$ are diffeomorphisms which commute with the bundle maps $q : P \to Q$ and $q' : P' \to Q'$, and $f : G \to G'$ is a morphism of Lie groups mapping $H$ to $H'$, and which satisfies the identity $$\tilde a'_{f(g)} \circ \tilde \varphi = \tilde \varphi \circ \tilde a_g$$ for all $g \in G$.

I think that this is the correct categorical description of principal homogeneous bundles; please correct me if I'm wrong. I selected the name $\operatorname{Klein}$ in homage to Felix Klein and his Erlangen Program.

It seems that such a bundle $\mathbf Q$ contains all the data on its symmetries. Namely, I think that its automorphism group $\operatorname{Aut}(\mathbf Q)$ is isomorphic to its Lie group Lie group $G = G(\mathbf Q)$?

It is easy to see that there is a natural map $K : G \hookrightarrow \operatorname{Aut}(\mathbf Q)$, in that each $u \in G$ corresponds to a unique automorphism $K_u \in \operatorname{Aut}(\mathbf Q)$. The morphism $K_u = (k_u, \tilde k_u, c_u)$ is defined by $$k_u(q) = a_u(q), \quad \tilde k_u(p) = \tilde a_u(p), \quad \mathrm{and} \quad c_u(g) = ugu^{-1}.$$ That is, the morphism $K_u$ acts by left-multiplication on both $Q$ and $P$, but by left-conjugation on $G$.

  • Is this map $K$ surjective? i.e., is $\operatorname{Aut}(\mathbf Q)$ isomorphic to $G = G(\mathbf Q)$?

If the answer is yes, then I think that this captures the notion of "internal symmetries" of the bundle, since these are the transformations which preserve the bundle structure.

However, I know that groupoids also show up to describe symmetries in a categorical setting, and I would be interested to hear more on that point of view.

  • How can groupoids be used to describe symmetries in this category?
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3 Answers 3

up vote 5 down vote accepted

No, in general $G=G(\mathbf{Q})$ can be strictly smaller that ${\rm Aut}(\mathbf{Q})$.

Let $G$ be a Lie group and $H\subset G$ be a Lie subgroup. Set $P=G$, $\ Q=G/H$, and define the maps in the obvious way. We get a principal homogeneous bundle $\mathbf{Q}=(Q,P,\dots)$ in your sense.

Now take $G=SL(n,\mathbb{C})$, $\ H=SO(n,\mathbb{C})$. Consider the automorphism $\tau$ of $G$ given by $\tau(g)= (g^T)^{-1}$, there $g^T$ denotes the transpose of $g$. Since $\tau$ takes $H$ to itself, it induces an automorphism of the principal homogeneous bundle $\mathbf{Q}$. It is well known that for $n\ge 3$, the automorphism $\tau$ of $G$ is not inner, i.e. is not of the form $c_u(g)=ugu^{-1}$, which gives the negative answer to your question.

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Oh yes, I should have caught that! Thank you so much for the good example, @Mikhail Borovoi. –  Tom LaGatta Feb 11 '13 at 1:50

Did you have a look at parabolic geometries. Your description reminds me of the flat models of parabolic geometries.

MR1352565 (96k:16039). Andreas Cap, Jan Slovak: Parabolic Geometries I: Background and General Theory. Mathematical Surveys and Monographs 154, Amer. Math. Soc. 2009, 628 pp.

Here $H$ is assumed to be a parabolic subgroup of a semisimple or reductive group $G$; you can do an astonishing amount of things with this.

A little more general, look at Cartan connections.

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The OP asks: How can groupoids be used to describe symmetries in this category? Here are some suggestions for starting.

A principal $G$-bundle $E \to B$ can also be desribed as a groupoid $P= EE^{-1}$ over $B$, a construction due to C. Ehresmann. Here $P(b,c)$ is the set of $G$-maps $E_b \to E_{c}$. So one may ask: what for groupoids generalises the well known inner automorphism map $G \to Aut (G)$ for a group $G$?

Now the category $Gpd$ of abstract groupoids is cartesian closed, this is one aspect of the utility of groupoids. We can write the exponential law as a bijection $$Gpd(G \times H,K) \cong Gpd(G, GPD(H,K)).$$ The objects of $GPD(H,K)$ are the morphisms, or functors, $H \to K$ and the arrows of $GPD(H,K)$ are the natural equivalences of functors. In the case of groups, these are just conjugacies of morphisms.

So for any groupoid $G$ there is an endomorphism object $END(G)$ which is a monoid object in groupoids, and this has a maximal subgroup object $AUT(G)$ which is a group object in the category of groupoids. However as shown in the paper available here, group objects in groupoids are equivalent to crossed modules, and the crossed module one obtains by this process is of the form $d: S(G) \to Aut(G)$ where is $S(G)$ is the group of admissible sections $\sigma$ of $G$ as defined by Ehresmann in his paper on topological and differentiable categories. Such a $\sigma $ is a section of say the source map $s$ such that $t\sigma$ is a bijection on $Ob(G)$. These have a multiplication defined by Ehresmann: $\sigma \tau (x)= \sigma (t\tau x) \tau)x)$ (or analogous, depending on conventions conventions). (Such $\sigma$ are called bisections in Mackenzie, K.C.H., General theory of Lie groupoids and Lie algebroids, London Mathematical Society Lecture Note Series, Volume 213. Cambridge University Press, Cambridge (2005). )

Can one use this framework to study the differentiable, or Lie, case?

I may be able to add more later.

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