# Appearances of 'exotic' compact Lie Groups

The structure theorem for compact Lie Groups states that all compact Lie groups are finite central quotients of a product of copies of $U(1)$ and simple compact Lie groups. And yet, as easy as arbitrary compact Lie groups are to describe, most Lie Groups one encounters are the various quotients of simple compact Lie groups and maybe some products of these groups (I will herein refer to such groups as standard Lie groups). The only compact examples which one encounters regularly that are not standard Lie groups are the unitary groups $U(n)$ (which are quotients of $U(1)\times SU(n)$) and $SO(4)$ (which is the diagonal $\mathbb{Z}/2\mathbb{Z}$ quotient of $Spin(4) \cong Spin(3)\times Spin(3)$).

I am currently trying to further expand my knowledge and understanding of compact Lie groups, so I am wondering:

Question: Has anyone encountered examples of non-standard Lie groups (other than the $U(n)$'s and $SO(4)$) in their research, as the autormorphism group of some object they were studying, or in some other way? If so, would you give a bit of description of the setting you were working in as well as a description of the non-standard group which appeared?

Although given a non-standard group, one can easily construct algebraic objects for which it is the automorphism group, I am more interested in instances of the reverse of this process wherein a non-standard group appears in the course of thinking about some other problem.

Edit: Since there still seems to be some misunderstanding of the intent of the question, to clarify the situation I am interested in, I am looking for groups of the form $G_1\times\ldots\times G_k/H$ where each $G_i$ is a compact simple Lie group, $k\geq 2$ and $H\subsetneq Z(G_1\times\ldots\times G_k)$ is not of the form $h_1\times \ldots \times h_k$ with $h_i\subseteq Z(G_i)$. So examples with multiple factors such as the the Structure Group of the Standard Model described by Theo are the sort of thing I'm looking for.

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I'm guessing you've only seen U(n) and SO(n) because you've only looked at the reals and the complex numbers. Look at the quaternions and the octonions and the other groups will pop up as automorphism groups of very natural objects. –  Tilman May 19 '11 at 21:46
Looking at the quaternions one usually gets $Sp(n)$ or perhaps $Sp(n)/2$, but these still fall under the umbrella of standard Lie groups. On the other hand, if say $Sp(n)\times Sp(m)$ mod its diagonal $\mathbb{Z}/2\mathbb{Z}$ for some $m,n$ somehow arose in someone's work, that is the kind of example I'm looking for. Similarly, the exceptional groups and their quotients arise from basic constructions using the octonions, but for example I have never seen any mention of more complicated groups like $E_6\times E_6$ mod its diagonal $\mathbb{Z}/3\mathbb{Z}$ coming up in any setting. –  ARupinski May 19 '11 at 22:33
I've seen orthogonal and symplectic similitude groups come up in the study of Theta correspondence, but i don't know enough to elaborate. Maybe Marty can say something. –  S. Carnahan May 20 '11 at 2:44
I think the most famous one is Berger's classification of holonomy groups of Riemannian manifolds. Check out: books.google.com/… Robert could say much more about this. –  Ian Agol May 20 '11 at 15:50
OK. Now that I see what you want, I agree that, in fact, Ian Agol's suggestion to look at the Riemannian holonomy groups is a very good one. I guess you'd regard Sp(n)Sp(1) (which is the quotient by the diagonal' $\mathbb{Z}_2$ in the center of Sp(n)×Sp(1)) as non-standard', and it does occur as holonomy. There are lots more in the exceptional symmetric spaces, of course. For example, $E_6\cdot S^1\subset SO(54)$, which is the holonomy of the symmetric space EVII (see Helgason's Table V), is the quotient of the product group by a diagonally embedded $\mathbb{Z}_3$. –  Robert Bryant May 21 '11 at 21:20

The structure group for the Standard Model of Particle Physics is almost the already-horrid $$G = \operatorname{SO}(3,1) \ltimes \mathbb R^4 \times \operatorname{SU}(3) \times \operatorname{SU}(2) \times \operatorname{U}(1)$$ (The non-compact part at the beginning you might throw out for the purposes of this question, and anyway it's almost certainly "wrong". For one, the universe does not have $\operatorname{SO}(3,1) \ltimes \mathbb R^4$ symmetry, because of gravity. For two, when you turn on a cosmological constant, you get things like anti de Sitter space, with structure group that is roughly $\operatorname{SO}(3,2)$.)

I say "almost", because in fact I have only described the Lie algebra of the symmetry group of standard particle physics. Some of this you already know. To account for spin, the non-compact part does not have $\operatorname{SO}(3,1)$ symmetry, but rather $\operatorname{Spin}(3,1)$. But in all of the real world, the fermion number operator (the non-identity element in the kernel of $\operatorname{Spin} \to \operatorname{SO}$) acts as $-1$ on all the adjoint-representations of the compact part. So actually, you should take the double cover of $G$, and mod out by a discrete group that encodes that, for example, there are no spin-$1$ particles that transform in as an $\operatorname{SU}(3)$ triplet. Also, you mod out by some finite groups to impose CPT theorem.

There may be other extensions and modding outs you need to do to get the symmetry group on the nose --- but the point is that even the compact part (which you get by breaking the global symmetry) is not just a product of these groups, but a quotient thereof.

Even more, there are four "accidental" Lie symmetries that follow from the standard model but aren't put in at the beginning. (CPT, on the other hand, was a discrete symmetry, and I've already sketched how to handle it, by taking a quotient.) These are "Baryon number", "Electron number", "Muon number", and "Tau number". They are "quantized", which is to say that each corresponds to an extra $\operatorname{U}(1)$ action. (This is a nice illustration of where the standard model is lying: in the standard model, neutrinos don't have mass, but in the real world, they do, and so among other things the three $\operatorname{U}(1)$ actions are actually all part of the same $\operatorname{SU}(3)$ action.)

If you would like to work out precisely the structure group of the standard model, take the Lie algebra of the group above (or just the compact part), and within its category of finite-dimensional representations, build the minimal subcategory that is closed under direct sums and tensor products and contains the representations listed under Wikipedia: Standard Model: Field Content. This is some Tannakian category, and you can run Tannaka-Krein reconstruction on it. You'll build some compact group, but it's not simply connected nor simple.

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Baryon number and the individual lepton numbers are not actually symmetries of the Standard Model viewed as a quantum field theory. They are symmetries of the classical theory but are spoiled by quantum effects known as anomalies. The only anomaly free global U(1) symmetry of the Standard Model is the difference of the baryon number and the total lepton number, B-L. –  Jeff Harvey May 20 '11 at 16:49
Very interesting example, even without the exact subgroup of the center which one quotients out being explicitly known. –  ARupinski May 20 '11 at 23:56
@Jeff Harvey! Oh, good to know, and my mistake. I was basing that part of my answer on Wikipedia. –  Theo Johnson-Freyd May 21 '11 at 15:59

There won't be any right' answer here because there are many different ways that one can come across groups in studying various algebra problems, but here are a few examples:

Maybe the most famous exotic case is when you classify the definite inner product algebras over $\mathbb{R}$: Such an algebra $\mathbb{A}$ is an $\mathbb{R}$-algebra with unit $1$ endowed with a positive definite inner product $\cdot$ that satisfies the multiplicative identity $xy\cdot xy = (x\cdot x) (y\cdot y)$. It's a classical theorem (due to A. Hurwitz (1898)) that the $\mathbb{R}$-dimension of such an algebra is one of $1$, $2$, $4$, or $8$, and the corresponding algebras are $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ (i.e., Hamilton's quaternions), or $\mathbb{O}$ (i.e., the octonions of Graves and Cayley). The group of automorphisms of $\mathbb{O}$ is the exceptional group $G_2$ (compact, connected, of dimension $14$).

$G_2$ and its noncompact dual, $G_2^\ast$ also arise as the stabilizer groups of generic' alternating $3$-forms in dimension $7$.

Similarly, Spin(7) as a subgroup of $GL(8,\mathbb{R})$ turns up as a stabilizer when you go to classify the alternating $4$-forms in dimension $8$.

The exceptional group $F_4$ arises as the autormorphisms of the `exceptional' Jordan algebra of dimension $27$. ('Exceptional' in this context means that it's the one irreducible example that doesn't fit into one of the standard series.)

Depending on your tastes and background, there are many more examples of this nature that might appeal to you.

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It's Hurwitz, not Dickson, I think. –  Mariano Suárez-Alvarez May 20 '11 at 12:48
@Mariano: Of course, you are right. I've fixed it by an edit. –  Robert Bryant May 20 '11 at 15:01

My favorite example: the maximal compact subgroup of the split real form of $E_8$ is $K=Spin(16)/[x]$ where $x^2=1$, but K is not isomorphic to SO(16) - K does not have an irreducible 16-dimensional representation. The center of Spin(16) is a Klein 4-group, say {1,x,y,xy}; the outer (diagram) automorphism of $Spin(16)$ interchanges x,y. The element $xy$ is fixed by this automorphism, and Spin(16)/[xy]≃SO(16).

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It does not sound like this is an answer to the question. The question is whether or not it has come up naturally in the course of your investigations into other things. –  Ryan Budney May 19 '11 at 19:55
@Ryan: Does $E_8$ count as an "other thing"? –  André Henriques May 19 '11 at 20:29
Perhaps I'm biased but I consider $E_8$ to have come out of the classification of Lie groups so it's about as native to the study of Lie groups as you can possibly be. –  Ryan Budney May 19 '11 at 20:32
Your $K$ is the group sometimes known as $SO(16)'$ which, while not isomorphic to $SO(16)$, is still a central quotient of a compact simple Lie group. –  ARupinski May 19 '11 at 22:37
It is true that K is standard in your sense. Non-standard groups arise all the time as subgroups of standard ones. For example $S[O(p)\times O(q)]\subset SO(p+q)$, or $SU(p)\times SU(q)/*\subset PSU(p+q)$. –  Jeffrey Adams May 20 '11 at 3:03

Berger showed the homogeneous space $SU(5)/Sp(2)\cdot S^1$ has a metric of positive sectional curvature. Here, $Sp(2)\cdot S^1 = Sp(2)\times_{\mathbb{Z}/2\mathbb{Z}} S^1$. More generally, Bazaikin found an infinite family of free isometric actions of $Sp(2)\cdot S^1$ on $SU(5)$ (with appropriate left invariant metric), for which infinitely many of the quotients inherit a metric of positive sectional curvature.

The fact that the group acting is $Sp(2)\cdot S^1$ and not $Sp(2)\times S^1$ caused a few headaches when it came time to compute the topology of these examples.

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You need the groups $Spin^c(n)=: (Spin(n)\times U(1))/C_2$ in order to define first $Spin^c$-structures on Riemannian manifolds, then Dirac operators. See the book: B. Lawson, L. Michelson: Spin geometry, Princeton University Press (1989).

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