# Exceptional isomorphisms of Lie groups

It is known that in low dimensions certain exceptional isomorphisms arise between Lie groups. I have read about some of them in some papers, but I have not been able to find a "systematic" treatment of such isomorphims. By "systematic", I mean a presentation of a list of the most relevant of them (in the real and complex case, for instance) with some proofs and explanations. So, my question is, Does there exists any relatively complete reference on this topic?

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Many of these isomorphisms can be organized around the theory of Clifford algebras and spin groups; the first chapter of Spin Geometry by Michelsohn and Lawson has a detailed discussion of this. I think some of the isomorphisms are best understood using the octonions, however, so Michelsohn and Lawson's discussion is unlikely to be comprehensive. –  Paul Siegel Nov 19 '11 at 14:37

The complete list is quite short -- at least, up to changing real form. I'll give the compact group version, up to isogeny.

A_1 = B_1 = C_1: $SU(2) \cong Spin(3) \cong U(1,{\mathbb H})$

For the first, the adjoint representation gives a map $SU(2)/Z \to O({\mathfrak su}(2))$, which by dimension count has image $SO(3)$. (It maps to $O(3)$ because we're preserving the Killing form.)

D_2 = A_1 x A_1: $SO(4) \cong (U(1,{\mathbb H}) \times U(1,{\mathbb H}))/Z_2$.

One way to think about this is for unit quaternions to act on all quaternions on left and right, with $(-1,-1)$ acting trivially.

B_2 = C_2: $U(2,{\mathbb H})\cong Spin(5)$

A_3 = D_3: $SU(4) \cong Spin(6)$

Both of these derive from the action of $SU(4)$ (and its subgroup $U(2,{\mathbb H})$) on $Alt^2 {\mathbb C}^4$, with image inside $SO(6)$ (respectively, $SO(5)$).

Then you get all the others by taking universal covers then modding out by (finite) central subgroups, complexifying, and taking other real forms. For example, the second one leads to $SO(3,1) \cong PSL_2({\mathbb C})$ in this way.

(There are more if you think about finite groups of Lie type, e.g. $PSL_2(7) \cong PSL_3(2)$.)

ADDED: I forgot to mention the exceptional isomorphisms between "different" real forms of the same complex Lie group. For example, $SU(p,q)$ is usually different from $SL(p+q,{\mathbb R})$, but as André Henriques points out, they agree for $p=q=1$.

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Hey Allen: do you know how to explain $PLS_2(7)\cong PLS_3(2)$? Given a 3-dimensional vector space over $\mathbb F_2$, there should be a way of constructing a projective line over $\mathbb F_7$. I can't even figure out what the underlying set of that projective line is. References welcome. –  André Henriques Nov 19 '11 at 19:09
My colleague G. Cornelissen pointed out to me that my question concerning $PSL_3(2)\cong PSL_2(7)$ has been posted (and answered) at this mathstackexchange post: math.stackexchange.com/questions/1401/… –  André Henriques Nov 20 '11 at 13:44
Here's one approach to the two linear avatars of the 168-element simple group: math.harvard.edu/~elkies/M155.09/h5.pdf (and yes, as noted in the stackexchange thread, there's more about it in my exposition "The Klein Quartic in Number Theory"). –  Noam D. Elkies Feb 25 '12 at 16:39

The question goes back to paper A: Cartan's 1914 classification of real simple Lie algebras (Collected papers, Part I, Vol.1) with essentially complete answers for the Lie algebras . A complete discussion together with explicit isomorphisms for all the cases (both the Lie algebras and the Lie groups ) is in my book B: Diff. Geometry, Lie Groups and Symmetric Spaces, pp. 517- 528.

Comments: On the first page of A (bottom) Cartan states that real forms of the same simple Lie algebra over C are in general commpletely determined by the signature of the Killing form. As shown in B p. 517 this is contradicted by so*(18) and so(12,6) as well as by su*(14) and su(9,5). Cartan's phrase "general" clearly and deliberately meant "almost all the time". However, at the end of the paper (p. 352-355) he identifies real forms on the basis of the signature statement (which would require some additional caution). As mentioned in B, p. 520 the isomorphism so*(8) ~ so(6,2) does not occur in Cartan's original list although in a later paper he shows that the corresponding symmetric spaces are isometric.

Generally the group isomorphism in B are computational although some have geometric interpretation like the isomorphism so(3)~su(2) which comes from the standard stereographic projection.

S. Helgason

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Pencilled inside the back cover of my copy of Knapp's book is a picture that helps me keep a synoptic view of (all?) real form isomorphisms. It is the analog of summarizing the complex isogenies (already explained by Allen) by the statement that the sequence $$SO(3,\Bbb C) \to SO(4,\Bbb C) \to SO(5,\Bbb C)\to SO(6,\Bbb C)$$ (where arrows denote the obvious inclusions) is the $Z_2$ quotient of $$Sp(1,\Bbb C) \to SL(2,\Bbb C)^2 \to Sp(2,\Bbb C)\to SL(4,\Bbb C).$$ What we get for real forms (and let's not forget, complex groups regarded as real) is that the diagram

\begin{array}{ccccccccccccc} O(3,3) &&&& O(4,2) &&&& O(5,1) &&&& O(6)\\\ &\nwarrow &&\nearrow &&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &\\\ && O(3,2) &&&& O(4,1) &&&& O(5) &&\\\ &\nearrow &&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &&&\\\ O(2,2) &&&& O(3,1) &&&& O(4) &&&& \\\ &\nwarrow &&\nearrow &&\nwarrow &&\nearrow && && &\\\ && O(2,1) &&&& O(3) &&&& &&\\\ &\nearrow &&\nwarrow &&\nearrow &&&& &&&\\\ O(1,1) &&&& O(2) &&&& &&&& \\ \end{array}

is the same (up to $\pi_0$ and $\pi_1$) as

\begin{array}{ccccccccccccc} SL(4,\Bbb R) &&&& SU(2,2) &&&& SU^*(4) &&&& SU(4)\\\ &\nwarrow &&\nearrow &&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &\\\ && Sp(2,\Bbb R) &&&& Sp(1,1) &&&& Sp(2) &&\\\ &\nearrow &&\nwarrow &&\nearrow &&\nwarrow &&\nearrow &&&\\\ SL(2,\Bbb R)^2 &&&& SL(2,\Bbb C) &&&& SU(2)^2 &&&& \\\ &\nwarrow &&\nearrow &&\nwarrow &&\nearrow && && &\\\ && Sp(1,\Bbb R) &&&& Sp(1) &&&& &&\\\ &\nearrow &&\nwarrow &&\nearrow &&&& &&&\\\ GL(1,\Bbb R) &&&& U(1) &&&& &&&& \\\ \end{array}

Moreover in the latter we could, of course, make any of the substitutions

$$Sp(1)=SU(2),\quad Sp(1,\Bbb C) = SL(2,\Bbb C),$$

$$Sp(1,\Bbb R) = SL(2,\Bbb R) = SU(1,1),$$

$$SU^*(4)=SL(2,\Bbb H).$$

At this point, a good exercise or "proof" is to replace each complex group (resp. real form) by its Dynkin (resp. Satake or Vogan) diagram.

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I'd replace 'is "the same" as' with 'is the $Z_2$ quotient of'. –  Allen Knutson Nov 20 '11 at 14:41
@Allen: done. I took that opportunity to rephras a bit. –  Francois Ziegler Nov 20 '11 at 17:35
And for proofs, see Paul Garrett's vignette linked at mathoverflow.net/a/136703/19276 –  Francois Ziegler Jul 14 '13 at 20:59

Allen already said almost everything that there is to say...

I'll just add one more that he did not mention explicitly: $PSL(2,\mathbb R)\cong PSU(1,1) \cong SO(2,1)_+$.

This group is the group of Moebius transformations of the circle.
It's is also the group of conformal transformations of the disc, and also the isometry group of the hyperbolic plane. The above three names for this Lie group are related to three models of the hyperbolic plane: The upper half plane model, the Poincare disk model, and the upper sheet of hyperboloid model.

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Oops! I forgot about isomorphisms of "different" real forms, and have now added that to my answer. –  Allen Knutson Nov 19 '11 at 20:50
Also: $SL(2,{\mathbb R}) = Sp({\mathbb R}^2)$, the $A_1 = C_1$ part of your real-form answer. –  Allen Knutson Nov 19 '11 at 20:53