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7 added 168 characters in body

Isomorphisms between complex and

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 groups form isomorphisms. It is the analog of summarizing the complex isogenies (like already explained by Allen) by the statement that the sequence$SL(2,\Bbb$SO(3,\Bbb C) \cong O_o(3,1)$to SO(4,\Bbb C) are often left out of \to SO(5,\Bbb C)\to SO(6,\Bbb C)(where arrows denote the discussion. If you include them, you can summarize a lot by saying: up obvious inclusions) is the$Z_2$quotient of Sp(1,\Bbb C) \to SL(2,\Bbb C)^2 \to Sp(2,\Bbb C)\to SL(4,\Bbb C).$\pi_0$and$\pi_1$, What we get for real forms (and let's not forget, complex groups regarded as real) is that the diagram is the same (up to$\pi_0$and$\pi_1$) as \Bbb GL(1,\Bbb R) &&&& U(1) &&&& &&&& \\ These are all "real forms" of the statement (already explained by Allen) thatSO(3,\Bbb C) \to SO(4,\Bbb C) \to SO(5,\Bbb C)\to SO(6,\Bbb C)is "the same" as Sp(1,\Bbb C) \to SL(2,\Bbb C)^2 \to Sp(2,\Bbb C)\to SL(4,\Bbb C). Moreover of course, in the second diagram latter we could, of course, make any of the substitutions At this point, a good exercise or "proof" is to replace each complex group (resp. real form) by its Dynkin (resp. paintedSatake or Vogan) Dynkin diagram. 6 deleted 13 characters in body Isomorphisms between complex and real groups (like$SL(2,\Bbb C) \cong O_o(3,1)$) are often left out of the discussion. If you include them, you can summarize a lot by saying: up to isogeny$\pi_0$and connected components,$\pi_1$, 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 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 &&&& &&&\\ \Bbb R &&&& U(1) &&&& &&&& \\ \end{array} These are all "real forms" of the statement (already explained by Allen) that $$SO(3,\Bbb C) \to SO(4,\Bbb C) \to SO(5,\Bbb C)\to SO(6,\Bbb C)$$ is "the same" as $$Sp(1,\Bbb C) \to SL(2,\Bbb C)^2 \to Sp(2,\Bbb C)\to SL(4,\Bbb C).$$ Moreover of course, in the second diagram we could 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 is to replace each complex group (resp. real form) by its (resp. painted) Dynkin diagram. 5 added 100 characters in body Isomorphisms between complex and real groups (like$SL(2,C) SL(2,\Bbb C) \cong O_o(3,1)\$) are often left out of the discussion. If you include them, you can summarize a lot by saying: up to isogeny and connected components, 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 as

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

These are all "real forms" of the statement (already explained by Allen) that $$SO(3,CSO(3,\Bbb C) \to SO(4,CSO(4,\Bbb C) \to SO(5,C)\to SO(6,CSO(5,\Bbb C)\to SO(6,\Bbb C)$$ is "the same" as $$Sp(1,CSp(1,\Bbb C) \to SL(2,C)^2 SL(2,\Bbb C)^2 \to Sp(2,C)\to SL(4,C)Sp(2,\Bbb C)\to SL(4,\Bbb C).$$ Moreover of course, in the second diagram we could make any of the substitutions

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

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

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

At this point, a good exercise is to replace each complex group (resp. real form) by its (resp. painted) Dynkin diagram.