I've been using the fact that Spin(3,2) is isomorphic to Sp(4, R) for a while, but I've never seen a proof. Can anyone point me in the direction of a good reference?
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2 Answers
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Here is a coordinate-free version. Let $(V, \psi)$ be a 4-dimensional symplectic space over a field, and let $\omega \in (\wedge^2 V)^{\ast} = \wedge^2(V^{\ast})$ be the nonzero 2-form arising from $\psi$. Then on the 6-dimensional vector space $W = \wedge^2(V)$ the kernel of $\omega$ is a hyperplane $H$ of dimension 5, and on $W$ there is a natural non-degenerate quadratic form $q$ valued in the line $\wedge^4(V)$ via $q(w) = (1/2)(w \wedge w)$ for $w \in W$ (e.g., if $w = e_1 \wedge e_2 + e_3 \wedge e_4$ for a basis $\{e_i\}$ of $W$ then $q(w) = e_1 \wedge e_2 \wedge e_3 \wedge e_4$); the definition of $q$ uses base change from the $\mathbf{Z}_{(2)}$-flat case for $(V,\psi)$ if $2$ isn't a unit.
By computing in linear coordinates of $V$ that "standardize" $\psi$ we see that $q$ is a split quadratic form on $W$, and the action of ${\rm{SL}}(V)$ on $W$ clearly preserves $q$ while the action of its subgroup ${\rm{Sp}}(V,\psi)$ preserves $H$. Hence, the action on $H$ defines a map $${\rm{Sp}}(V,\psi) \rightarrow {\rm{O}}(q|_H),$$ so this lands inside ${\rm{SO}}(q|_H)$ and as such defines a homomorphism $${\rm{Sp}}_4 = {\rm{Sp}}(V,\psi) \rightarrow {\rm{SO}}(q|_H) = {\rm{SO}}_5.$$
This map kills the center $\mu_2$ inside ${\rm{Sp}}_4$, and thereby identifies ${\rm{Sp}}_4$ as the degree-2 "simply connected" central cover of ${\rm{SO}}_5$ (in the sense of algebraic groups). Hence, this uniquely lifts to an isomorphism of ${\rm{Sp}}_4$ onto ${\rm{Spin}}_5$.
A nice feature of this conceptual construction is that it kills two birds with one stone: if we don't restrict to $H$ and instead work with the entire 6-dimensional $W$ then a similar construction defines the isomorphism of ${\rm{SL}}_4 = {\rm{SL}}(V)$ onto ${\rm{Spin}}_6$.
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2$\begingroup$ To avoid confusion, I think you should perhaps write $\operatorname{SO}_{3,2}$ and $\operatorname{Spin}_{3,2}$ for $\operatorname{SO}_5$ and $\operatorname{Spin}_5$, respectively, and similarly, $\operatorname{Spin}_{3,3}$ for $\operatorname{Spin}_6$ in the last line. $\endgroup$ Commented Jul 15, 2013 at 10:20
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2$\begingroup$ @Jose: I am using standard notation among algebraists who study algebraic groups over an arbitrary field $k$, where non-degenerate quadratic forms are not classified by a "signature", but typically a lot more data. One writes ${\rm{SO}}(q)$ and ${\rm{Spin}}(q)$ for the connected semisimple $k$-groups associated to a non-degenerate quadratic space $(V,q)$, and when $q$ is split with $V$ of rank $n$ then they are denoted ${\rm{SO}}_n$ and ${\rm{Spin}}_n$. For $k=\mathbf{R}$, one writes ${\rm{SO}}(r,s)$ to denote ${\rm{SO}}(q)(k)$ for $q$ of signature $(r,s)$. So I prefer to leave it as is. $\endgroup$ Commented Jul 15, 2013 at 12:36
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$\begingroup$ I do agree it's good to learn how to avoid "choice-of-coordinates" silliness. But/and some of the intrinsification is better understood after one has done things an ad-hoc way. (I think we all know this, although it is not high-rep to admit it.) Then there is the field-specific issue of "classification" of (non-degenerate) quadratic forms over a particular field, which is surely not solvable in general... But, over any algebraically closed field of char not 2 (such as complex...) it's just dimension, and a slightly non-trivial result: over $\mathbb R$, it's "signature". Not "just algebra". $\endgroup$ Commented Jul 15, 2013 at 23:55
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$\begingroup$ @paul: even in char. 2 everything works perfectly well based on char-free definitions, as it must if there is to be a reasonable theory over $\mathbf{Z}$ for linking it up with Chevalley groups. The "correct" dichotomy for SO and Spin groups is not char. 2 versus char. $\ne 2$ but rather $\dim V$ being even or odd (i.e., B versus D). $\endgroup$ Commented Jul 16, 2013 at 1:19
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1$\begingroup$ @paul: Via algebraic-group techniques (Bruhat-Tits, cohomology, group schemes), the char-free classification over non-arch. local fields $k$ can be done with a dichotomy based on the parity of $n$. The cases $n \le 2$ have one cohomological invariant (and the relevant algebraic groups are not semisimple). If $n \ge 5$ then isotropicity holds for the quadratic space because absolutely simple semisimple $k$-groups not of type A must be isotropic (due to Bruhat-Tits theory). So by Witt cancellation, one is brought to $n=3, 4$. Exceptional isomorphisms yield two concrete cohomological invariants. $\endgroup$ Commented Jul 17, 2013 at 14:27
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By coincidence, I put up a protraced working-out of several of these examples today, at http://www.math.umn.edu/~garrett/m/v/sporadic_isogenies.pdf
Edit: Over algebraically closed fields (especially in char not $2$, which I want to not think about), these sporadic isogenies are easy to write down. In principle, but I think not in practice for most of us, to see what happens over not algebraically closed fields is a question of "Galois cohomology", as in Weil's "Algebras and classical groups" paper. For me, it's much easier to use a few coordinates. E.g., although $SU(4)\rightarrow SO(6)$ and $SU(2,2)\rightarrow SO(4,2)$, apparently $SU(3,1)$ does not map to $SO(p,q)$ with $p+q=6$. Meanwhile, $SL_2(\mathbb H)\rightarrow SO(5,1)$. Seems weird to me.
answered Jul 14, 2013 at 20:37
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3$\begingroup$ For completeness: Paul lets $Sp_4(R)$ act on the 16-dimensional matrix algebra $M_4(R)$ by conjugation. The usual bilinear form $tr(xy)$ is invariant. The Lie algebra $RI_4+sp_4(R)$ is an invariant subspace on which this form is nondegenerate ($sp_4(R)$ being the 10-dimensional Lie algebra), so its orthogonal is also nondegenerate, it is thus (16-(10+1))-dimensional, i.e. 5-dimensional. Explicit coordinates show the signature is (3,2). This defines a 2-to-1 map from $Sp_4(R)$ onto $SO(3,2)$. $\endgroup$– YCorCommented Jul 14, 2013 at 21:57
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$\begingroup$ @YvesCornulier, thx for filling in! :) $\endgroup$ Commented Jul 15, 2013 at 23:51
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2$\begingroup$ @paulgarrett: $SU(3,1)$ that you ask about maps to $SO^*(6)$ (a.k.a. $SO(3,\mathbb H)$), according to Helgason (§X.6.4) or Besse (Einstein Manifolds, p. 201). You may need to add a third section "Over $\mathbb H$"! $\endgroup$ Commented Jul 23, 2013 at 4:30
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1$\begingroup$ Hi, I hope you find this comment (for your 10 years ago answer). Are the exceptional isogenies also hold over nonarchimedean fields? It seems like most of the arguments of proofs does not actively use archimedeaness of $\mathbb{R}$ and $\mathbb{C}$, so we may replace $\mathbb{C}/\mathbb{R}$ with quadratic extensions of local fields $E/F$ - am I right? $\endgroup$ Commented May 7, 2023 at 18:00
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1$\begingroup$ @SeewooLee, yes, using quadratic extensions of the base field, and/or the unique quaternion division algebra of $p$-adic local fields, gives similar results. But/and instead of "signature", a slightly different classification is needed... $\endgroup$ Commented May 7, 2023 at 21:15