4
$\begingroup$

A (compact) torus is a Lie group isomorphic to the product of finitely many circles: $T^n = S^1 \times \cdots \times S^1$. Such groups are extremely important in Lie theory, Differential Geometry, Topological Groups, Harmonic Analysis etc. They can also be characterized as the compact connected Abelian Lie groups.

Of course, $S^1$ is the multiplicative group $U(1)$ of units in the field of complex numbers. In some contexts, it is interesting to consider a product of the form $S^3 \times \cdots \times S^3$, where $S^3$ as a Lie group is viewed as the multiplicative (non-Abelian) group $Sp(1)$ of units in the algebra of quaternions. (Also, $S^1$ and $S^3$ are only spheres that are Lie groups.) Is there a name for groups of this kind? I thought of 'quaternionic tori'...

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ Your title asks about $\mathrm{Sp}(1)$, but the body asks about $S^3$. What is the former? $\endgroup$
    – LSpice
    Commented Jun 27, 2016 at 18:48
  • $\begingroup$ I have edited the text to make it clear, OK? $\endgroup$
    – Claudio Gorodski
    Commented Jun 27, 2016 at 19:14
  • $\begingroup$ It's certainly OK with me—I wasn't being snide, but genuinely didn't know. Is this somehow related to the symplectic group? (Maybe it's the compact form?) $\endgroup$
    – LSpice
    Commented Jun 27, 2016 at 19:20
  • 2
    $\begingroup$ Sure, the (compact) symplectic group $Sp(n)$ is the group of linear transformations of $\mathbb H^n$, viewed as a right $\mathbb H$-module, that preserve the scalar product $x\cdot y = \Re\{x\bar y\}$. $\endgroup$
    – Claudio Gorodski
    Commented Jun 27, 2016 at 19:49
  • 1
    $\begingroup$ @LSpice, I gather that topologists (e.g.) by default do refer to the compact real form, which I'd denote something like $Sp^*(n)$ to distinguish it from the maximally-non-compact real form that automorphic formists take as default. The slightly obscure groups $Sp^*(p,q)$ with signatures (there is an inertia theorem...) are occasionally useful in contriving archimedean phenomena... $\endgroup$
    – paul garrett
    Commented Jun 27, 2016 at 22:47

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .