42 votes

Is it possible to "get" quaternions without specifically postulating them?

Let $S_2$ denote the set of all integers that are representable as a sum of two squares. Then $S_2$ is closed under multiplication, because of the identity $$(a^2 + b^2)(\alpha^2 + \beta^2) = (a\...
Timothy Chow's user avatar
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31 votes

Is it possible to "get" quaternions without specifically postulating them?

Quaternions—in the sense of objects that obey the relations of the quaternion group—arise automatically in Lie Theory. The most elementary noncommutative continuous group is $SO(3)$, and if you want ...
Buzz's user avatar
  • 1,360
26 votes
Accepted

Why is the automorphism group of the octonions $G_2$ instead of $SO(7)$

The quaternions are generated by any two imaginary elements $x$ and $y$ that are orthonormal, i.e., $\bigl(1,\, x,\, y,\, xy\bigr)$ is an orthonormal basis of the quaternions. Moreover, the ...
Robert Bryant's user avatar
18 votes
Accepted

How many distinct quaternions have a given prime norm $p$?

Depends on what you mean by "quaternions" and "distinct". If you mean quaternions $a + b{\bf i} + c{\bf j} + d{\bf k}$, with $a,b,c,d$ all integers, then you're asking to count solutions of $a^2+b^2+c^...
Noam D. Elkies's user avatar
15 votes
Accepted

Automorphisms and isometries of the quaternions

In other words, what you are asking is whether every $f\colon\mathbb{H}\to\mathbb{H}$ in $\mathit{SO}_4$ takes the form $x\mapsto \bar u x v$ where $u,v$ are unit quaternions (the connection with your ...
Gro-Tsen's user avatar
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15 votes

Is it possible to "get" quaternions without specifically postulating them?

The group algebra over $\mathbb{R}$ of every finite group is a direct sum of simple algebras. If you decide to compute this decomposition for some small groups, then you will discover that for the 8-...
Richard Stanley's user avatar
13 votes

Automorphisms and isometries of the quaternions

This is a standard result in representation theory: Let the quadratic form on $\mathbb{H}$ be $\langle x,x\rangle> = x\bar x$ (which is positive definite). (Note that I am considering $\mathbb{...
Robert Bryant's user avatar
13 votes
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Representing a number as a sum of four squares and factorization

I think the reason is that there are $p+1$ distinct ways of writing an odd prime $p$ as the sum of four squares up to sign changes; these correspond to the same number of elements of the Lipschitz ...
John Voight's user avatar
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11 votes
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Maximal compact subgroup of $\mathrm{SL}(2,\mathbb{H})$

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SO{SO}\DeclareMathOperator\Sp{Sp}$YCor's comment contains the essential idea needed for the proof, but maybe a few more ...
Robert Bryant's user avatar
11 votes

Diagonalizing quaternionic unitary matrices

This follows from the general fact that, in a compact connected Lie group, every element is conjugate to an element in a maximal torus (and all maximal tori are conjugate). This result is proved in ...
Robert Bryant's user avatar
11 votes
Accepted

Covolumes of unit groups of division algebras

First, you need to avoid definite quaternion algebras over $\mathbb{Q}$: in this case, the unit groups are finite, so the index cannot grow with $N$. With that out of the way, your algebra $D$ ...
John Voight's user avatar
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11 votes
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Lifting of map from $S^3$ to itself

No. The reason this holds in the complex case (any degree $0$ map $S^1 \to S^1$ factors through $\exp: \mathbb{R}i\to S^1$) is that in that case the exponential map is a covering map, in particular a ...
Achim Krause's user avatar
  • 8,584
10 votes

Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$

The theta function associated to your quadratic form (here $\theta(\tau)\theta(3\tau)^2$) belongs to the modular form space $M_{3/2}(\Gamma_0(12))$, and you are in luck because there are no cusp forms,...
Henri Cohen's user avatar
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10 votes

Principled construction of the quaternions

$\def\RR{\mathbb{R}}$This is an answer to flesh out what I wrote in comments. Let $V$ be a real vector space with positive definite inner product. For a unit vector $\vec{u} \in V$, let $s_{\vec{u}}$ ...
David E Speyer's user avatar
10 votes
Accepted

Is there a definition of $\log(x)$ for quaternion/octonion $x$?

In characteristic zero, you can always just use the power series to define $\log(1+x)$ with a radius of convergence of $1$, since your algebras are at least power associative. It will satisfy $\exp(\...
Dave Benson's user avatar
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9 votes
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Circle Action on Quaternionic Projective Space

$\mathbb{HP}^n\cong \mathrm{Sp}(n+1)/(\mathrm{Sp}(n)\times \mathrm{Sp}(1))$ is a symmetric space, so every one-parameter subgroup of $\mathrm{Sp}(n+1)$ acts on it. As was noted in the comments, such ...
Bertram Arnold's user avatar
9 votes

Automorphic quotient for quaternion algebras

Weil's "Adeles and algebraic groups" proves this for general division algebras. It is labelled "Fujisaki's Lemma", and is a natural division-algebra generalization of the analogue for number fields ...
paul garrett's user avatar
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9 votes

Inner forms of $GL(2)$

If you have a group $G$ over $k$ such that $G_{\overline{k}}$ is isomorphic to $GL_2$ and, and each element of $\operatorname{Gal}(\overline{k}|k)$ acts on this $GL_2$ by the standard action times ...
Will Sawin's user avatar
  • 135k
9 votes

Is it possible to "get" quaternions without specifically postulating them?

They can be deduced via Dirac Belt Trick, see Understanding Quaternions and the Dirac Belt Trick by Mark Staley. This Trick is also known as Plate Trick.
Alexey Ustinov's user avatar
8 votes
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Do these definitions of integrable quaternionic structure agree?

These two 'definitions' do not agree. Also, you should be careful about your choice of sources. Most differential geometers use the terminology 'almost quaternionic' to mean that the structure group ...
Robert Bryant's user avatar
8 votes
Accepted

The Hilbert symbols of quaternion algebras over a totally real field

Yes. First choose $a$. You can take any $a$ such that $K = k(\sqrt{a})$ is a splitting field of $B$, so by Grunwald--Wang (or elementary congruences and sign conditions) you can enforce $\sigma(a)>...
Aurel's user avatar
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8 votes

Ramification of quaternion algebras over $\mathbb Q$

I'm not sure how to define "classically" here (Hensel's $p$-adics hail from around 1890s), but the Hilbert symbol gives a way to do this without $p$-adics. (See Chapter 12 of my book, http:/...
John Voight's user avatar
  • 2,929
8 votes

How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?

A quaternion algebra with center $F$ that is not a division ring is isomorphic to ${\rm M}_2(F)$. See Theorem 4.21 and Corollary 4.24 here. Let's show when $F = \mathbf F_p$ (field of order $p$) for $...
KConrad's user avatar
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7 votes
Accepted

A symmetric-like group and the quaternion group $Q_8$

To simplify typing I will call these groups $G_n$ rather than $\tilde{S}_n$. Note that $G_2$ is just the free product of two groups of order $2$, so is the infinite dihedral group, and I conjecture ...
Derek Holt's user avatar
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7 votes
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Exceptional isomorphism with Spin(6,2)?

To put the problem to a rest, I add my comment as an answer which is, in a nutshell, $Spin(6,2)\cong Spin(4,\mathbb H)$. The existence of this isomorphism follows from the isomorphism of the Satake ...
Friedrich Knop's user avatar
7 votes
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Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$

One of the beautiful (and sometimes flummoxing) aspects of the theory of ternary quadratic forms is that you can find answers and questions coming from many different points of view! Using modular ...
John Voight's user avatar
  • 2,929
7 votes
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Idempotent functions on Sp(1)

As Venkataramana says, this is a natural candidate for the Peter-Weyl theorem: Let $G$ be a compact group. Let $\{ V_i \}_{i \in I}$ be the set of isomorphism classes of irreducible complex ...
David E Speyer's user avatar
7 votes
Accepted

Generalizing contour integration to quaternions and bicomplex numbers

I will address the commutative case. For every finite-dimensional commutative associative unital $\mathbb{C}$-algebra $\mathcal{A}$ there exists a complete analogue of the theory of Complex Analysis ...
M.G.'s user avatar
  • 6,683
7 votes

How to prove that a quaternion algebra over ℤₚ is isomorphic to Mat₂(ℤₚ) for p prime?

At the risk of being redundant and repeating earlier answers, let me mention that this is explicitly contained in the book "Elementary number theory, group theory and Ramanujan graphs" by ...
Tom De Medts's user avatar
  • 6,494
6 votes
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Computing Tamagawa number of torus in Quaternion algebra

Here are some more details. As John Voight said, the quaternion algebra is kind of irrelevant here. If $\gamma$ is a regular semisimple element, then its centralizer is a torus ${\mathbf T}$ over ${\...
Marty's user avatar
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