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\...
- 82.6k
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 ...
- 1,422
27
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 ...
- 108k
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^...
- 79.9k
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 ...
- 32.5k
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-...
- 50.8k
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{...
- 108k
13
votes
Accepted
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 ...
- 3,009
11
votes
Accepted
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 ...
- 108k
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 ...
- 108k
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 $D$ ...
- 3,009
11
votes
Accepted
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 ...
- 10.8k
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,...
- 13.1k
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 $s_{\vec{u}}$ ...
- 156k
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(\...
- 16.2k
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 ...
- 23k
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 ...
- 148k
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.
- 12.3k
8
votes
Accepted
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 ...
- 108k
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)>...
- 5,382
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:/...
- 3,009
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 $...
- 50.6k
7
votes
Accepted
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 ...
- 3,009
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 ...
- 37.4k
7
votes
Accepted
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 ...
- 156k
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 ...
- 7,127
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 ...
- 6,614
6
votes
Accepted
About some property of automorphism of octonions
Seen as a map of $8$-dimensional Euclidean vector spaces, $f$ is obviously (special) orthogonal, so we can find an orthonormal basis on which is has a block diagonal form of $2\times 2$ rotation ...
- 32.5k
6
votes
Accepted
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 ${\...
- 13.3k
6
votes
Automorphic quotient for quaternion algebras
What does ``automorphic quotient'' mean?
Does my book (http://quatalg.org), Main Theorem 38.4.3 (a theorem of Hey) answer your question? A quaternion algebra $B$ over a field $F$ is either ...
- 3,009
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