# Tag for quaternion algebras?

A quaternion algebra is a generalization of the classical Hamiltonian quaternions. It is a $4$-dimensional algebra over a field $F$ with basis $1,i,j,ij$ subject to the relations $$i^2 = a, \quad j^2 = b, \quad ji = -ij$$ for some $a,b \in F^\times$. (This is the definition for $\operatorname{char}(F) \neq 2$, anyway.)

There are quite a few questions on quaternion algebras, such as

Brauer Group of $\mathbb{Q}_2$

Proof that Quaternion Algebras are simple

Equivalent conditions of quaternion matrix algebra

Quaternion algebras over finite fields up to isomorphism

Quaternion algebra of characteristic 2?

Isomorphism of quaternion algebra

among others. Some of these posts are tagged with (quaternions), but according to its description the (quaternions) tag only seems to apply to the Hamilton quaternions:

For questions about the quaternions: a noncommutative four dimensional division algebra over the real numbers.

(The full description is here.) At the moment, there are 648 questions with a (quaternions) tag. Most probably refer to the Hamiltonian quaternions, but some (such as a couple of the above links) are about general quaternion algebras.

I see three possible courses of action:

(1) Expand the scope of the (quaternions) tag to include quaternion algebras.

(2) Create a new (quaternion-algebras) tag separate from the existing (quaternions) tag.

(3) Do nothing, if one thinks that there are not enough questions on this topic to merit a new tag and that quaternion algebras are too different from the Hamiltonian quaternions for an expansion of the existing tag description.

Which do you think is the best choice? I am leaning toward (2) because I get the sense that many questions with a (quaternions) tag are about interpreting Hamiltonian quaternions as rotations, while most questions about quaternion algebras relate to number theory.

• Props for asking! I would also consider having such questions under division-algebras. I do know that not all quaternion algebras are division rings. And division-algebras is a small tag itself (there would be more questions carrying that tag, but I keep purging questions about junior high "algebra" questions about "division"). – Jyrki Lahtonen Jan 13 '17 at 6:05
• But, I don't know what is best. – Jyrki Lahtonen Jan 13 '17 at 6:06
• @JyrkiLahtonen Thanks for pointing out the division algebras tag. I'd be wary of using it for quaternion algebras for the reason you state, as it might cause misunderstandings. There is a slightly larger (algebras) tag, which would work, I guess. I'm also surprised that there's no central simple algebras tag. – Viktor Vaughn Jan 13 '17 at 6:20
• I agree that central simple algebras would give a wider umbrella as a tag. And less likely to be used accidentally :-) Anyway, I have some doubts because division-algebras has not been a particularly successful tag. Waiting for more users to chime in! – Jyrki Lahtonen Jan 13 '17 at 6:28
• Perhaps it would make sense to post your suggestion to do (2) also as an answer. Upvotes on this question could mean: +1 It is good thing to bring this up. (After all, many users will not read your question until the last paragraph.) But they might also mean: +1 I agree with the suggestion made at the end of the question. (Just a suggestion - which should I probably made sooner, seeing that the existing answer already is at score 7.) – Martin Sleziak Jan 16 '17 at 4:10
• @MartinSleziak Yes, you're right. It seems a bit late in the game to add it as an answer now, but I'll add one. (And one for (3) as well.) – Viktor Vaughn Jan 16 '17 at 22:45

I agree with course of action (1): Expand the scope of to include quaternion algebras.

Partially because they are closely related, partially because this matches existing usage patterns, and partially because there are not enough questions of this type to merit a new tag.

Completely as a side note, the preface to Hamilton's lectures on quaternions is well worth reading.

It's my opinion that the only reason modern readers ever find Hamilton's theoretical approach to quaternions difficult, is that they haven't really understood his theory of algebraic numbers first. That set of ideas is explained in his preface.

(The danger is to skim over the text because the equations look familiar. They might look familiar, but it's his interpretation that's unique.)

• Ah, I meant (3) to really be do nothing, so I've edited to that effect. – Viktor Vaughn Jan 13 '17 at 6:17

I agree with course of action (2): Create a new (quaternion-algebras) tag separate from the existing tag.

Expanding the existing tag is not suitable because most existing questions with this tag refer to the Hamiltonian quaternions, and often deal with interpreting Hamiltonian quaternions as rotations.

I agree with course of action (3): Do nothing.

There are not enough questions on this topic to merit a new tag and quaternion algebras are too different from the Hamiltonian quaternions for an expansion of the description for the tag.