Quaternion algebra

In

), i.e. for a suitable field extension K of F, ${\displaystyle A\otimes _{F}K}$ is

isomorphic

to the 2 × 2 matrix algebra over K.

The notion of a quaternion algebra can be seen as a generalization of Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over ${\displaystyle F=\mathbb {R} }$, and indeed the only one over ${\displaystyle \mathbb {R} }$ apart from the 2 × 2 real matrix algebra, up to isomorphism. When ${\displaystyle F=\mathbb {C} }$, then the biquaternions form the quaternion algebra over F.

Structure

Quaternion algebra here means something more general than the algebra of Hamilton's quaternions. When the coefficient field F does not have characteristic 2, every quaternion algebra over F can be described as a 4-dimensional F-vector space with basis ${\displaystyle \{1,i,j,k\}}$, with the following multiplication rules:

${\displaystyle i^{2}=a}$

${\displaystyle j^{2}=b}$

${\displaystyle ij=k}$

${\displaystyle ji=-k}$

where a and b are any given nonzero elements of F. From these rules we get:

${\displaystyle k^{2}=ijij=-iijj=-ab}$

The classical instances where ${\displaystyle F=\mathbb {R} }$ are Hamilton's quaternions (a = b = −1) and split-quaternions (a = −1, b = +1). In split-quaternions, ${\displaystyle k^{2}=+1}$ and ${\displaystyle jk=-i}$, differing from Hamilton's equations.

The algebra defined in this way is denoted (a,b)_F or simply (a,b).^[3] When F has characteristic 2, a different explicit description in terms of a basis of 4 elements is also possible, but in any event the definition of a quaternion algebra over F as a 4-dimensional central simple algebra over F applies uniformly in all characteristics.

A quaternion algebra (a,b)_F is either a

The norm form

${\displaystyle N(t+xi+yj+zk)=t^{2}-ax^{2}-by^{2}+abz^{2}\ }$

defines a structure of

C(a,b) defined by

${\displaystyle ax^{2}+by^{2}=z^{2}\ }$

has a point (x,y,z) with coordinates in F in the split case.[5]

Application

Quaternion algebras are applied in

.

Classification

It is a theorem of Frobenius that there are only two real quaternion algebras: 2 × 2 matrices over the reals and Hamilton's real quaternions.

In a similar way, over any local field F there are exactly two quaternion algebras: the 2 × 2 matrices over F and a division algebra. But the quaternion division algebra over a local field is usually not Hamilton's quaternions over the field. For example, over the p-adic numbers Hamilton's quaternions are a division algebra only when p is 2. For odd prime p, the p-adic Hamilton quaternions are isomorphic to the 2 × 2 matrices over the p-adics. To see the p-adic Hamilton quaternions are not a division algebra for odd prime p, observe that the congruence x² + y² = −1 mod p is solvable and therefore by Hensel's lemma — here is where p being odd is needed — the equation

x² + y² = −1

is solvable in the p-adic numbers. Therefore the quaternion

xi + yj + k

has norm 0 and hence doesn't have a multiplicative inverse.

One way to classify the F-algebra isomorphism classes of all quaternion algebras for a given field F is to use the one-to-one correspondence between isomorphism classes of quaternion algebras over F and isomorphism classes of their norm forms.

To every quaternion algebra A, one can associate a quadratic form N (called the norm form) on A such that

${\displaystyle N(xy)=N(x)N(y)}$

for all x and y in A. It turns out that the possible norm forms for quaternion F-algebras are exactly the Pfister 2-forms.

Quaternion algebras over the rational numbers

Quaternion algebras over the rational numbers have an arithmetic theory similar to, but more complicated than, that of quadratic extensions of ${\displaystyle \mathbb {Q} }$.

Let ${\displaystyle B}$ be a quaternion algebra over ${\displaystyle \mathbb {Q} }$ and let ${\displaystyle \nu }$ be a

of ${\displaystyle \mathbb {Q} }$, with completion ${\displaystyle \mathbb {Q} _{\nu }}$ (so it is either the p-adic numbers ${\displaystyle \mathbb {Q} _{p}}$ for some prime p or the real numbers ${\displaystyle \mathbb {R} }$). Define ${\displaystyle B_{\nu }:=\mathbb {Q} _{\nu }\otimes _{\mathbb {Q} }B}$, which is a quaternion algebra over ${\displaystyle \mathbb {Q} _{\nu }}$. So there are two choices for ${\displaystyle B_{\nu }}$: the 2 × 2 matrices over ${\displaystyle \mathbb {Q} _{\nu }}$ or a division algebra.

We say that ${\displaystyle B}$ is split (or unramified) at ${\displaystyle \nu }$ if ${\displaystyle B_{\nu }}$ is isomorphic to the 2 × 2 matrices over ${\displaystyle \mathbb {Q} _{\nu }}$. We say that B is non-split (or ramified) at ${\displaystyle \nu }$ if ${\displaystyle B_{\nu }}$ is the quaternion division algebra over ${\displaystyle \mathbb {Q} _{\nu }}$. For example, the rational Hamilton quaternions is non-split at 2 and at ${\displaystyle \infty }$ and split at all odd primes. The rational 2 × 2 matrices are split at all places.

A quaternion algebra over the rationals which splits at ${\displaystyle \infty }$ is analogous to a

and one which is non-split at ${\displaystyle \infty }$ is analogous to an in an order of a rational quaternion algebra: it is infinite if the quaternion algebra splits at ${\displaystyle \infty }$^{[citation needed]} and it is finite otherwise^{[citation needed]}, just as the unit group of an order in a quadratic ring is infinite in the real quadratic case and finite otherwise.

The number of places where a quaternion algebra over the rationals ramifies is always even, and this is equivalent to the

over the rationals. Moreover, the places where B ramifies determines B up to isomorphism as an algebra. (In other words, non-isomorphic quaternion algebras over the rationals do not share the same set of ramified places.) The product of the primes at which B ramifies is called the discriminant of B.

See also

• Composition algebra
• Cyclic algebra
• Octonion algebra
• Hurwitz quaternion order
• Hurwitz quaternion

Notes

References

• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge:
  Zbl 1137.12001
  .
• Zbl 1068.11023
  .

Further reading

• Knus, Max-Albert;
  Zbl 0955.16001
  .
• Maclachlan, Colin; Ried, Alan W. (2003). The Arithmetic of Hyperbolic 3-Manifolds. New York: Springer-Verlag.
  MR 1937957
  . See chapter 2 (Quaternion Algebras I) and chapter 7 (Quaternion Algebras II).
• Chisholm, Hugh, ed. (1911). "Algebra" . Encyclopædia Britannica (11th ed.). Cambridge University Press. (See section on quaternions.)
• Quaternion algebra at Encyclopedia of Mathematics.