Quaternion Algebra

In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes the matrix algebra by extending scalars (=tensoring with a field extension), i.e. for a suitable field extension K of F, is isomorphic to the 2×2 matrix algebra over K.

The notion of a quaternion algebra can be seen as a generalization of the Hamilton quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over (the real number field), and indeed the only one over ℝ apart from the 2×2 real matrix algebra, up to isomorphism.

Read more about Quaternion Algebra:  Structure, Application, Classification, Quaternion Algebras Over The Rational Numbers

Other articles related to "quaternion algebra, algebra, quaternion algebras, quaternions":

Bolza Surface - Quaternion Algebra
... Following MacLachlan and Reid, the quaternion algebra can be taken to be the algebra over generated as an associative algebra by generators i,j and relations with an appropriate choice of an order ...
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 ... Let be a quaternion algebra over and let be a place of, with completion (so it is either the p-adic numbers for some prime p or the real numbers ) ... Define, which is a quaternion algebra over ...
(2,3,7) Triangle Group - Constructions - Quaternion Algebra
... The (2,3,7) triangle group admits a presentation in terms of the group of quaternions of norm 1 in a suitable order in a quaternion algebra ... group is the quotient of the group of quaternions by its center ±1 ... group is a subgroup of the group of norm 1 elements in the quaternion algebra generated as an associative algebra by the pair of generators i,j and relations i2 = j2 = η, ij = −ji ...
Arithmetic Hyperbolic 3-manifold - Trace Fields
... The quaternion algebra of a Kleinian group Γ is the subring of M(2, C) generated by the trace field and the elements of Γ, and is a 4-dimensional simple algebra over the trace field if ... The invariant quaternion algebra of Γ is the quaternion algebra of Γ(2) ... The quaternion algebra may be split, in other words a matrix algebra this happens whenever Γ is non-elementary and has a parabolic element, in particular if it is a ...
Weil Conjectures - Weil Cohomology
... The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2 dimensional vector space over the ... However a quaternion algebra over the rationals cannot act on a 2 dimensional vector space over the rationals ... coefficient field being the reals or the p-adic numbers, because the quaternion algebra is still a division algebra over these fields ...

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