In mathematics, a quaternion algebra over a field F is a central simple algebra A over F[1][2] that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, 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 Hamilton's quaternions to an arbitrary base field. The Hamilton quaternions are a quaternion algebra (in the above sense) over , and indeed the only one over apart from the 2 × 2 real matrix algebra, up to isomorphism. When , then the biquaternions form the quaternion algebra over F.
^See Pierce. Associative algebras. Springer. Lemma at page 14.
^See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2.
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space. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by H . {\displaystyle \mathbb {H} .} Quaternions are not a...
In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form q = a + b i + c...
In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus...
These relations, discovered by W. R. Hamilton, also generate the quaternions as an algebra over the real numbers. Another presentation of Q8 is Q 8 = ⟨ a...
of versors with quaternion multiplication forms a group, and the set of versors is a 3-sphere in the 4-dimensional quaternionalgebra. Hamilton denoted...
simple algebra A then dimF S divides dimF A. Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternionalgebra; in fact...
structure of a distinguished subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems...
algebra of quaternions and the non-commutative Hamilton product. In particular, when the Hamilton product of two vectors (that is, pure quaternions with...
the dual quaternionalgebra to 2D geometry. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which we...
"geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quaternionalgebra. Adding...
chooses a suitable Hurwitz quaternion order Q H u r {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }} in the quaternionalgebra, Γ(I) is then the group of...
Unit quaternions, known as versors, provide a convenient mathematical notation for representing spatial orientations and rotations of elements in three...
algebra of (real) quaternions. In other words, the biquaternions are just the complexification of the quaternions. Viewed as a complex algebra, the biquaternions...
Hamiltonian quaternions. Hamilton's innovation consisted of expressing quaternions as an algebra over R. The formulae for the multiplication of quaternions are...
with quadratic form x2 − y2, quaternions and split-quaternions, octonions and split-octonions. Every composition algebra has an associated bilinear form...
than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition...
{\displaystyle (2,3,8)} group does not have a realization in terms of a quaternionalgebra, but the ( 3 , 3 , 4 ) {\displaystyle (3,3,4)} group does. Under the...
expressions in terms of quaternionalgebra, all using exponentiation. Real powers of a quaternion are defined in terms of the quaternion exponential function...
there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4). Wedderburn's...
quaternionalgebras. The center of that ring is R × R {\displaystyle \mathbb {R} \times \mathbb {R} } , and hence it has the structure of an algebra over...
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd...
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