quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra...

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...

of versors with quaternion multiplication forms a group, and the set of versors is a 3-sphere in the 4-dimensional quaternion algebra. Hamilton denoted...

In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of odd...

In mathematics, the dual quaternions are an 8-dimensional real algebra isomorphic to the tensor product of the quaternions and the dual numbers. Thus...

In abstract algebra, the split-quaternions or coquaternions form an algebraic structure introduced by James Cockle in 1849 under the latter name. They...

Eugene Dickson. The algebras produced by this process are known as Cayley–Dickson algebras, for example complex numbers, quaternions, and octonions. These...

quaternion algebra to 2D geometry are discussed. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which...

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...

chooses a suitable Hurwitz quaternion order Q H u r {\displaystyle {\mathcal {Q}}_{\mathrm {Hur} }} in the quaternion algebra, Γ(I) is then the group of...

quaternions of norm 1 in a suitable order in a quaternion algebra. More specifically, the triangle group is the quotient of the group of quaternions by...

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...

functional analysis, Banach *-algebras and C*-algebras are special types of Banach algebras with involutions. In a quaternion algebra, an (anti-)involution is...

simple algebra A then dimF S divides dimF A. Every 4-dimensional central simple algebra over a field F is isomorphic to a quaternion algebra; in fact...

structure of a distinguished subspace. As K-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems...

Hamiltonian quaternions. Hamilton's innovation consisted of expressing quaternions as an algebra over R. The formulae for the multiplication of quaternions are...

"geometric algebras"). Clifford defined the Clifford algebra and its product as a unification of the Grassmann algebra and Hamilton's quaternion algebra. Adding...

there are three such algebras: the reals themselves (dimension 1), the field of complex numbers (dimension 2), and the quaternions (dimension 4). Wedderburn's...

than the modern approach, which emphasizes quaternions' algebraic properties. Mathematically, quaternions discussed differ from the modern definition...

Cartan–Brauer–Hua theorem. A cyclic algebra, introduced by L. E. Dickson, is a generalization of a quaternion algebra. A semisimple module is a direct sum...

product depends on selection of a γ from k. Given q and Q from a quaternion algebra over k, the octonion is written q + Qe. Another octonion may be written...

are a special class of Kleinian groups constructed using orders in quaternion algebras. They are particular instances of arithmetic groups. An arithmetic...

expressions in terms of quaternion algebra, all using exponentiation. Real powers of a quaternion are defined in terms of the quaternion exponential function...

with quadratic form x2 − y2, quaternions and split-quaternions, octonions and split-octonions. Every composition algebra has an associated bilinear form...

{\displaystyle \mathbf {H} \oplus \mathbf {H} } , the direct sum of two quaternion algebras – van der Waerden 1985 Split-octonions Clifford, W. K. (1873). "Preliminary...

{\displaystyle (2,3,8)} group does not have a realization in terms of a quaternion algebra, but the ( 3 , 3 , 4 ) {\displaystyle (3,3,4)} group does. Under the...

quaternion algebras. The center of that ring is R × R {\displaystyle \mathbb {R} \times \mathbb {R} } , and hence it has the structure of an algebra over...

quaternion algebra over R, which was an experimental algebra before the adoption of Minkowski space for special relativity. More classes of algebras:...