Structure
Quaternion algebra here means something more general than the algebra of Hamilton 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, with the following multiplication rules:
- i2 = a
- j2 = b
- ij = k
- ji = −k
where a and b are any given nonzero elements of F. A short calculation shows k2 = −ab. (The Hamilton quaternions are the case where and a = b = −1.) The algebra defined in this way is denoted (a,b)F or simply (a,b). 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 division algebra or isomorphic to the matrix algebra of 2×2 matrices over F: the latter case is termed split. The norm form
defines a structure of division algebra if and only if the norm is an anisotropic quadratic form, that is, zero only on the zero element. The conic C(a,b) defined by
has a point (x,y,z) with coordinates in F in the split case.
Read more about this topic: Quaternion Algebra
Famous quotes containing the word structure:
“Agnosticism is a perfectly respectable and tenable philosophical position; it is not dogmatic and makes no pronouncements about the ultimate truths of the universe. It remains open to evidence and persuasion; lacking faith, it nevertheless does not deride faith. Atheism, on the other hand, is as unyielding and dogmatic about religious belief as true believers are about heathens. It tries to use reason to demolish a structure that is not built upon reason.”
—Sydney J. Harris (1917–1986)
“There is no such thing as a language, not if a language is anything like what many philosophers and linguists have supposed. There is therefore no such thing to be learned, mastered, or born with. We must give up the idea of a clearly defined shared structure which language-users acquire and then apply to cases.”
—Donald Davidson (b. 1917)
“Each structure and institution here was so primitive that you could at once refer it to its source; but our buildings commonly suggest neither their origin nor their purpose.”
—Henry David Thoreau (1817–1862)