In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form
where the squares of i, j, and k are +1 and distinct elements of {i, j, k} multiply with the anti-commutative property.
The four-dimensional algebra of hyperbolic quaternions incorporates some of the features of the older and larger algebra of biquaternions. They both contain subalgebras isomorphic to the split-complex number plane. Furthermore, just as the quaternion algebra H can be viewed as a union of complex planes, so the hyperbolic quaternion algebra is a union of split-complex number planes sharing the same real line.
It was Alexander Macfarlane who promoted this concept in the 1890s as his Algebra of Physics, first through the American Association for the Advancement of Science in 1891, then through his 1894 book of five Papers in Space Analysis, and in a series of lectures at Lehigh University in 1900.
Like the quaternions, the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4. A linear combination
is a hyperbolic quaternion when and are real numbers and the basis set has these products:
Using the distributive property, these relations can be used to multiply any two hyperbolic quaternions.
Unlike the ordinary quaternions, the hyperbolic quaternions are not associative. For example, , while . In fact, this example shows that the hyperbolic quaternions are not even an alternative algebra.
The first three relations show that products of the (non-real) basis elements are anti-commutative. Although this basis set does not form a group, the set
forms a quasigroup. One also notes that any subplane of the set M of hyperbolic quaternions that contains the real axis forms a plane of split-complex numbers. If
is the conjugate of , then the product
is the quadratic form used in spacetime theory. In fact, for events p and q, the bilinear form
arises as the negative of the real part of the hyperbolic quaternion product pq*, and is used in Minkowski space.
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In abstract algebra, the biquaternions are the numbers w + x i + y j + z k, where w, x, y, and z are complex numbers, or variants thereof, and the elements of {1, i, j, k} multiply as in the quaternion group and commute with their coefficients. There are three types of biquaternions corresponding to complex numbers and the variations thereof: Biquaternions when the coefficients are complex numbers. Split-biquaternions when the coefficients are split-complex numbers. Dual quaternions when the coefficients are dual numbers.
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term abstract algebra was coined in the early 20th century to distinguish it from older parts of algebra, and more specifically from elementary algebra, the use of variables to represent numbers in computation and reasoning.
William Kingdon Clifford (4 May 1845 - 3 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics, geometry, and computing.
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