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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク hyperbolic quaternion ： ウィキペディア英語版 hyperbolic quaternion In abstract algebra, the algebra of hyperbolic quaternions is a nonassociative algebra over the real numbers with elements of the form :$q\; =\; a\; +\; bi\; +\; cj\; +\; dk,\; \backslash quad\; a,b,c,d\; \backslash in\; \backslash Bbb\; \backslash !$ where multiplication is determined with rules that are similar to (but different from) multiplication in the quaternions. 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 (see below). ==Algebraic structure== Like the quaternions, the set of hyperbolic quaternions form a vector space over the real numbers of dimension 4. A linear combination :$q\; =\; a+bi+cj+dk$ is a hyperbolic quaternion when $a,\; b,\; c,$ and $d$ are real numbers and the basis set $\backslash$ has these products: :$ij=k=-ji$ :$jk=i=-kj$ :$ki=j=-ik$ :$i^2=+1=j^2=k^2$ 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, $(ij)j\; =\; kj\; =\; -i$, while $i(jj)\; =\; i$. 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 :$\backslash$ 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 :$q^$ *=a-bi-cj-dk is the conjugate of $q$, then the product :$q(q^$ *)=a^2-b^2-c^2-d^2 is the quadratic form used in spacetime theory. In fact, the bilinear form called the Minkowski inner product arises as the negative of the real part of the hyperbolic quaternion product ''pq'' * : :$-p\_0q\_0\; +\; p\_1q\_1\; +\; p\_2q\_2\; +\; p\_3q\_3$. Note that the set of units U = is not closed under multiplication. See the references (external link) for details. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「hyperbolic quaternion」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.