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In mathematics, Clifford algebras are a type of associative algebra. As ''K''-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems.〔W. K. Clifford, "Preliminary sketch of bi-quaternions, Proc. London Math. Soc. Vol. 4 (1873) pp. 381-395〕〔W. K. Clifford, ''Mathematical Papers'', (ed. R. Tucker), London: Macmillan, 1882.〕 The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English geometer William Kingdon Clifford. The most familiar Clifford algebra, or orthogonal Clifford algebra, is also referred to as ''Riemannian Clifford algebra''.〔see for ex. Z. Oziewicz, Sz. Sitarczyk: ''Parallel treatment of Riemannian and symplectic Clifford algebras''. In: Artibano Micali, Roger Boudet, Jacques Helmstetter (eds.): ''Clifford Algebras and their Applications in Mathematical Physics'', Kluwer Academic Publishers, ISBN 0-7923-1623-1, 1992, (p. 83 )〕 ==Introduction and basic properties== A Clifford algebra is a unital associative algebra that contains and is generated by a vector space ''V'' over a field ''K'', where ''V'' is equipped with a quadratic form ''Q''. The Clifford algebra ''C''ℓ(''V'', ''Q'') is the "freest" algebra generated by ''V'' subject to the condition〔Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in index theory) sometimes use a different choice of sign in the fundamental Clifford identity. That is, they take One must replace ''Q'' with −''Q'' in going from one convention to the other.〕 : where the product on the left is that of the algebra, and the 1 is its multiplicative identity. The definition of a Clifford algebra endows the algebra with more structure than a "bare" ''K''-algebra: specifically it has a designated or privileged subspace ''V''. Such a subspace cannot in general be uniquely determined given only a ''K''-algebra isomorphic to the Clifford algebra. If the characteristic of the ground field ''K'' is not 2, then one can rewrite this fundamental identity in the form : where : is the symmetric bilinear form associated with ''Q'', via the polarization identity. The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property, as done below. Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if it is not true that a quadratic form determines a symmetric bilinear form, or that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not 2, and are false if this condition is removed. 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「Clifford algebra」の詳細全文を読む スポンサード リンク
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