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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 Quantum differential calculus ： ウィキペディア英語版 Quantum differential calculus In quantum geometry or noncommutative geometry a quantum differential calculus or noncommutative differential structure on an algebra $A$ over a field $k$ means the specification of a space of differential forms over the algebra. The algebra $A$ here is regarded as a coordinate ring but it is important that it may be noncommutative and hence not an actual algebra of coordinate functions on any actual space, so this represents a point of view replacing the specification for an actual space of a differentiable structure. In ordinary differential geometry one can multiply differential 1-forms by functions form the left and the right and has an exterior derivative. Correspondingly, a first order quantum differential calculus means at least the following: 1. An $A$-$A$-bimodule $\backslash Omega^1$ over $A$, i.e. one can multiply elements of $\backslash Omega^1$ by elements of $A$ in an associative way: : $a(\backslash omega\; b)=(a\backslash omega)b,\backslash \; \backslash forall\; a,b\backslash in\; A,\backslash \; \backslash omega\backslash in\backslash Omega^1$. 2. A linear map $:A\backslash to\backslash Omega^1$ obeying the Leibniz rule : $(ab)=a(b)+(a)b,\backslash \; \backslash forall\; a,b\backslash in\; A$ 3. $\backslash Omega^1=\backslash$ 4. (optional connectedness condition) $\backslash ker\backslash \; =k1$ The last condition is not always imposed but holds in ordinary geometry when the manifold is connected. It says that the only functions killed by $$ are constant functions. An ''exterior algebra'' or ''differential graded algebra'' structure over $A$ means a compatible extension of $\backslash Omega^1$ to include analogues of higher order differential forms : $\backslash Omega=\backslash oplus\_n\backslash Omega^n,\backslash \; :\backslash Omega^n\backslash to\backslash Omega^$ obeying a graded-Leibniz rule with respect to an associative product on $\backslash Omega$ and obeying $^2=0$. Here $\backslash Omega^0=A$ and it is usually required that $\backslash Omega$ is generated by $A,\backslash Omega^1$. The product of differential forms is called the exterior or wedge product and often denoted $\backslash wedge$. The noncommutative or quantum de Rham cohomology is defined as the cohomology of this complex. A higher order differential calculus can mean an exterior algebra or it can mean the partial specification of one up to some highest degree and with products that would result in a degree beyond the highest being unspecified. The above definition lies at the crossroads of two approaches to noncommutative geometry. In the Connes approach a more fundamental object is a replacement for the Dirac operator in the form of a spectral triple, and an exterior algebra can be constructed from this data. In the quantum groups approach to noncommutative geometry one starts with the algebra and a choice of first order calculus but constrained by covariance under a quantum group symmetry. ==Note== The above definition is minimal and gives something more general than classical differential calculus even when the algebra $A$ is commutative or functions on an actual space. This is because we do ''not'' demand that : $a(b)=()a,\backslash \; \backslash forall\; a,b\backslash in\; A$ since this would imply that $(ab-ba)=0,\backslash \; \backslash forall\; a,b\backslash in\; A$, which would violate axiom 4 when the algebra was noncommutative. As a byproduct, this enlarged definition includes finite difference calculi and quantum differential calculi on finite sets and finite groups (finite group Lie algebra theory). 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「Quantum differential calculus」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.