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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 Weyl algebra ： ウィキペディア英語版 Weyl algebra In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), :$f\_n(X)\; \backslash partial\_X^n\; +\; f\_(X)\; \backslash partial\_X^\; +\; \backslash cdots\; +\; f\_1(X)\; \backslash partial\_X\; +\; f\_0(X).$ More precisely, let ''F'' be a field, and let ''F''() be the ring of polynomials in one variable, ''X'', with coefficients in ''F''. Then each ''fi'' lies in ''F''(). ''∂X'' is the derivative with respect to ''X''. The algebra is generated by ''X'' and ''∂X'' . The Weyl algebra is an example of a simple ring that is not a matrix ring over a division ring. It is also a noncommutative example of a domain, and an example of an Ore extension. The Weyl algebra is a quotient of the free algebra on two generators, ''X'' and ''Y'', by the ideal generated by elements of the form :$YX\; -\; XY\; -\; 1~.$ The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The ''n''-th Weyl algebra, ''An'', is the ring of differential operators with polynomial coefficients in ''n'' variables. It is generated by ''Xi'' and ''∂Xi''. Weyl algebras are named after Hermann Weyl, who introduced them to study the Heisenberg uncertainty principle in quantum mechanics. It is a quotient of the universal enveloping algebra of the Heisenberg algebra, the Lie algebra of the Heisenberg group, by setting the central element of the Heisenberg algebra (namely ''()'') equal to the unit of the universal enveloping algebra (called ''1'' above). The Weyl algebra is also referred to as the symplectic Clifford algebra.〔Jacques Helmstetter, Artibano Micali: ''Quadratic Mappings and Clifford Algebras'', Birkhäuser, 2008, ISBN 978-3-7643-8605-4 (p. xii )〕〔Rafał Abłamowicz: ''Clifford algebras: applications to mathematics, physics, and engineering'' (dedicated to Pertti Lounesto), Progress in Mathematical Physics, Birkhäuser Boston, 2004, ISBN 0-8176-3525-4. Foreword, (p. xvi )〕〔Z. Oziewicz, Cz. Sitarczyk: ''Parallel treatment of Riemannian and symplectic Clifford algebras'', pp.83-96. In: Artibano Micali, Roger Boudet, Jacques Helmstetter (eds.): ''Clifford algebras and their applications in mathematical physics'', Kluwer, 1989, ISBN 0-7923-1623-1, (p. 92 )〕 Weyl algebras represent the same structure for symplectic bilinear forms that Clifford algebras represent for non-degenerate symmetric bilinear forms.〔 == Generators and relations == One may give an abstract construction of the algebras ''An'' in terms of generators and relations. Start with an abstract vector space ''V'' (of dimension 2''n'') equipped with a symplectic form ω. Define the Weyl algebra ''W''(''V'') to be :$W(V)\; :=\; T(V)\; /\; (\backslash !(\; v\; \backslash otimes\; u\; -\; u\; \backslash otimes\; v\; -\; \backslash omega(v,u),\; \backslash text\; v,u\; \backslash in\; V\; )\backslash !),$ where ''T''(''V'') is the tensor algebra on ''V'', and the notation $(\backslash !(\; )\backslash !)$ means "the ideal generated by". In other words, ''W''(''V'') is the algebra generated by ''V'' subject only to the relation . Then, ''W(V)'' is isomorphic to ''An'' via the choice of a Darboux basis for . 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「Weyl algebra」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.