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| In [[abstract algebra]], the '''Weyl algebra''' is the [[ring (mathematics)|ring]] of [[differential operator]]s with [[polynomial]] coefficients (in one variable),
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| :<math> f_n(X) \partial_X^n + f_{n-1}(X) \partial_X^{n-1} + \cdots + f_1(X) \partial_X + f_0(X).</math>
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| More precisely, let ''F'' be a [[field (mathematics)|field]], and let ''F''[''X''] be the [[polynomial ring|ring of polynomials]] in one variable, ''X'', with coefficients in ''F''. Then each ''f<sub>i</sub>'' lies in ''F''[''X'']. ''∂<sub>X</sub>'' is the [[derivative]] with respect to ''X''. The algebra is generated by ''X'' and ''∂<sub>X</sub>''.
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| 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 (ring theory)|domain]], and an example of an [[Ore extension]].
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| The Weyl algebra is a [[quotient ring|quotient]] of the [[free algebra]] on two generators, ''X'' and ''Y'', by the [[ideal (ring theory)|ideal]] generated by elements of the form
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| :<math>YX - XY - 1.\ </math>
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| The Weyl algebra is the first in an infinite family of algebras, also known as Weyl algebras. The '''''n''-th Weyl algebra''', ''A<sub>n</sub>'', is the ring of differential operators with polynomial coefficients in ''n'' variables. It is generated by ''X<sub>i</sub>'' and <math>\part_{X_i}</math>.
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| Weyl algebras are named after [[Hermann Weyl]], who introduced them to study the [[Werner Heisenberg|Heisenberg]] [[uncertainty principle]] in [[quantum mechanics]]. It is a [[quotient ring|quotient]] of the [[universal enveloping algebra]] of the [[Heisenberg algebra]], the [[Lie algebra]] of the [[Heisenberg group]], by setting the element ''1'' of
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| the Lie algebra equal to the unit ''1'' of the universal enveloping algebra.
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| The Weyl algebra is also referred to as the '''symplectic Clifford algebra'''.<ref name="helmstetter-2008-p12">Jacques Helmstetter, Artibano Micali: ''Quadratic Mappings and Clifford Algebras'', Birkhäuser, 2008, ISBN 978-3-7643-8605-4 [http://books.google.com/books?id=x_VfARQsSO8C&pg=PR12 p. xii]</ref><ref name="ablamowicz-Pxvi">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, [http://books.google.com/books?id=b6mbSCv_MHMC&pg=PR16 p. xvi]</ref><ref>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, [http://books.google.com/books?id=FhU9QpPIscoC&pg=PA92 p. 92]</ref> Weyl algebras represent the same structure for [[bilinear form]]s that (orthogonal) [[Clifford algebra]]s represent for [[quadratic form]]s.<ref name="helmstetter-2008-p12"/>
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| == Generators and relations ==
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| One may give an abstract construction of the algebras ''A<sub>n</sub>'' 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
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| :<math>W(V) := T(V) / (\!( v \otimes u - u \otimes v - \omega(v,u), \text{ for } v,u \in V )\!),</math>
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| where ''T''(''V'') is the [[tensor algebra]] on ''V'', and the notation <math>(\!( )\!)</math> means "the [[ideal (ring theory)|ideal]] generated by". In other words, ''W''(''V'') is the algebra generated by ''V'' subject only to the relation ''vu'' − ''uv'' = ω(''v'', ''u''). Then, ''W(V)'' is isomorphic to ''A<sub>n</sub>'' via the choice of a Darboux basis for ω.
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| === Quantization ===
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| The algebra ''W''(''V'') is a [[quantization (physics)|quantization]] of the [[symmetric algebra]] Sym(''V''). If ''V'' is over a field of characteristic zero, then ''W''(''V'') is naturally isomorphic to the underlying vector space of the [[symmetric algebra]] Sym(''V'') equipped with a deformed product – called the Groenewold–[[Moyal product]] (considering the symmetric algebra to be polynomial functions on ''V''*, where the variables span the vector space ''V'', and replacing <math>i \hbar</math> in the Moyal product formula with 1). The isomorphism is given by the symmetrization map from Sym(''V'') to ''W''(''V''):
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| :<math>a_1 \cdots a_n \mapsto \frac{1}{n!} \sum_{\sigma \in S_n} a_{\sigma(1)} \otimes \cdots \otimes a_{\sigma(n)}.</math>
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| If one prefers to have the <math>i\hbar</math> and work over the complex numbers, one could have instead defined the Weyl algebra above as generated by ''X''<sub>''i''</sub> and <math>i \hbar \part_{X_i}</math> (as is frequently done in [[quantum mechanics]]).
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| Thus, the Weyl algebra is a quantization of the symmetric algebra, which is essentially the same as the [[Moyal product|Moyal quantization]] (if for the latter one restricts to polynomial functions), but the former is in terms of generators and relations (considered to be differential operators) and the latter is in terms of a deformed multiplication.
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| In the case of [[exterior algebra]]s, the analogous quantization to the Weyl one is the [[Clifford algebra]], which is also referred to as the ''orthogonal Clifford algebra''.<ref name="ablamowicz-Pxvi"/><ref>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, [http://books.google.com/books?id=FhU9QpPIscoC&pg=PA83 p. 83]</ref>
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| ==Properties of the Weyl algebra==
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| In the case that the ground field ''F'' has characteristic zero, the ''n''th Weyl algebra is a [[simple ring|simple]] [[Noetherian ring|Noetherian]] [[domain (ring theory)|domain]]. It has [[global dimension]] ''n'', in contrast to the ring it deforms, Sym(''V''), which has global dimension 2''n''.
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| It has no finite dimensional representations; although this follows from simplicity, it can be more directly shown by taking the trace σ(''X'') and σ(''Y'') for some finite dimensional representation σ (where {{nowrap|1=[''X'',''Y''] = 1}}).
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| :<math> tr([\sigma(X),\sigma(Y)])=tr(1)</math>
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| Since the trace of a commutator is zero, and the trace of the identity is the dimension of the matrix, the representation must be zero dimensional.
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| In fact, there are stronger statements than the absence of finite-dimensional representations. To any f.g. ''A''_''n''-module ''M'', there is a corresponding subvariety ''Char(M)'' of {{nowrap|''V'' × ''V''*}} called the 'characteristic variety' whose size roughly corresponds to the size of ''M'' (a finite-dimensional module would have zero-dimensional characteristic variety). Then [[Bernstein's inequality (mathematical analysis)|Bernstein's inequality]] states that for ''M'' non-zero,
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| :<math>\dim(\operatorname{char}(M))\geq n</math>
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| An even stronger statement is [[Gabber's theorem]], which states that ''Char(M)'' is a [[Lagrangian submanifold|co-isotropic]] subvariety of {{nowrap|''V'' × ''V''*}} for the natural symplectic form.
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| ===Positive characteristic===
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| The situation is considerably different in the case of a Weyl algebra over a field of [[characteristic (algebra)|characteristic]] {{nowrap|''p'' > 0}}. In this case, for any element ''D'' of the Weyl algebra, the element ''D<sup>p</sup>'' is central, and so the Weyl algebra has a very large center. In fact, it is a finitely-generated module over its center; even more so, it is an [[Azumaya algebra]] over its center. As a consequence, there are many finite-dimensional representations which are all built out of simple representations of dimension ''p''.
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| == Generalizations ==
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| For more details about this quantization in the case ''n'' = 1 (and an extension using the [[Fourier transform]] to integrable ("most") functions, not just polynomial functions), see [[Weyl quantization]].
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| Weyl algebras and Clifford algebras admit a further structure of a [[*-algebra]], and can be unified as even and odd terms of a [[superalgebra]], as discussed in [[CCR and CAR algebras]].
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| ==References==
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| * M. Rausch de Traubenberg, M. J. Slupinski, A. Tanasa, ''[http://arxiv.org/abs/math/0504224 Finite-dimensional Lie subalgebras of the Weyl algebra]'', (2005) ''(Classifies subalgebras of the one dimensional Weyl algebra over the complex numbers; shows relationship to [[SL(2,C)]])''
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| * [[Tsit Yuen Lam]], ''A first course in noncommutative rings''. Volume 131 of [[Graduate texts in mathematics]]. 2ed. Springer, 2001. p. 6. ISBN 978-0-387-95325-0
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| {{Reflist}}
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| [[Category:Algebras]]
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| [[Category:Differential operators]]
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| [[Category:Ring theory]]
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