|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| In [[abstract algebra]], the '''Weyl algebra''' is the [[ring (mathematics)|ring]] of [[differential operator]]s with [[polynomial]] coefficients (in one variable),
| | Regardless of the [http://prometeu.net Clash of Clans hack tool]; there are unquestionably also hack tools with [http://Search.usa.gov/search?query=respect respect] to other games. People can check out all of the hacks and obtain those which they need. It is sure which will have lost with regards to fun once they keep the hack tool at their disposal.<br><br>The most important upsides of video gaming applications can include fun, entertainment and even education. The downsides range in [http://www.bbc.Co.uk/search/?q=addictive+game addictive game] play in which to younger individuals seeing and after that hearing things they are undoubtedly not old enough in order for. With luck, that this ideas presented within doing this article can help customers manage video games clearly within your home with everyone's benefit.<br><br>A great clash of clans hack equipment moreover tricks could be unbelievably handy in the give good results that you are an enthusiastic player of the diversion. The finest hack apparatuses tackle just around any doing work platform - Android and Apple inc iOS. You possibly can exploit the obtain that you can use on your apple iphone 3gs or Android telephones.<br><br>Many are a group towards coders that loves with regard to play Cof. We are going to are continuously developing Cheats to speed up Levelling easily and to produce more gems for free. Without our hacks it does take you ages to assist you to reach your level.<br><br>Waiting for game of the particular season editions of very important titles. These often come out per the four seasons or higher after your current initial headline, but take into account a lot of specific down-loadable and extra happy which was released in steps once the primary headline. These video titles supply a good more bang for all of the buck.<br><br>You will notice for yourself that your Money Compromise of Clans i fairly effective, completely invisible by the manager of the game, remarkably absolutely no price!<br><br>Pc games or computer games have increased in popularity nowadays, just with the younger generation, but also with people today as well. There are plenty games available, ranging coming from the intellectual to the seven days a week - your options may very well be limitless. Online role playing games are amongst the most popular games anywhere on the earth. With this popularity, plenty consumers are exploring and wanting to find ways to go through the whole game as rather quickly as they can; causitive factors of using computer How to break into in clash of clans range from simply endeavoring to own your own close stare at you living in awe, or getting a tremendous amount of game money a person really can sell later, or simply just into rid the game within the fun factor for the opposite players. |
| | |
| :<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> | |
| | |
| 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>''.
| |
| | |
| 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]].
| |
| | |
| 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
| |
| | |
| :<math>YX - XY - 1.\ </math>
| |
| | |
| 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>. | |
| | |
| 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
| |
| the Lie algebra equal to the unit ''1'' of the universal enveloping algebra.
| |
| | |
| 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"/>
| |
| | |
| == Generators and relations ==
| |
| 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
| |
| | |
| :<math>W(V) := T(V) / (\!( v \otimes u - u \otimes v - \omega(v,u), \text{ for } v,u \in V )\!),</math>
| |
| | |
| 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.
| |
| | |
| === Quantization ===
| |
| 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''):
| |
| | |
| :<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>
| |
| | |
| 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]]).
| |
| | |
| 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.
| |
| | |
| 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>
| |
| | |
| ==Properties of the Weyl algebra==
| |
| 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''.
| |
| | |
| 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}}).
| |
| :<math> tr([\sigma(X),\sigma(Y)])=tr(1)</math>
| |
| 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.
| |
| | |
| 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,
| |
| :<math>\dim(\operatorname{char}(M))\geq n</math>
| |
| 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.
| |
| | |
| ===Positive characteristic===
| |
| 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''.
| |
| | |
| == Generalizations ==
| |
| 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]].
| |
| | |
| 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]].
| |
| | |
| ==References==
| |
| * 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)]])''
| |
| | |
| * [[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
| |
| | |
| {{Reflist}}
| |
| | |
| [[Category:Algebras]]
| |
| [[Category:Differential operators]]
| |
| [[Category:Ring theory]]
| |
Regardless of the Clash of Clans hack tool; there are unquestionably also hack tools with respect to other games. People can check out all of the hacks and obtain those which they need. It is sure which will have lost with regards to fun once they keep the hack tool at their disposal.
The most important upsides of video gaming applications can include fun, entertainment and even education. The downsides range in addictive game play in which to younger individuals seeing and after that hearing things they are undoubtedly not old enough in order for. With luck, that this ideas presented within doing this article can help customers manage video games clearly within your home with everyone's benefit.
A great clash of clans hack equipment moreover tricks could be unbelievably handy in the give good results that you are an enthusiastic player of the diversion. The finest hack apparatuses tackle just around any doing work platform - Android and Apple inc iOS. You possibly can exploit the obtain that you can use on your apple iphone 3gs or Android telephones.
Many are a group towards coders that loves with regard to play Cof. We are going to are continuously developing Cheats to speed up Levelling easily and to produce more gems for free. Without our hacks it does take you ages to assist you to reach your level.
Waiting for game of the particular season editions of very important titles. These often come out per the four seasons or higher after your current initial headline, but take into account a lot of specific down-loadable and extra happy which was released in steps once the primary headline. These video titles supply a good more bang for all of the buck.
You will notice for yourself that your Money Compromise of Clans i fairly effective, completely invisible by the manager of the game, remarkably absolutely no price!
Pc games or computer games have increased in popularity nowadays, just with the younger generation, but also with people today as well. There are plenty games available, ranging coming from the intellectual to the seven days a week - your options may very well be limitless. Online role playing games are amongst the most popular games anywhere on the earth. With this popularity, plenty consumers are exploring and wanting to find ways to go through the whole game as rather quickly as they can; causitive factors of using computer How to break into in clash of clans range from simply endeavoring to own your own close stare at you living in awe, or getting a tremendous amount of game money a person really can sell later, or simply just into rid the game within the fun factor for the opposite players.