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{{for|the universal enveloping W* algebra of a C* algebra|Sherman–Takeda theorem}}
== 」その後、このアイデアの彼らの冷たい心の当事者は驚きました ==
In [[mathematics]], for any [[Lie algebra]] ''L'' one can construct its '''universal enveloping algebra''' ''U''(''L''). This construction passes from the non-[[associative]] structure ''L'' to a (more familiar, and possibly easier to handle) [[unital algebra|unital]] [[associative algebra]] which captures the important properties of ''L''.


Any associative algebra ''A'' over the [[field (mathematics)|field]] ''K'' becomes a Lie algebra over ''K'' with the '''Lie bracket''':
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:[''a'',''b''] = ''ab'' &minus; ''ba''.
== '最後に、宇宙の意味は何ですか ==


That is, from an associative product, one can construct a Lie bracket by taking the [[commutator]] with respect to that associative product. Denote this Lie algebra by ''A<sub>L</sub>''.
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Construction of the universal enveloping algebra attempts to reverse this process: to a given Lie algebra ''L'' over ''K'', find the "most general" unital associative ''K''-algebra ''A'' such that the Lie algebra ''A<sub>L</sub>'' contains ''L''; this algebra ''A'' is ''U''(''L''). The important constraint is to preserve the representation theory: the [[representation of a Lie algebra|representations]] of ''L'' correspond in a one-to-one manner to the [[module (mathematics)|module]]s over ''U''(''L''). In a typical context where ''L'' is acting by ''[[infinitesimal transformation]]s'', the elements of ''U''(''L'') act like [[differential operator]]s, of all orders. Next to Lie algebras, the construction of the universal enveloping algebra has been generalized for Malcev algebras,<ref>J.M. Perez-Izquierdo, I.P. Shestakov: ''An envelope for Malcev algebras'', Journal of Algebra 272 (2004) 379-393.</ref> Bol algebras <ref>J.M. Perez-Izquierdo: ''An envelope for Bol algebras'', Journal of Algebra 284 (2005) 480-493.</ref> and left alternative algebras.<ref>Rukavicka Josef: ''An envelope for left alternative algebras'', International Journal of Algebra, Vol. 7, 2013, no. 10, 455-462, [http://www.m-hikari.com/ija/ija-2013/ija-9-12-2013/rukavickaIJA9-12-2013.pdf]</ref>
== They Palace family Maoshang ==


==Motivation==
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An important topic in Lie algebras studies and probably the main source of their appearance in applications is [[Lie algebra representation|representation of the Lie algebra]]. A representation ρ assigns to any element ''x'' of a Lie algebra a linear operator ρ(''x''). The space of linear operators is not only a Lie algebra, but also an associative algebra and so one can consider products ρ(''x'')ρ(''y''). The main point to introduce  the universal enveloping  algebra is to study such products in various representations of a Lie algebra. One obstacle can be immediately seen in a naive attempt to do this: properties of products drastically depend on the representation, not only on the Lie algebra itself. For example for one representation we might have ρ(''x'')ρ(''y'') = 0, while in another representation this product may not be zero.
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Nevertheless it appears to be true that certain properties are '''universal''' for all representations, i.e. they hold true for all representations simultaneously. The universal enveloping algebra is a way to grasp all such properties and only them.
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==Universal property==
== panacea ==
Let ''X'' be any Lie algebra over ''K''.  Given a unital associative ''K''-algebra ''U'' and a Lie algebra homomorphism: ''h'': ''X'' → ''U<sub>L</sub>'', (notation as above) we say that ''U'' is the '''universal enveloping algebra''' of ''X'' if it satisfies the following [[universal property]]: for any unital associative ''K''-algebra ''A'' and Lie algebra homomorphism ''f'': ''X'' → ''A<sub>L</sub>'' there exists a ''unique'' unital algebra homomorphism ''g'': ''U'' → ''A'' such that: ''f''(-) = ''g<sub>L</sub>'' (''h''(-)).


This is the universal property expressing that the [[functor]] sending ''X'' to its universal enveloping algebra is [[left adjoint]] to the functor sending a unital associative algebra ''A'' to its Lie algebra ''A<sub>L</sub>''.
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==Direct construction==
== you can communicate heaven Makino family ==
From this universal property, one can prove that ''if'' a Lie algebra has a universal enveloping algebra, then this enveloping algebra is uniquely determined by ''L'' (up to a unique algebra isomorphism). By the following construction, which suggests itself on general grounds (for instance, as part of a pair of [[adjoint functors]]), we establish that indeed every Lie algebra does have a universal enveloping algebra.


Starting with the [[tensor algebra]] ''T''(''L'') on the [[vector space]] underlying ''L'', we take ''U''(''L'') to be the quotient of ''T''(''L'') made by imposing the relations
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: <math> a \otimes b - b \otimes a = [a,b]</math>
<ul>
 
 
for all ''a'' and ''b'' in (the image in ''T''(''L'') of) ''L'', where the bracket on the [[Sides of an equation|RHS]]  means the given Lie algebra product, in ''L''.
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Formally, we define
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:<math>U(L) = T(L)/I</math>
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where ''I'' is the two-sided [[Ideal (ring theory)|ideal]] of ''T''(''L'') generated by elements of the form
</ul>
 
: <math> a\otimes b - b \otimes a - [a,b], \quad a,b \in L.</math>
 
The natural map ''L'' → ''T''(''L'') descends to a map ''h'' : ''L'' → ''U''(''L''), and this is the Lie algebra homomorphism used in the universal property given above.
 
The analogous construction for [[Lie superalgebra]]s is straightforward.
 
==Examples in particular cases==
If ''L'' is ''abelian'' (that is, the bracket is always 0), then ''U''(''L'') is commutative; if a [[basis (linear algebra)|basis]] of the [[vector space]] ''L'' has been chosen, then ''U''(''L'') can be identified with the [[polynomial]] algebra over ''K'', with one variable per basis element.
 
If ''L'' is the Lie algebra corresponding to the [[Lie group]] ''G'', ''U''(''L'') can be identified with the algebra of left-invariant [[differential operator]]s (of all orders) on ''G''; with ''L'' lying inside it as the left-invariant [[vector field]]s as first-order differential operators.
 
To relate the above two cases: if ''L'' is a vector space ''V'' as abelian Lie algebra, the left-invariant differential operators are the constant coefficient operators, which are indeed a polynomial algebra in the [[partial derivative]]s of first order.
 
The [[center of an algebra|center]] of ''U''(''L'') is called ''Z''(''L'') and consists of the left- and right- invariant differential operators; this in the case of ''G'' not commutative will often not be generated by first-order operators (see for example [[Casimir operator]] of a semi-simple Lie algebra).
 
Another characterisation in Lie group theory is of ''U''(''L'') as the [[convolution]] algebra of [[Distribution (mathematics)|distribution]]s [[Support (mathematics)#Support of a distribution|support]]ed only at the [[identity element]] ''e'' of ''G''.
 
The algebra of differential operators in ''n'' variables with polynomial coefficients may be obtained starting with the Lie algebra of the [[Heisenberg group]]. See [[Weyl algebra]] for this; one must take a quotient, so that the central elements of the Lie algebra act as prescribed scalars.
 
==Further description of structure==
 
The fundamental [[Poincaré–Birkhoff–Witt theorem]] gives a precise description of ''U''(''L''); the most important consequence is that ''L'' can be viewed as a [[linear subspace]] of ''U''(''L''). More precisely: the canonical map ''h'' : ''L'' → ''U''(''L'') is always [[injective]]. Furthermore, ''U''(''L'') is generated as a unital associative algebra by ''L''.
 
''L'' acts on itself by the Lie algebra [[Adjoint representation of a Lie algebra|adjoint representation]], and this action can be extended to a  [[representation of a Lie algebra|representation]] of ''L'' on ''U''(''L''): ''L'' acts as an algebra of [[derivation (abstract algebra)|derivation]]s on ''T''(''L''), and this action respects the imposed relations, so it actually acts on ''U''(''L''). (This is the purely infinitesimal way of looking at the invariant differential operators mentioned above.)
 
Under this representation, the elements of ''U''(''L'') ''invariant'' under the action of ''L'' (i.e. such that any element of ''L'' acting on them gives zero) are called ''invariant elements''. They are generated by the [[Casimir invariant]]s.
 
As mentioned above, the construction of universal enveloping algebras is part of a pair of adjoint functors. ''U'' is a functor from the category of Lie algebras over ''K'' to the category of unital associative ''K''-algebras. This functor is left adjoint to the functor which maps an algebra ''A'' to the Lie algebra ''A<sub>L</sub>''. The universal enveloping algebra construction is not exactly inverse to the formation of ''A<sub>L</sub>'': if we start with an associative algebra ''A'', then ''U''(''A<sub>L</sub>'') is ''not'' equal to ''A''; it is much bigger.
 
The facts about representation theory mentioned earlier can be made precise as follows: the [[abelian category]] of all [[representation of a Lie algebra|representations]] of ''L'' is [[isomorphism of categories|isomorphic]] to the abelian category of all left modules over ''U''(''L'').
 
The construction of the [[group algebra]] for a given [[group (mathematics)|group]] is in many ways analogous to constructing the universal enveloping algebra for a given Lie algebra. Both constructions are universal and translate representation theory into module theory. Furthermore, both group algebras and universal enveloping algebras carry natural [[coalgebra|comultiplications]] which turn them into [[Hopf algebra]]s.
 
The center of the universal enveloping algebra of a simple Lie algebra is described by the [[Harish-Chandra isomorphism]].
 
==See also==
*[[Poincaré–Birkhoff–Witt theorem]]
*[[Harish-Chandra homomorphism]]
 
==References==
{{reflist}}
*{{Citation | last1=Dixmier | first1=Jacques | title=Enveloping algebras | origyear=1974 | url=http://books.google.com/books?isbn=0821805606 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-0560-2 | mr=0498740 | year=1996 | volume=11}}
 
[[Category:Lie algebras]]
[[Category:Ring theory]]
[[Category:Hopf algebras]]
[[Category:Representation theory]]

Latest revision as of 08:15, 3 May 2014

」その後、このアイデアの彼らの冷たい心の当事者は驚きました

自分の斬首の敵に出フライは、手を振って暗殺を所有する必要があります prada 新作 財布。
パーティー清は自然に歩くことを継続して進めてささやい冷たい心臓のどの側面、知らない、目が正面を見て、「その前にあなたはまだ開いているが、悪魔の神の祭壇に近い、夜叉は精神的な、より多くのを覚えています。レルムおよび重い肉10は、神にも何も変わっていないだけ超自然ファムに入って、一般の人々の世界を壊し、月を求めることができると私は言っていた、あなたは超自然的なファム踏み入れると、私は意志さえあれば、何も、清魏があなたと結婚するしてみましょう。 プラダ人気財布 '
一瞬のためにご自身を低温側「下の妹が、結婚して私はまた、ほとんどの妹を気にしない結婚していない」、もちろん、約ささやいた私の心に、表面上に発現することはできません prada トートバッグ。
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'最後に、宇宙の意味は何ですか

いったん放棄物事ダウン、またはベビー宇宙の遺産に依存する。 プラダ 財布 リボン '
側前駆コールドミラーと悪の魔女を連想させる「宇宙の遺産は？ prada 新作 財布 '：'最後に、宇宙の意味は何ですか？ プラダ 財布 定価 '
「宇宙、世界の前に5崩壊天の最後で、私はどのくらいの時代の前に知らない」絶妙な妖精の銅像と言った：「当時、3000の世界はまだ生産されていない、全世界の宇宙だけでなく、宇宙は後でお守りした後、宇宙が形成された状態であるかを推測することはできませんない星が。、星、スターフィールド、3000世界の無数のと同様に、カルマ、さまざまに変えていない。 prada ピンク 財布 「繊細な妖精の銅像が説明する：「しかし進化のこの世界は、幸運のことですが、私は問題ははるかに届くから幸運のレルムです熟考、それは」状態であるかを説明することは困難である？ 問題をじっくり考えることができます
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They Palace family Maoshang

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rip it!
ten arrows just startled, took each other's Body Gangqi tear.
'unfeeling Arrow Road! realm with the life of the world!'
Ouyang Figure felt changjon terrorist, who once appeared in a lot of magic, screaming, waving his hands and forged a round after round of the French and Indian, barely withstood changjon.
However, in the Palace 相关的主题文章：

panacea

Defeated China days are, but three thousand six hundred centuries, but I can ah! '
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'here is not suitable for cultivation, you practice the Qing Emperor wooden Wong Gong, Reiki need to absorb a lot of 長財布 prada wood, プラダ ピンク 財布 there is no And this strength is thin, but there is no panacea, coming after those masters, they can not get out. might also Shou Yuan do not wait, they starved to prada ピンク 財布 death. '
'Yan' poured cold water splashed down.
Fanghan Li carved to look around, and then violent breathing, found this magnificent palace, what strength are not, gold, wood, water, fire, earth, and so strength is not, indeed can not practice supernatural powers, let alone What nourishing herbs, panacea, and in addition prada 財布 人気 to the weathered bones, is deserted palace.
looked back to the prada トートバッグ road, square cold even found it 'too yuan Rays' are gone. outside
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you can communicate heaven Makino family

Some ancient door, the 長財布 プラダ reason prada 財布 2014 there プラダの財布 are big names from fairy angel into practice, but also secretly stored Yuanshi of gas. Obtained heaven reward. For example Makino vast, war 王天君 blood, you can communicate heaven Makino family, so the family of heaven through mysterious channels, through down the gas Yuanshi of blessing to his body.
side cold now, although all blood Makino was vast, but it is impossible to communicate heaven Makino family, so that they reward under Yuanshi gas down.
but good is God, プラダ 迷彩 財布 the プラダ 財布 修理 ball can learn Tai Chi.
'cold side, what you do, it is too arrogant, I Divine door, though always ignore things, but you really could not understand this and other Magic outrageous behavior, you are still too evildoer than Protoss , is the World's a mysterious yellow tumor. Fengyun you have to pull out a malignant tumor, the Protoss amulet, will ride out. ' 相关的主题文章：

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