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| {{for|the universal enveloping W* algebra of a C* algebra|Sherman–Takeda theorem}}
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| 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''.
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| 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'' − ''ba''.
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| 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>
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| ==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==
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| 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''(-)).
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| 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==
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| 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.
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| 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>
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| 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
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| : <math> a\otimes b - b \otimes a - [a,b], \quad a,b \in L.</math>
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| 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.
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| The analogous construction for [[Lie superalgebra]]s is straightforward.
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| ==Examples in particular cases==
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| 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.
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| 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.
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| 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.
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| 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).
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| 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''.
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| 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.
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| ==Further description of structure== | |
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| 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''.
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| ''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.)
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| 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.
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| 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.
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| 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'').
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| 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.
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| The center of the universal enveloping algebra of a simple Lie algebra is described by the [[Harish-Chandra isomorphism]].
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| ==See also==
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| *[[Poincaré–Birkhoff–Witt theorem]]
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| *[[Harish-Chandra homomorphism]]
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| ==References==
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| {{reflist}}
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| *{{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}}
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| [[Category:Lie algebras]]
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| [[Category:Ring theory]]
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| [[Category:Hopf algebras]]
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| [[Category:Representation theory]]
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