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| In mathematics, '''Lie algebra cohomology''' is a [[cohomology]] theory for [[Lie algebras]]. It was defined by {{harvs|txt|last=Chevalley|last2=Eilenberg|author1-link=Claude Chevalley|author2-link=Samuel Eilenberg|year=1948}} in order to give an algebraic construction of the cohomology of the underlying [[topological space]]s of compact [[Lie groups]]. In the paper above, a specific complex, called the [[Koszul complex]], is defined for a [[module (mathematics)|module]] over a Lie algebra, and its cohomology is taken in the normal sense.
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| ==Motivation==
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| If ''G'' is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the [[de Rham cohomology]] of the complex of differential forms on ''G''. This can be replaced by the complex of equivariant differential forms, which can in turn be identified with the exterior algebra of the Lie algebra, with a suitable differential. The construction of this differential on an exterior algebra makes sense for any Lie algebra, so is used to define Lie algebra cohomology for all Lie algebras. More generally one uses a similar construction to define Lie algebra cohomology with coefficients in a module.
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| ==Definition==
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| Let <math>\mathfrak g</math> be a Lie algebra over a commutative ring ''R'' with [[universal enveloping algebra]] <math>U\mathfrak g</math>, and let ''M'' be a representation of <math>\mathfrak g</math> (equivalently, a <math>U\mathfrak g</math>-module). Considering ''R'' as a trivial representation of <math>\mathfrak g</math>, one defines the cohomology groups
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| ::<math>\mathrm{H}^n(\mathfrak{g}; M) := \mathrm{Ext}^n_{U\mathfrak{g}}(R, M)</math>
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| (see [[Ext functor]] for the definition of Ext). Equivalently, these are the right [[derived functors]] of the left exact invariant submodule functor
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| ::<math>M \mapsto M^{\mathfrak{g}} := \{ m \in M \mid gm = 0\ \text{ for all } g \in \mathfrak{g}\}.</math> | |
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| Analogously, one can define Lie algebra homology as
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| ::<math>\mathrm{H}_n(\mathfrak{g}; M) := \mathrm{Tor}_n^{U\mathfrak{g}}(R, M)</math>
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| (see [[Tor functor]] for the definition of Tor), which is equivalent to the left derived functors of the right exact [[coinvariant]]s functor
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| ::<math> M \mapsto M_{\mathfrak{g}} := M / \mathfrak{g} M.</math>
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| Some important basic results about the cohomology of Lie algebras include [[Whitehead's lemma (Lie algebras)|Whitehead's lemma]]s, [[Weyl's theorem on complete reducibility|Weyl's theorem]], and the [[Levi decomposition]] theorem.
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| ==Cohomology in small dimensions==
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| The zeroth cohomology group is (by definition) just the invariants of the Lie algebra acting on the module:
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| ::<math>H^0(\mathfrak{g}; M) =M^{\mathfrak{g}} = \{ m \in M \mid gm = 0\ \text{ for all } g \in \mathfrak{g}\}.</math>
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| The first cohomology group is the space Der of derivations modulo the space Ider of inner derivations
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| ::<math>H^1(\mathfrak{g}; M) =Der(\mathfrak{g}, M)/Ider(\mathfrak{g}, M)</math>
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| where a derivation is a map ''d'' from the Lie algebra to ''M'' such that
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| ::<math>d[x,y] = xdy-ydx~</math> | |
| and is called inner if it is given by
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| ::<math>dx = xa~</math> | |
| for some ''a'' in ''M''.
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| The second cohomology group
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| ::<math>H^2(\mathfrak{g}; M)</math>
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| is the space of equivalence classes of Lie algebra extensions
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| ::<math>0\rightarrow M\rightarrow \mathfrak{h}\rightarrow\mathfrak{g}\rightarrow 0</math>
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| of the Lie algebra by the module ''M''.
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| There do not seem to be any similar easy interpretations for the higher cohomology groups.
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| ==See also==
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| * [[BRST formalism]] in theoretical physics.
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| ==References==
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| *{{Citation | last1=Chevalley | first1=Claude | last2=Eilenberg | first2=Samuel | author2-link=Samuel Eilenberg | title=Cohomology Theory of Lie Groups and Lie Algebras | jstor=1990637 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | mr=0024908 | year=1948 | journal=[[Transactions of the American Mathematical Society]] | issn=0002-9947 | volume=63 | issue=1 | pages=85–124 | doi=10.2307/1990637}}
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| *{{Citation | last1=Hilton | first1=P. J. | last2=Stammbach | first2=U. | title=A course in homological algebra | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-94823-2 | mr=1438546 | year=1997 | volume=4}}
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| *{{Citation | last1=Knapp | first1=Anthony W. | title=Lie groups, Lie algebras, and cohomology | publisher=[[Princeton University Press]] | series=Mathematical Notes | isbn=978-0-691-08498-5 | mr=938524 | year=1988 | volume=34}}
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| ==External links==
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| {{scholarpedia|title=An introduction to Lie algebra cohomology|urlname=An_introduction_to_Lie_algebra_cohomology}}
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| [[Category:Cohomology theories]]
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| [[Category:Homological algebra]]
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| [[Category:Lie algebras]]
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Nice meet up with you, i am Ambrose Schwan. The job she's been occupying remember is a data processing officer but she's always wanted her own business. I am really fond of to ride horses when compared to would never give upward. Texas is where he's been living for changing times. Check out her website here: http://www.vimeo.com/91956936
Feel free to visit my weblog :: stoya (www.vimeo.com)