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{{Lie groups |Representation}}

In the [[mathematics|mathematical]] field of [[representation theory]], a '''Lie algebra representation''' or '''representation of a Lie algebra''' is a way of writing a [[Lie algebra]] as a set of [[matrix (mathematics)|matrices]] (or [[endomorphism]]s of a [[vector space]]) in such a way that the Lie bracket is given by the [[commutator]].

The notion is closely related to that of a [[representation of a Lie group]]. Roughly speaking, the representations of Lie algebras are the differentiated form of representations of Lie groups, while the representations of the [[universal cover]] of a Lie group are the integrated form of the representations of its Lie algebra.

In the study of representations of a Lie algebra, a particular [[ring (mathematics)|ring]], called the [[universal enveloping algebra]], associated with the Lie algebra plays a decisive role. The universality of this construction of this ring says that the category of representations of a Lie algebra is the same as the category of modules over its enveloping algebra.

==Formal definition==
A '''representation''' of a [[Lie algebra]] <math>\mathfrak g</math> is a [[Lie algebra homomorphism]]
:<math>\rho\colon \mathfrak g \to \mathfrak{gl}(V)</math>
from <math>\mathfrak g</math> to the Lie algebra of [[endomorphism]]s on a [[vector space]] ''V'' (with the [[commutator]] as the Lie bracket), sending an element ''x'' of <math>\mathfrak g</math> to an element ''ρ''''x'' of <math>\mathfrak{gl}(V)</math>.

Explicitly, this means that
:<math>\rho_{[x,y]} = [\rho_x,\rho_y] = \rho_x\rho_y - \rho_y\rho_x\,</math>
for all ''x,y'' in <math>\mathfrak g</math>. The vector space ''V'', together with the representation ρ, is called a '''<math>\mathfrak g</math>-module'''. (Many authors abuse terminology and refer to ''V'' itself as the representation).

The representation <math>\rho</math> is said to be '''faithful''' if it is injective.

One can equivalently define a <math>\mathfrak g</math>-module as a vector space ''V'' together with a [[bilinear map]] <math>\mathfrak g \times V\to V</math> such that
:<math>[x,y]\cdot v = x\cdot(y\cdot v) - y\cdot(x\cdot v)</math>
for all ''x,y'' in <math>\mathfrak g</math> and ''v'' in ''V''. This is related to the previous definition by setting ''x'' ⋅ ''v'' = ρ''x'' (v).

==Examples ==
===Adjoint representations===
{{main|Adjoint representation of a Lie algebra}}
The most basic example of a Lie algebra representation is the adjoint representation of a Lie algebra <math>\mathfrak{g}</math> on itself:
:<math>\textrm{ad}:\mathfrak{g} \to \mathfrak{gl}(\mathfrak{g}), \quad x \mapsto \operatorname{ad}_x, \quad \operatorname{ad}_x(y) = [x, y].</math>
Indeed, by virtue of the [[Jacobi identity]], <math>\operatorname{ad}</math> is a Lie algebra homomorphism.

===Infinitesimal Lie group representations===
A Lie algebra representation also arises in nature. If φ: ''G'' → ''H'' is a [[homomorphism]] of (real or complex) [[Lie group]]s, and <math>\mathfrak g</math> and <math>\mathfrak h</math> are the [[Lie algebra]]s of ''G'' and ''H'' respectively, then the [[pushforward (differential)|differential]] <math>d \phi: \mathfrak g \to \mathfrak h</math> on [[tangent space]]s at the identities is a Lie algebra homomorphism. In particular, for a finite-dimensional vector space ''V'', a [[representation of Lie groups]]
:<math>\phi: G\to \mathrm{GL}(V)\,</math>
determines a Lie algebra homomorphism
:<math>d \phi: \mathfrak g \to \mathfrak{gl}(V)</math>
from <math>\mathfrak g</math> to the Lie algebra of the [[general linear group]] GL(''V''), i.e. the endomorphism algebra of ''V''.

For example, let <math>c_g(x) = gxg^{-1}</math>. Then the differential of <math>c_g: G \to G</math> at the identity is an element of <math>\mathrm{GL}(\mathfrak{g})</math>. Denoting it by <math>\operatorname{Ad}(g)</math> one obtains a representation <math>\operatorname{Ad}</math> of ''G'' on the vector space <math>\mathfrak{g}</math>. Applying the preceding, one gets the Lie algebra representation <math>d\operatorname{Ad}</math>. It can be shown that <math>d\operatorname{Ad} = \operatorname{ad}.</math>

A partial converse to this statement says that every representation of a finite-dimensional (real or complex) Lie algebra lifts to a unique representation of the associated [[simply connected]] Lie group, so that representations of simply-connected Lie groups are in one-to-one correspondence with representations of their Lie algebras.

== Basic concepts ==
Let <math>\mathfrak{g}</math> be a [[Lie algebra]]. Let ''V'', ''W'' be <math>\mathfrak{g}</math>-modules. Then a linear map <math>f: V \to W</math> is a homomorphism of <math>\mathfrak{g}</math>-modules if it is <math>\mathfrak{g}</math>-equivariant; i.e., <math>f(xv) = xf(v)</math> for any <math>x \in \mathfrak{g}, v \in V</math>. If ''f'' is bijective, <math>V, W</math> are said to be ''equivalent''. Similarly, many other constructions from module theory in abstract algebra carry over to this setting: submodule, quotient, subquotient, direct sum, Jordan-Hölder series, etc.

Let ''V'' be a <math>\mathfrak{g}</math>-module. Then ''V'' is said to be ''semisimple'' or ''completely reducible'' if it satisfies the following equivalent conditions: (cf. [[semisimple module]])
# ''V'' is a direct sum of simple modules.
# ''V'' is the sum of its simple submodules.
# Every submodule of ''V'' is a [[direct summand]]: for every submodule ''W'' of ''V'', there is a complement ''P'' such that ''V'' = ''W'' ⊕ ''P''.

If <math>\mathfrak{g}</math> is a finite-dimensional [[semisimple Lie algebra]] over a field of characteristic zero and ''V'' is finite-dimensional, then ''V'' is semisimple ([[Weyl's complete reducibility theorem]]).<ref>{{harvnb|Dixmier|1977|loc=Theorem 1.6.3}}</ref> A Lie algebra is said to be [[Reductive Lie algebra|reductive]] if the adjoint representation is semisimple. Thus, a semisimple Lie algebra is reductive. An element ''v'' of ''V'' is said to be <math>\mathfrak{g}</math>-invariant if <math>xv = v</math> for all <math>x \in \mathfrak{g}</math>. The set of all invariant elements is denoted by <math>V^\mathfrak{g}</math>. <math>V \mapsto V^\mathfrak{g}</math> is a left-exact functor.

==Basic constructions==
If we have two representations, with ''V''1 and ''V''2 as their underlying vector spaces and ·[·]1 and ·[·]2 as the representations, then the product of both representations would have ''V''1 ⊗ ''V''2 as the underlying vector space and

:<math>x[v_1\otimes v_2]=x[v_1]\otimes v_2+v_1\otimes x[v_2] .</math>

If ''L'' is a real Lie algebra and ρ: ''L'' × ''V''→ ''V'' is a complex representation of it, we can construct another representation of ''L'' called its dual representation as follows.

Let ''V''∗ be the dual vector space of ''V''. In other words, ''V''∗ is the set of all linear maps from ''V'' to '''C''' with addition defined over it in the usual linear way, but scalar multiplication defined over it such that <math>(z\omega)[X]=\bar{z}\omega[X]</math> for any ''z'' in '''C''', ω in ''V''∗ and ''X'' in ''V''. This is usually rewritten as a contraction with a [[sesquilinear]] form &lang;·,·&rang;. i.e. &lang;ω,''X''&rang; is defined to be ω[''X''].

We define <math>\bar{\rho}</math> as follows:

:&lang;<math>\bar{\rho}</math>(''A'')[ω],''X''&rang; + &lang;ω, ρ''A''[''X'']&rang; = 0,

for any ''A'' in ''L'', ω in ''V''∗ and ''X'' in ''V''. This defines <math>\bar{\rho}</math> uniquely.

Let <math>V, W</math> be <math>\mathfrak{g}</math>-modules, <math>\mathfrak{g}</math> a Lie algebra. Then <math>\operatorname{Hom}(V, W)</math> becomes a <math>\mathfrak{g}</math>-module by setting <math>(x \cdot f)(v) = x f(v) - f (x v)</math>. In particular, <math>\operatorname{Hom}_\mathfrak{g}(V, W) = \operatorname{Hom}(V, W)^\mathfrak{g}</math>. Since any field becomes a <math>\mathfrak{g}</math>-module with a trivial action, taking ''W'' to be the base field, the dual vector space <math>V^*</math> becomes a <math>\mathfrak{g}</math>-module.

== Enveloping algebras ==
To each Lie algebra <math>\mathfrak{g}</math> over a field ''k'', one can associate a certain [[ring (mathematics)|ring]] called the [[universal enveloping algebra]] of <math>\mathfrak{g}</math>. The construction is universal and consequently (along with the PBW theorem) representations of <math>\mathfrak{g}</math> corresponds in one-to-one with [[algebra representation]]s of universal enveloping algebra of <math>\mathfrak{g}</math>. The construction is as follows.<ref>{{harvnb|Jacobson|1962}}</ref> Let ''T'' be the [[tensor algebra]] of the vector space <math>\mathfrak{g}</math>. Thus, by definition, <math>T = \oplus_{n=0}^\infty \otimes_1^n \mathfrak{g}</math> and the multiplication on it is given by <math>\otimes</math>. Let <math>U(\mathfrak{g})</math> be the [[quotient ring]] of ''T'' by the ideal generated by elements <math>[x, y] - x \otimes y + y \otimes x</math>. Since <math>U(\mathfrak{g})</math> is an associative [[algebra over a field|algebra]] over the field ''k'', it can be turned into a Lie algebra via the commutator <math>[x, y] = x y - yx</math> (omitting <math>\otimes</math> from the notation). There is a canonical morphism of Lie algebras <math>\mathfrak{g} \to U(\mathfrak{g})</math> obtained by restricting <math>T \to U(\mathfrak{g})</math> to degree one piece. The [[PBW theorem]] implies that the canonical map is actually injective. Note if <math>\mathfrak{g}</math> is [[abelian Lie algebra|abelian]], then <math>U(\mathfrak{g})</math> is the symmetric algebra of the vector space <math>\mathfrak{g}</math>.

Since <math>\mathfrak{g}</math> is a module over itself via adjoint representation, the enveloping algebra <math>U(\mathfrak{g})</math> becomes a <math>\mathfrak{g}</math>-module by extending the adjoint representation. But one can also use the left and right [[regular representation]] to make the enveloping algebra a <math>\mathfrak{g}</math>-module; namely, with the notation <math>l_x(y) = xy, x \in \mathfrak{g}, y \in U(\mathfrak{g})</math>, the mapping <math>x \mapsto l_x</math> defines a representation of <math>\mathfrak{g}</math> on <math>U(\mathfrak{g})</math>. The right regular representation is defined similarly.

== Induced representation ==
Let <math>\mathfrak{g}</math> be a finite-dimensional Lie algebra over a field of characteristic zero and <math>\mathfrak{h} \subset \mathfrak{g}</math> a subalgebra. <math>U(\mathfrak{h})</math> acts on <math>U(\mathfrak{g})</math> from the right and thus, for any <math>\mathfrak{h}</math>-module ''W'', one can form the left <math>U(\mathfrak{g})</math>-module <math>U(\mathfrak{g}) \otimes_{U(\mathfrak{h})} W</math>. It is a <math>\mathfrak{g}</math>-module denoted by <math>\operatorname{Ind}_\mathfrak{h}^\mathfrak{g} W</math> and called the <math>\mathfrak{g}</math>-module induced by ''W''. It satisfies (and is in fact characterized by) the universal property: for any <math>\mathfrak{g}</math>-module ''E''
:<math>\operatorname{Hom}_\mathfrak{g}(\operatorname{Ind}_\mathfrak{h}^\mathfrak{g} W, E) \simeq \operatorname{Hom}_\mathfrak{h}(W, \operatorname{Res}^\mathfrak{g}_\mathfrak{h} E)</math>.
Furthermore, <math>\operatorname{Ind}_\mathfrak{h}^\mathfrak{g}</math> is an exact functor from the category of <math>\mathfrak{h}</math>-modules to the category of <math>\mathfrak{g}</math>-modules. These uses the fact that <math>U(\mathfrak{g})</math> is a free right module over <math>U(\mathfrak{h})</math>. In particular, if <math>\operatorname{Ind}_\mathfrak{h}^\mathfrak{g} W</math> is simple (resp. absolutely simple), then ''W'' is simple (resp. absolutely simple). Here, a <math>\mathfrak{g}</math>-module ''V'' is absolutely simple if <math>V \otimes_k F</math> is simple for any field extension <math>F/k</math>.

The induction is transitive: <math>\operatorname{Ind}_\mathfrak{h}^\mathfrak{g} \simeq \operatorname{Ind}_\mathfrak{h'}^\mathfrak{g} \circ \operatorname{Ind}_\mathfrak{h}^\mathfrak{h'}</math>
for any Lie subalgebra <math>\mathfrak{h'} \subset \mathfrak{g}</math> and any Lie subalgebra <math>\mathfrak{h} \subset \mathfrak{h}'</math>. The induction commutes with restriction: let <math>\mathfrak{h} \subset \mathfrak{g}</math> be subalgebra and <math>\mathfrak{n}</math> an ideal of <math>\mathfrak{g}</math> that is contained in <math>\mathfrak{h}</math>. Set <math>\mathfrak{g}_1 = \mathfrak{g}/\mathfrak{n}</math> and <math>\mathfrak{h}_1 = \mathfrak{h}/\mathfrak{n}</math>. Then <math>\operatorname{Ind}^\mathfrak{g}_\mathfrak{h} \circ \operatorname{Res}_\mathfrak{h} \simeq \operatorname{Res}_\mathfrak{g} \circ \operatorname{Ind}^\mathfrak{g_1}_\mathfrak{h_1}</math>.

== Representations of a semisimple Lie algebra ==
Let <math>\mathfrak{g}</math> be a finite-dimensional semisimple Lie algebra over a field of characteristic zero. (in the solvable or nilpotent case, one studies [[primitive ideal]]s of the enveloping algebra; cf. Dixmier for the definitive account.)

The category of modules over <math>\mathfrak{g}</math> turns out to be too large especially for homological algebra methods to be useful: it was realized that a smaller subcategory [[category O]] is a better place for the representation theory in the semisimple case in zero characteristic. For instance, the category O turned out to be of a right size to formulate the celebrated [[BGG reciprocity]].<ref>http://mathoverflow.net/questions/64931/why-the-bgg-category-o</ref>

== <math>(\mathfrak{g}, K)</math>-module ==
{{main|(g,K)-module|Harish-Chandra module}}

One of the most important applications of Lie algebra representations is to the representation theory of real reductive Lie group. The application is based on the idea that if <math>\pi</math> is a Hilbert-space representation of, say, a connected real semisimple linear Lie group ''G'', then it has two natural actions: the complexification <math>\mathfrak{g}</math> and the connected [[maximal compact subgroup]] ''K''. The <math>\mathfrak{g}</math>-module structure of <math>\pi</math> allows algebraic especially homological methods to be applied and <math>K</math>-module structure allows harmonic analysis to be carried out in a way similar to that on connected compact semisimple Lie groups.

==Classification==
===Finite-dimensional representations of semisimple Lie algebras===
{{Expand section|date=December 2009}}
{{details|Weight (representation theory)}}
Similarly to how [[semisimple Lie algebra]]s can be classified, the finite-dimensional representations of semisimple Lie algebras can be classified. This is a classical theory, widely regarded as beautiful, and a standard reference is {{Harv|Fulton|Harris|1992}}.

Briefly, finite-dimensional representations of a semisimple Lie algebra are [[Semisimple Lie algebra|completely reducible]], so it suffices to classify irreducible (simple) representations. Semisimple Lie algebras are classified in terms of the [[Weight (representation theory)|weights]] of the adjoint representation, the so-called [[root system]]; in a similar manner all finite-dimensional irreducible representations can be understood in terms of weights; see [[weight (representation theory)]] for details.

==Representation on an algebra==
If we have a Lie superalgebra ''L'', then a representation of ''L'' on an algebra is a (not necessarily [[associative]]) [[graded algebra|'''Z'''2 graded]] [[algebra over a field|algebra]] ''A'' which is a representation of ''L'' as a '''Z'''2 [[graded vector space]] and in addition, the elements of ''L'' acts as [[Derivation (abstract algebra)|derivation]]s/[[antiderivation]]s on ''A''.

More specifically, if ''H'' is a [[pure element]] of ''L'' and ''x'' and ''y'' are [[pure element]]s of ''A'',

:''H''[''xy''] = (''H''[''x''])''y'' + (−1)''xH''''x''(''H''[''y''])

Also, if ''A'' is [[unital algebra|unital]], then

:''H''[1] = 0

Now, for the case of a '''representation of a Lie algebra''', we simply drop all the gradings and the (−1) to the some power factors.

A Lie (super)algebra is an algebra and it has an [[adjoint endomorphism|adjoint representation]] of itself. This is a representation on an algebra: the (anti)derivation property is the [[superJacobi identity|super]][[Jacobi identity]].

If a vector space is both an [[associative algebra]] and a [[Lie algebra]] and the adjoint representation of the Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a [[Poisson algebra]]. The analogous observation for Lie superalgebras gives the notion of a [[Poisson superalgebra]].

==See also==
*[[Quillen's lemma]] - analog of Schur's lemma
*[[Verma module]]
*[[Geometric quantization]]
*[[Kazhdan–Lusztig conjectures]]
*[[Representation of a Lie superalgebra]]
*[[Whitehead's lemma (Lie algebras)]]

== Notes ==
{{reflist}}

==References==
*Bernstein I.N., Gelfand I.M., Gelfand S.I., "Structure of Representations that are generated by vectors of highest weight," Functional. Anal. Appl. 5 (1971)
*{{citation|last=Dixmier|first=J.|title=Enveloping Algebras|publisher=North-Holland|publication-place=Amsterdam, New York, Oxford|year=1977|isbn=0-444-11077-1}}.
*A. Beilinson and J. Bernstein, "Localisation de g-modules," C. R. Acad. Sci. Paris Sér. I Math., vol. 292, iss. 1, pp. 15-18, 1981.
* {{Fulton-Harris}}
* D. Gaitsgory, [http://www.math.harvard.edu/~gaitsgde/267y/index.html Geometric Representation theory, Math 267y, Fall 2005]
* Ryoshi Hotta, Kiyoshi Takeuchi, Toshiyuki Tanisaki, ''D-modules, perverse sheaves, and representation theory''; translated by Kiyoshi Takeuch
* J.Humphreys, ''Introduction to Lie algebras and representation theory'', Birkhäuser, 2000.
*N. Jacobson, ''Lie algebras'', Courier Dover Publications, 1979.

{{DEFAULTSORT:Lie Algebra Representation}}
[[Category:Representation theory of Lie algebras]]

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