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''in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813901.png" />''
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A homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813902.png" /> of a [[Lie algebra|Lie algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813903.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813904.png" /> into the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813905.png" /> of all linear transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813906.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813907.png" />. Two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813909.png" /> are called equivalent (or isomorphic) if there is an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139010.png" /> for which
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139011.png" /></td> </tr></table>
+
''in a vector space  $  V $''
  
for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139013.png" />. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139014.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139015.png" /> is called finite-dimensional if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139016.png" />, and irreducible if there are no subspaces in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139017.png" />, distinct from the null subspace and all of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139018.png" />, that are invariant under all operators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139020.png" />.
+
A homomorphism  $  \rho $
 +
of a [[Lie algebra|Lie algebra]]  $  L $
 +
over a field  $  k $
 +
into the algebra  $  \mathfrak g \mathfrak l ( V) $
 +
of all linear transformations of  $  V $
 +
over  $  k $.  
 +
Two representations  $  \rho _ {1} : L \rightarrow \mathfrak g \mathfrak l ( V _ {1} ) $
 +
and  $  \rho _ {2} : L \rightarrow \mathfrak g \mathfrak l ( V _ {2} ) $
 +
are called equivalent (or isomorphic) if there is an isomorphism  $  \alpha : V _ {1} \rightarrow V _ {2} $
 +
for which
  
For given representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139022.png" /> one constructs the representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139023.png" /> (the direct sum) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139024.png" /> (the tensor product) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139025.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139027.png" />, by putting
+
$$
 +
\alpha ( \rho _ {1} ( l) v _ {1} )  = \
 +
\rho _ {2} ( l) \alpha ( v _ {1} )
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139028.png" /></td> </tr></table>
+
for arbitrary  $  l \in L $,
 +
$  v _ {1} \in V _ {1} $.
 +
A representation  $  \rho $
 +
in  $  V $
 +
is called finite-dimensional if  $  \mathop{\rm dim}  V < \infty $,
 +
and irreducible if there are no subspaces in  $  V $,
 +
distinct from the null subspace and all of  $  V $,
 +
that are invariant under all operators  $  \rho ( l) $,
 +
$  l \in L $.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139029.png" /></td> </tr></table>
+
For given representations  $  \rho _ {1} : L \rightarrow \mathfrak g \mathfrak l ( V _ {1} ) $
 +
and  $  \rho _ {2} : L \rightarrow \mathfrak g \mathfrak l ( V _ {2} ) $
 +
one constructs the representations  $  \rho _ {1} \oplus \rho _ {2} $(
 +
the direct sum) and  $  \rho _ {1} \otimes \rho _ {2} $(
 +
the tensor product) of  $  L $
 +
into  $  V _ {1} \oplus V _ {2} $
 +
and  $  V _ {1} \otimes V _ {2} $,
 +
by putting
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139032.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139033.png" /> is a representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139034.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139035.png" />, then the formula
+
$$
 +
( \rho _ {1} \oplus \rho _ {2} )
 +
( l) ( v _ {1} , v _ {2} )  = \
 +
( \rho _ {1} ( l) v _ {1} ,\
 +
\rho _ {2} ( l) v _ {2} ),
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139036.png" /></td> </tr></table>
+
$$
 +
( \rho _ {1} \otimes \rho _ {2} ) ( l) v _ {1} \otimes v _ {2}  = \rho _ {1} ( l) v _ {1} \otimes v _ {2} + v _ {1} \otimes \rho _ {2} ( l) v _ {2} ,
 +
$$
  
defines a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139037.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139038.png" /> in the space dual to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139039.png" />; it is called the contragredient representation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139040.png" />.
+
where  $  v _ {1} \in V _ {1} $,
 +
$  v _ {2} \in V _ {2} $,
 +
$  l \in L $.
 +
If  $  \rho $
 +
is a representation of $  L $
 +
in $  V $,
 +
then the formula
  
Every representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139041.png" /> can be uniquely extended to a representation of the [[Universal enveloping algebra|universal enveloping algebra]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139042.png" />; this gives an isomorphism between the category of representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139043.png" /> and the category of modules over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139044.png" />. In particular, to a representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139045.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139046.png" /> corresponds the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139047.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139048.png" /> — the kernel of the extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139049.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139050.png" /> is irreducible, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139051.png" /> is a primitive ideal. Conversely, every primitive ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139052.png" /> can be obtained in this manner from an (in general, non-unique) irreducible representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139054.png" />. The study of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139055.png" /> of primitive ideals, endowed with the Jacobson topology, is an essential part of the representation theory of Lie algebras. It has been studied completely in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139056.png" /> is a finite-dimensional solvable algebra and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139057.png" /> is an algebraically closed field of characteristic zero (cf. [[#References|[2]]]).
+
$$
 +
\langle  \rho  ^ {*} ( l) u , v \rangle  = - \langle  u , \rho ( l) v \rangle
 +
$$
  
Finite-dimensional representations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero have been studied most extensively [[#References|[6]]], [[#References|[3]]], [[#References|[5]]]. When the field is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139058.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139059.png" />, these representations are in one-to-one correspondence with the analytic finite-dimensional representations of the corresponding simply-connected (complex or real) Lie group. In this case every representation of a solvable Lie algebra contains a one-dimensional invariant subspace (cf. [[Lie theorem|Lie theorem]]). Any representation of a semi-simple Lie algebra is totally reduced, i.e. is isomorphic to a direct sum of irreducible representations. The irreducible representations of a semi-simple Lie algebra have been completely classified: the classes of isomorphic representations correspond one-to-one to the dominant weights; here, a weight, i.e. an element of the dual space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139060.png" /> of a Cartan subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139061.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139062.png" />, is called dominant if its values on a canonical basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139063.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139064.png" /> are non-negative integers (cf. [[Cartan theorem|Cartan theorem]] on the highest weight vector). For a description of the structure of an irreducible representation by its corresponding dominant weight (its highest weight) see [[Multiplicity of a weight|Multiplicity of a weight]]; [[Character formula|Character formula]].
+
defines a representation $  \rho  ^ {*} $
 +
of $  L $
 +
in the space dual to $  V $;  
 +
it is called the contragredient representation with respect to  $  \rho $.
  
An arbitrary element (not necessarily a dominant weight) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139065.png" /> also determines an irreducible linear representation of a semi-simple Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139066.png" /> with highest weight <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139067.png" />. This representation is, however, infinite-dimensional (cf. [[Representation with a highest weight vector|Representation with a highest weight vector]]). The corresponding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139068.png" />-modules are called Verma modules (cf. [[#References|[2]]]). A complete classification of the irreducible infinite-dimensional representations of semi-simple Lie algebras has not yet been obtained (1991).
+
Every representation of  $  L $
 +
can be uniquely extended to a representation of the [[Universal enveloping algebra|universal enveloping algebra]]  $  U ( L) $;
 +
this gives an isomorphism between the category of representations of  $  L $
 +
and the category of modules over  $  U ( L) $.
 +
In particular, to a representation  $  \rho $
 +
of  $  L $
 +
corresponds the ideal  $  \mathop{\rm ker}  \widetilde \rho  $
 +
in  $  U ( L) $—
 +
the kernel of the extension  $  \widetilde \rho  $.  
 +
If  $  \rho $
 +
is irreducible,  $  \mathop{\rm ker}  \widetilde \rho  $
 +
is a primitive ideal. Conversely, every primitive ideal in  $  U ( L) $
 +
can be obtained in this manner from an (in general, non-unique) irreducible representation $  \rho $
 +
of $  L $.  
 +
The study of the space  $  \mathop{\rm Prim}  U ( L) $
 +
of primitive ideals, endowed with the Jacobson topology, is an essential part of the representation theory of Lie algebras. It has been studied completely in case  $  L $
 +
is a finite-dimensional solvable algebra and  $  k $
 +
is an algebraically closed field of characteristic zero (cf. [[#References|[2]]]).
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139069.png" /> is an algebraically closed field of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139070.png" />, then irreducible representations of a finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139071.png" /> are always finite-dimensional and their dimensions are bounded by a constant depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139072.png" />. If the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139073.png" /> has a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139074.png" />-structure (cf. [[Lie p-algebra|Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139075.png" />-algebra]]), then the constant is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139076.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139077.png" /> is the minimum possible dimension of an annihilator of a linear form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139078.png" /> in the co-adjoint representation [[#References|[4]]]. The following construction is used for the description of the set of irreducible representations in this case. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139079.png" /> be the centre of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139080.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139081.png" /> be the affine algebraic variety (of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139082.png" />) whose algebra of regular functions coincides with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139083.png" /> (a Zassenhaus variety). The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139084.png" /> makes it possible to assign a point on the Zassenhaus variety to each irreducible representation. The mapping thus obtained is surjective, the pre-image of any point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139085.png" /> is finite and for the points of an open everywhere-dense subset this pre-image consists of one element [[#References|[7]]]. A complete description of all irreducible representations has been obtained for nilpotent Lie algebras (cf. [[#References|[8]]]) and certain individual examples (cf. [[#References|[9]]], [[#References|[10]]]). Most varied results have also been obtained for special types of representations.
+
Finite-dimensional representations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero have been studied most extensively [[#References|[6]]], [[#References|[3]]], [[#References|[5]]]. When the field is  $  \mathbf R $
 +
or  $  \mathbf C $,
 +
these representations are in one-to-one correspondence with the analytic finite-dimensional representations of the corresponding simply-connected (complex or real) Lie group. In this case every representation of a solvable Lie algebra contains a one-dimensional invariant subspace (cf. [[Lie theorem|Lie theorem]]). Any representation of a semi-simple Lie algebra is totally reduced, i.e. is isomorphic to a direct sum of irreducible representations. The irreducible representations of a semi-simple Lie algebra have been completely classified: the classes of isomorphic representations correspond one-to-one to the dominant weights; here, a weight, i.e. an element of the dual space  $  H  ^  \star  $
 +
of a Cartan subalgebra  $  H $
 +
of  $  L $,
 +
is called dominant if its values on a canonical basis  $  h _ {1} \dots h _ {r} $
 +
of  $  H $
 +
are non-negative integers (cf. [[Cartan theorem|Cartan theorem]] on the highest weight vector). For a description of the structure of an irreducible representation by its corresponding dominant weight (its highest weight) see [[Multiplicity of a weight|Multiplicity of a weight]]; [[Character formula|Character formula]].
 +
 
 +
An arbitrary element (not necessarily a dominant weight)  $  \lambda \in H  ^ {*} $
 +
also determines an irreducible linear representation of a semi-simple Lie algebra  $  L $
 +
with highest weight  $  \lambda $.
 +
This representation is, however, infinite-dimensional (cf. [[Representation with a highest weight vector|Representation with a highest weight vector]]). The corresponding  $  U ( L) $-
 +
modules are called Verma modules (cf. [[#References|[2]]]). A complete classification of the irreducible infinite-dimensional representations of semi-simple Lie algebras has not yet been obtained (1991).
 +
 
 +
If  $  k $
 +
is an algebraically closed field of characteristic $  p > 0 $,  
 +
then irreducible representations of a finite-dimensional Lie algebra $  L $
 +
are always finite-dimensional and their dimensions are bounded by a constant depending on $  n = \mathop{\rm dim}  L $.  
 +
If the algebra $  L $
 +
has a $  p $-
 +
structure (cf. [[Lie p-algebra|Lie $  p $-
 +
algebra]]), then the constant is $  p ^ {( n - r)/2 } $,  
 +
where r $
 +
is the minimum possible dimension of an annihilator of a linear form on $  L $
 +
in the co-adjoint representation [[#References|[4]]]. The following construction is used for the description of the set of irreducible representations in this case. Let $  Z ( L) $
 +
be the centre of $  U ( L) $
 +
and let $  M _ {L} $
 +
be the affine algebraic variety (of dimension $  \mathop{\rm dim}  M _ {L} = n $)  
 +
whose algebra of regular functions coincides with $  Z ( L) $(
 +
a Zassenhaus variety). The mapping $  \rho \mapsto  \mathop{\rm ker} ( \rho \mid  _ {Z( L) }  ) $
 +
makes it possible to assign a point on the Zassenhaus variety to each irreducible representation. The mapping thus obtained is surjective, the pre-image of any point of $  M _ {L} $
 +
is finite and for the points of an open everywhere-dense subset this pre-image consists of one element [[#References|[7]]]. A complete description of all irreducible representations has been obtained for nilpotent Lie algebras (cf. [[#References|[8]]]) and certain individual examples (cf. [[#References|[9]]], [[#References|[10]]]). Most varied results have also been obtained for special types of representations.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Dixmier,  "Enveloping algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.A. Mil'ner,  "Maximal degree of irreducible Lie algebra representations over a field of positive characteristic"  ''Funct. Anal. Appl.'' , '''14''' :  2  (1980)  pp. 136–137  ''Funkts. Anal. i Prilozhen.'' , '''14''' :  2  (1980)  pp. 67–68</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Secr. Math. Univ. Paris  (1955)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Zassenhaus,  "The representations of Lie algebras of prime characteristic"  ''Proc. Glasgow Math. Assoc.'' , '''2'''  (1954)  pp. 1–36</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B.Yu. Veisfeiler,  V.G. Kats,  "Irreducible representations of Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139086.png" />-algebras"  ''Funct. Anal. Appl.'' , '''5''' :  2  (1971)  pp. 111–117  ''Funkts. Anal. i Prilozhen.'' , '''5''' :  2  (1971)  pp. 28–36</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J.C. Jantzen,  "Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren"  ''Math. Z.'' , '''140''' :  1  (1974)  pp. 127–149</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.N. Rudakov,  "On the representation of the classical Lie algebras in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139087.png" />"  ''Math. USSR Izv.'' , '''4'''  (1970)  pp. 741–749  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''34''' :  4  (1970)  pp. 735–743</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Dixmier,  "Enveloping algebras" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  N. Jacobson,  "Lie algebras" , Interscience  (1962)  ((also: Dover, reprint, 1979))</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  A.A. Mil'ner,  "Maximal degree of irreducible Lie algebra representations over a field of positive characteristic"  ''Funct. Anal. Appl.'' , '''14''' :  2  (1980)  pp. 136–137  ''Funkts. Anal. i Prilozhen.'' , '''14''' :  2  (1980)  pp. 67–68</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> , ''Theórie des algèbres de Lie. Topologie des groupes de Lie'' , ''Sem. S. Lie'' , '''Ie année 1954–1955''' , Secr. Math. Univ. Paris  (1955)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  H. Zassenhaus,  "The representations of Lie algebras of prime characteristic"  ''Proc. Glasgow Math. Assoc.'' , '''2'''  (1954)  pp. 1–36</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  B.Yu. Veisfeiler,  V.G. Kats,  "Irreducible representations of Lie <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139086.png" />-algebras"  ''Funct. Anal. Appl.'' , '''5''' :  2  (1971)  pp. 111–117  ''Funkts. Anal. i Prilozhen.'' , '''5''' :  2  (1971)  pp. 28–36</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  J.C. Jantzen,  "Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren"  ''Math. Z.'' , '''140''' :  1  (1974)  pp. 127–149</TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top">  A.N. Rudakov,  "On the representation of the classical Lie algebras in characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139087.png" />"  ''Math. USSR Izv.'' , '''4'''  (1970)  pp. 741–749  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''34''' :  4  (1970)  pp. 735–743</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
For a study of Prim <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139088.png" /> for semi-simple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r08139089.png" />, see [[#References|[a2]]].
+
For a study of Prim $  U( L) $
 +
for semi-simple $  L $,  
 +
see [[#References|[a2]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Humphreys,  "Introduction to Lie algebras and representation theory" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. Jantzen,  "Einhüllende Algebren halbeinfacher Lie-Algebren" , Springer  (1983)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J.E. Humphreys,  "Introduction to Lie algebras and representation theory" , Springer  (1972)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.C. Jantzen,  "Einhüllende Algebren halbeinfacher Lie-Algebren" , Springer  (1983)</TD></TR></table>

Latest revision as of 08:11, 6 June 2020


in a vector space $ V $

A homomorphism $ \rho $ of a Lie algebra $ L $ over a field $ k $ into the algebra $ \mathfrak g \mathfrak l ( V) $ of all linear transformations of $ V $ over $ k $. Two representations $ \rho _ {1} : L \rightarrow \mathfrak g \mathfrak l ( V _ {1} ) $ and $ \rho _ {2} : L \rightarrow \mathfrak g \mathfrak l ( V _ {2} ) $ are called equivalent (or isomorphic) if there is an isomorphism $ \alpha : V _ {1} \rightarrow V _ {2} $ for which

$$ \alpha ( \rho _ {1} ( l) v _ {1} ) = \ \rho _ {2} ( l) \alpha ( v _ {1} ) $$

for arbitrary $ l \in L $, $ v _ {1} \in V _ {1} $. A representation $ \rho $ in $ V $ is called finite-dimensional if $ \mathop{\rm dim} V < \infty $, and irreducible if there are no subspaces in $ V $, distinct from the null subspace and all of $ V $, that are invariant under all operators $ \rho ( l) $, $ l \in L $.

For given representations $ \rho _ {1} : L \rightarrow \mathfrak g \mathfrak l ( V _ {1} ) $ and $ \rho _ {2} : L \rightarrow \mathfrak g \mathfrak l ( V _ {2} ) $ one constructs the representations $ \rho _ {1} \oplus \rho _ {2} $( the direct sum) and $ \rho _ {1} \otimes \rho _ {2} $( the tensor product) of $ L $ into $ V _ {1} \oplus V _ {2} $ and $ V _ {1} \otimes V _ {2} $, by putting

$$ ( \rho _ {1} \oplus \rho _ {2} ) ( l) ( v _ {1} , v _ {2} ) = \ ( \rho _ {1} ( l) v _ {1} ,\ \rho _ {2} ( l) v _ {2} ), $$

$$ ( \rho _ {1} \otimes \rho _ {2} ) ( l) v _ {1} \otimes v _ {2} = \rho _ {1} ( l) v _ {1} \otimes v _ {2} + v _ {1} \otimes \rho _ {2} ( l) v _ {2} , $$

where $ v _ {1} \in V _ {1} $, $ v _ {2} \in V _ {2} $, $ l \in L $. If $ \rho $ is a representation of $ L $ in $ V $, then the formula

$$ \langle \rho ^ {*} ( l) u , v \rangle = - \langle u , \rho ( l) v \rangle $$

defines a representation $ \rho ^ {*} $ of $ L $ in the space dual to $ V $; it is called the contragredient representation with respect to $ \rho $.

Every representation of $ L $ can be uniquely extended to a representation of the universal enveloping algebra $ U ( L) $; this gives an isomorphism between the category of representations of $ L $ and the category of modules over $ U ( L) $. In particular, to a representation $ \rho $ of $ L $ corresponds the ideal $ \mathop{\rm ker} \widetilde \rho $ in $ U ( L) $— the kernel of the extension $ \widetilde \rho $. If $ \rho $ is irreducible, $ \mathop{\rm ker} \widetilde \rho $ is a primitive ideal. Conversely, every primitive ideal in $ U ( L) $ can be obtained in this manner from an (in general, non-unique) irreducible representation $ \rho $ of $ L $. The study of the space $ \mathop{\rm Prim} U ( L) $ of primitive ideals, endowed with the Jacobson topology, is an essential part of the representation theory of Lie algebras. It has been studied completely in case $ L $ is a finite-dimensional solvable algebra and $ k $ is an algebraically closed field of characteristic zero (cf. [2]).

Finite-dimensional representations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero have been studied most extensively [6], [3], [5]. When the field is $ \mathbf R $ or $ \mathbf C $, these representations are in one-to-one correspondence with the analytic finite-dimensional representations of the corresponding simply-connected (complex or real) Lie group. In this case every representation of a solvable Lie algebra contains a one-dimensional invariant subspace (cf. Lie theorem). Any representation of a semi-simple Lie algebra is totally reduced, i.e. is isomorphic to a direct sum of irreducible representations. The irreducible representations of a semi-simple Lie algebra have been completely classified: the classes of isomorphic representations correspond one-to-one to the dominant weights; here, a weight, i.e. an element of the dual space $ H ^ \star $ of a Cartan subalgebra $ H $ of $ L $, is called dominant if its values on a canonical basis $ h _ {1} \dots h _ {r} $ of $ H $ are non-negative integers (cf. Cartan theorem on the highest weight vector). For a description of the structure of an irreducible representation by its corresponding dominant weight (its highest weight) see Multiplicity of a weight; Character formula.

An arbitrary element (not necessarily a dominant weight) $ \lambda \in H ^ {*} $ also determines an irreducible linear representation of a semi-simple Lie algebra $ L $ with highest weight $ \lambda $. This representation is, however, infinite-dimensional (cf. Representation with a highest weight vector). The corresponding $ U ( L) $- modules are called Verma modules (cf. [2]). A complete classification of the irreducible infinite-dimensional representations of semi-simple Lie algebras has not yet been obtained (1991).

If $ k $ is an algebraically closed field of characteristic $ p > 0 $, then irreducible representations of a finite-dimensional Lie algebra $ L $ are always finite-dimensional and their dimensions are bounded by a constant depending on $ n = \mathop{\rm dim} L $. If the algebra $ L $ has a $ p $- structure (cf. Lie $ p $- algebra), then the constant is $ p ^ {( n - r)/2 } $, where $ r $ is the minimum possible dimension of an annihilator of a linear form on $ L $ in the co-adjoint representation [4]. The following construction is used for the description of the set of irreducible representations in this case. Let $ Z ( L) $ be the centre of $ U ( L) $ and let $ M _ {L} $ be the affine algebraic variety (of dimension $ \mathop{\rm dim} M _ {L} = n $) whose algebra of regular functions coincides with $ Z ( L) $( a Zassenhaus variety). The mapping $ \rho \mapsto \mathop{\rm ker} ( \rho \mid _ {Z( L) } ) $ makes it possible to assign a point on the Zassenhaus variety to each irreducible representation. The mapping thus obtained is surjective, the pre-image of any point of $ M _ {L} $ is finite and for the points of an open everywhere-dense subset this pre-image consists of one element [7]. A complete description of all irreducible representations has been obtained for nilpotent Lie algebras (cf. [8]) and certain individual examples (cf. [9], [10]). Most varied results have also been obtained for special types of representations.

References

[1] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[2] J. Dixmier, "Enveloping algebras" , North-Holland (1977) (Translated from French)
[3] N. Jacobson, "Lie algebras" , Interscience (1962) ((also: Dover, reprint, 1979))
[4] A.A. Mil'ner, "Maximal degree of irreducible Lie algebra representations over a field of positive characteristic" Funct. Anal. Appl. , 14 : 2 (1980) pp. 136–137 Funkts. Anal. i Prilozhen. , 14 : 2 (1980) pp. 67–68
[5] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[6] , Theórie des algèbres de Lie. Topologie des groupes de Lie , Sem. S. Lie , Ie année 1954–1955 , Secr. Math. Univ. Paris (1955)
[7] H. Zassenhaus, "The representations of Lie algebras of prime characteristic" Proc. Glasgow Math. Assoc. , 2 (1954) pp. 1–36
[8] B.Yu. Veisfeiler, V.G. Kats, "Irreducible representations of Lie -algebras" Funct. Anal. Appl. , 5 : 2 (1971) pp. 111–117 Funkts. Anal. i Prilozhen. , 5 : 2 (1971) pp. 28–36
[9] J.C. Jantzen, "Zur Charakterformel gewisser Darstellungen halbeinfacher Gruppen und Lie-Algebren" Math. Z. , 140 : 1 (1974) pp. 127–149
[10] A.N. Rudakov, "On the representation of the classical Lie algebras in characteristic " Math. USSR Izv. , 4 (1970) pp. 741–749 Izv. Akad. Nauk SSSR Ser. Mat. , 34 : 4 (1970) pp. 735–743

Comments

For a study of Prim $ U( L) $ for semi-simple $ L $, see [a2].

References

[a1] J.E. Humphreys, "Introduction to Lie algebras and representation theory" , Springer (1972)
[a2] J.C. Jantzen, "Einhüllende Algebren halbeinfacher Lie-Algebren" , Springer (1983)
How to Cite This Entry:
Representation of a Lie algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Representation_of_a_Lie_algebra&oldid=48518
This article was adapted from an original article by A.N. Rudakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article