View source for Representation of a Lie algebra

''in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081390/r0813901.png" />''

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

<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>

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" />.

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

<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>

<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>

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

<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>

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" />.

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]]]).

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]].

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).

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.

====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>



====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]]].

====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>