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Representation of a Lie algebra

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in a vector space

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