Representation of a Lie algebra

in a vector space

A homomorphism of a Lie algebra over a field into the algebra of all linear transformations of over . Two representations and are called equivalent (or isomorphic) if there is an isomorphism for which

for arbitrary , . A representation in is called finite-dimensional if , and irreducible if there are no subspaces in , distinct from the null subspace and all of , that are invariant under all operators , .

For given representations and one constructs the representations (the direct sum) and (the tensor product) of into and , by putting

where , , . If is a representation of in , then the formula

defines a representation of in the space dual to ; it is called the contragredient representation with respect to .

Every representation of can be uniquely extended to a representation of the universal enveloping algebra ; this gives an isomorphism between the category of representations of and the category of modules over . In particular, to a representation of corresponds the ideal in — the kernel of the extension . If is irreducible, is a primitive ideal. Conversely, every primitive ideal in can be obtained in this manner from an (in general, non-unique) irreducible representation of . The study of the space 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 is a finite-dimensional solvable algebra and 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 or , 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 of a Cartan subalgebra of , is called dominant if its values on a canonical basis of 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) also determines an irreducible linear representation of a semi-simple Lie algebra with highest weight . This representation is, however, infinite-dimensional (cf. Representation with a highest weight vector). The corresponding -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 is an algebraically closed field of characteristic , then irreducible representations of a finite-dimensional Lie algebra are always finite-dimensional and their dimensions are bounded by a constant depending on . If the algebra has a -structure (cf. Lie -algebra), then the constant is , where is the minimum possible dimension of an annihilator of a linear form on in the co-adjoint representation [4]. The following construction is used for the description of the set of irreducible representations in this case. Let be the centre of and let be the affine algebraic variety (of dimension ) whose algebra of regular functions coincides with (a Zassenhaus variety). The mapping 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 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 for semi-simple , 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)