Representation of a Lie algebra

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

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