# 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. ).

Finite-dimensional representations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero have been studied most extensively , , . 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. ). 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 . 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 . A complete description of all irreducible representations has been obtained for nilpotent Lie algebras (cf. ) and certain individual examples (cf. , ). Most varied results have also been obtained for special types of representations.

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