Definition 2

Let g be a Lie algebra over K and U its enveloping algebra. A unitary left module over U is called a left g-module, or simply a g-module.

If M is a g-module and x ∈ U, x_M will denote the homothety of M defined by x (cf. Algebra, Chapter VIII, § 1, no. 2).

A unitary right module over U is called a right g-module. Such a module is identified with a left U^0-module, that is (§ 2, no. 4) with a left g^0-module.

Let φ be the principal antiautomorphism of U. If M is a right g-module, a left g-module structure is defined on M by writing a.m = m.φ(a) for m ∈ M and a ∈ U.

The notions and results of the theory of modules can be translated into the language of representations:

(1) Two representations ρ and ρ' of g on M and M' are called similar or isomorphic if the g-modules M and M' are isomorphic. For this it is necessary and sufficient that there exist an isomorphism u of the K-module M onto the K-module M' such that

for all x ∈ g.

(2) For all i ∈ I, let ρ_i be a representation of g on M_i. Let M be the g-module the direct sum of the g-modules M_i. There is a corresponding representation ρ of g on M, called the direct sum of the ρ_i and denoted by $\sum_{i \in I} \rho_i$ (or $\rho_1 + \cdots + \rho_n$ in the case of n representations $\rho_1, \ldots, \rho_n$). If $m = (m_i)_{i \in I}$ is an element of M and x ∈ g, then $\rho(x).m = (\rho_i(x).m_i)_{i \in I}$.

(3) A representation ρ of g on M is called simple or irreducible if the associated g-module is simple. It amounts to the same to say that there exists no sub-K-module of M (other than {0} and M) stable under all the $\rho(x), x \in g$. A class of simple g-modules (Algebra, Chapter VIII, § 3, no. 2) defines a class of simple representations of g.

(4) A representation ρ of g on M is called semi-simple or completely reducible if the associated g-module is semi-simple. It amounts to the same to say that ρ is similar to a direct sum of simple representations or that every sub-K-module of

M stable under the $\rho(x)$ ($x \in g$) has a supplement stable under the $\rho(x)$ ($x \in g$) (cf. Algebra, Chapter VIII, § 3, no. 3).

(5) Let $\delta$ be a class of simple representations of $g$ corresponding to a class C of simple $g$-modules. On the other hand let $\rho$ be a representation of $g$ on M. The isotypical component $M_C$ of species C of the $g$-module M (Algebra, Chapter VIII, § 3, no. 4) is also called the isotypical component of M of species $\delta$. This component is the sum of the sub-K-modules of M stable under the $\rho(x)$ and on which the $\rho(x)$ induce a representation of class $\delta$; it is the direct sum of certain of these submodules; if $M_C$ is of length $n$, $\rho$ is said to contain $\delta$ n times. The sum of the different $M_C$ is direct; it is equal to M if and only if $\rho$ is semi-simple.

(6) Let $\rho, \rho'$ be two representations of $g$. $\rho'$ is called a subrepresentation (resp. quotient representation) of $\rho$ if the module of $\rho'$ is a submodule (resp. quotient module) of the module of $\rho$.

Let M be a K-module. The zero representation of $g$ on M defines on M a $g$-module structure. With this structure M is called a trivial $g$-module.

Let M be a $g$-module. The quotient $g$-modules of the sub-$g$-modules of M are also the sub-$g$-modules of the quotient modules of M: they are obtained by considering two sub-$g$-modules U, $U'$ of M such that $U \supset U'$ and forming the $g$-module $U/U'$. Then if all the simple modules of the above type are isomorphic to a given simple $g$-module N, M is called a pure $g$-module of species N. If $\rho$ and $\sigma$ are the representations of $g$ corresponding to M and N, we also say that $\rho$ is pure of species $\sigma$.

Let $M'$ be a sub-$g$-module of M. For M to be pure of species N, it is necessary and sufficient that $M'$ and $M/M'$ be pure of species N. For the condition is obviously necessary. Suppose that it holds and let U, $U'$ be sub-$g$-modules of M such that $U' \subset U$ and $U/U'$ is simple; let $\phi$ be the canonical homomorphism of M onto $M/M'$; if $\phi(U) \neq \phi(U')$, $U/U'$ is isomorphism to $\phi(U)/\phi(U')$ and hence isomorphic to N; if $\phi(U) = \phi(U')$, then $U \subset U' + M'$, hence $U/U'$ is isomorphic to a simple submodule of $(U' + M')/U'$ and the latter module is itself isomorphic to $M'/(U' \cap M')$; hence $U/U'$ is again isomorphic to N, so that M is pure of species N.

Henceforth let M be a $g$-module and suppose that the set of sub-$g$-modules of M which are pure of species N admits a maximal element $M'$. Then every submodule $M''$ of M which is pure of species N is contained in $M'$. For $M''/(M' \cap M'')$ and $M'$ are pure of species N, hence $M' + M''$ is pure of species N by the above and hence $M' + M'' \subset M'$.

Suppose that the $g$-module M admits a Jordan-Hölder series $(M_i)_{0 \leq i \leq n}$. For M to be pure of species N, it is necessary and sufficient that $M_0/M_1, M_1/M_2, \ldots, M_{n-1}/M_n$ be isomorphic to N; for the condition is obviously necessary and its sufficiency follows immediately by induction on $n$ from what we have seen above.