Proposition 2

Let $g$ be a Lie algebra over $K$ and $M_i$ a $g$-module ($1 \leq i \leq n$). On the tensor product $M_1 \otimes_K M_2 \otimes \cdots \otimes M_n$, there exists one and only one $g$-module structure such that

for all $x \in g, m_1 \in M_1, \ldots, m_n \in M_n$.

The corresponding representation is called the tensor product of the given representations of $g$ on the $M_i$.

In particular, if $M$ is a $g$-module, Proposition 2 defines a $g$-module structure on each $M_p = \bigotimes^p M$ and hence on the tensor algebra $T$ of $M$.

Formula (3) shows that, for all $x \in g, x_T$ is the unique derivation of the algebra $T$ which extends $x_M$. We know (§ 2, no. 8) that $x_T$ defines on passing to the quotient a derivation of the symmetric algebra $S$ of $M$. Hence $S$ can be considered as a quotient $g$-module of $T$ and the $x_S$ are derivations of $S$.

Still more particularly, consider $g$ as a $g$-module by means of the adjoint representation of $g$. Let $U$ be the enveloping algebra of $g$. By Proposition 7 of § 2, $x_M$ defines on passing to the quotients a derivation of $U$ which is just the inner derivation defined by $\sigma(x)$ ($\sigma$ denoting the canonical mapping of $g$ into $U$). Then $U$ can be considered as a quotient $g$-module of $T$. If $K$ is a field of characteristic 0, the canonical isomorphism of $S$ onto $U$ is a $g$-module isomorphism (§ 2, no. 8).