Definition 1

Let $g$ be a Lie algebra over $K$, $T$ the tensor algebra of the $K$-module $g$ and $J$ the two-sided ideal of $T$ generated by the tensors $x \otimes y - y \otimes x - [x, y]$ where $x \in g, y \in g$. The associative algebra $U = T/J$ is called the enveloping algebra of $g$. The restriction to $g$ of the canonical mapping of $T$ onto $U$ is called the canonical mapping of $g$ into $U$.

Let $T_+$ be the two-sided ideal of $T$ consisting of the tensors whose component of order 0 is zero. Let $T_0 = K.1$ be the set of elements of $T$ of order 0. Let $U_+$ and $U_0$ be the canonical images of $T_+$ and $T_0$ in $U$. As $J \subset T_+$, the decomposition into a direct sum $T = T_0 + T_+$ implies a decomposition into a direct sum $U = U_0 + U_+$. The algebra $U$ therefore has a unit element distinct from 0 and $U_0 = K.1$. For all $x \in U$, the component of $x$ in $U_0$ is called the constant term of $x$. The elements with constant term zero form a two-sided ideal of $U$, namely the two-sided ideal $U^+$ generated by the canonical image of $g$ in $U$.

The associative algebra $U$ is generated by 1 and the canonical image of $g$ in $U$.

If $x \in g$ and $y \in g$, $x \otimes y - y \otimes x$ and $[x, y]$ are congruent in $T$ modulo $J$; hence, if $\sigma_0$ denotes the canonical mapping of $g$ into $U$,

in $U$. In other words, $\sigma_0$ is an $\alpha$-mapping of $g$ into $U$.