I found two terms:
- Enveloping algebras, and
- Universal enveloping algebras
I read the answer to the following question Universal enveloping algebra, but I am seeking more details and a connection between them, if possible.
As in An Introduction to Homological Algebra (Page 83), let $A$ be a $k$-algebra, where $k$ is a commutative ring, then its enveloping algebra is $$ A^e = A \otimes_k A^{\text{op}}. $$
In Enveloping Algebras (Page 66), let $\mathcal{T}$ be the tensor algebra of the vector space $\mathfrak{g}$ ($\mathfrak{g}$ is a Lie algebra), i.e. $$ \mathcal{T} = T^0 \oplus T^1 \oplus \dots \oplus T^n \oplus \dots, $$ where $T^n = \mathfrak{g} \otimes \mathfrak{g} \otimes \dots \otimes \mathfrak{g}$ ($n$ times); in particular, $T^0 = k \cdot 1$ and $T^1 = \mathfrak{g}$; the product in $\mathcal{T}$ is simply tensor multiplication.
Let $J$ be the two-sided ideal of $\mathcal{T}$ generated by the tensors $$ x \otimes y - y \otimes x - [x,y], $$ where $x, y \in \mathfrak{g}$. The associative algebra $\mathcal{T}/J$ is termed the universal enveloping algebra of $\mathfrak{g}$ and is denoted by $U(\mathfrak{g})$.
NB: I know that every associative algebra with additive $+$ and multiplicative $\times$ laws can be turned into a Lie algebra by defining the Lie bracket using them.
Thanks.
