I am learning universal enveloping algebras. Here are two questions I can not understand.
- Let $A$ be an arbitrary associative algebra over a field k with an identity. I have seen that " The universal enveloping algebra $A^e$ of $A$ is $A \otimes_k A^{op}$, where $A^{op}$ is an algebra anti-isomorphism to $A$". I have not find the definition of universal enveloping algebra of an associative algebra using universal property, so I can not understand why $A^{e}=A \otimes_kA^{op}$. (An associative algebra $A$ is a Lie algebra $A^-$ under the commutator bracket, so the universal enveloping algebra of $A$ is the universal enveloping algebra of $A^-$? If is, could anyone tell me how to show it is isomorphic to $A \otimes_k A^{op}$? If not, could anyone tell me the definition of universal enveloping algebras of an associative algebra using universal property? )
- I have seen that "Let $A$ be an algebra in a variety of algebras $\mathcal{M}$, then the notation of an A-bimodule in $\mathcal{M}$ is equivalent to the notation of a left module over the associative algebra $U(A)$, where $U(A)$ is the universal enveloping algebra of $A$". I know the left module category of a lie algebra $\mathcal{g}$ is equivalent to the category of left module category of its enveloping algebra $U(\mathcal{g})$, why in $\mathcal{M}$ we use $A$-bimodule not left $A$-module?
Best Answer
They share the same name but $A^e$ is not the enveloping álgebra of a Lie algebra. The property that it has is that the category of k-symmetric $A$-bimodules is isomorphic to the category of left $A^e$-modules. The functor giving the isomorphism at the level of objects preserves the underlying k-module, it is the identity on maps, and the actions are related by $$(a\otimes b)m=amb$$ ($a,b$ in $A$, $m$ in the left $A^e$-module or $A$-bimodule, depending on which side of the equivalemce you want to define).