# Universal enveloping algebra

I am learning universal enveloping algebras. Here are two questions I can not understand.

1. 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? )
2. 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?

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).