About the structure of a Hopf algebra on universal enveloping algebras of Lie algebras We know that the universal enveloping algebra construction provides a functor from Lie algebras to cocommutative Hopf algebras which is left adjoint to the primitive functor. Furthermore, if we restrict to connected cocommutative Hopf algebras over a field of characteristic zero, it becomes an equivalence by Milnor-Moore Theorem.
Now let consider the diagonal map $L \to L \times L$, where $L$ is a Lie algebra. I do not know how this diagonal map defines a structure of Hopf algebra on universal enveloping algebra $U(L)$? Moreover, let consider the augmentation ideal of $U(L)$ and $\operatorname{gr} U(L)$ be its grading associated to filtration by augmentation ideal, can we show that $\operatorname{gr} U(L)$ is a primitively generated Hopf algebra? Your assistance with understanding the details behind the scenes of the above mentioned concepts will be highly appreciated.
 A: For every two Lie algebras $$ and $$, their direct sum $ ⊕ $ as vector spaces can again be made into a Lie algebra via the bracket
$$
 [(x_1, y_1), (x_2, y_2)] = ( [x_1, x_2] , [y_1, y_2] ) \,.
$$
The inclusion maps
$$
 i \colon  \longrightarrow  ⊕  \,,
 \quad
 j \colon  \longrightarrow  ⊕ 
$$
given by $i(x) = (x, 0)$ and $j(y) = (0, y)$ are homomorphisms of Lie algebras.
The induced homomorphisms of algebras
$$
 \mathrm{U}(i) \colon \mathrm{U}() \longrightarrow \mathrm{U}( ⊕ ) \,,
 \quad
 \mathrm{U}(j) \colon \mathrm{U}() \longrightarrow \mathrm{U}( ⊕ )
$$
can be combined into a single homomorphism of algebras
$$
 Φ \colon \mathrm{U}() ⊗ \mathrm{U}() \longrightarrow \mathrm{U}( ⊕ ) \,.
$$
The homomorphism $Φ$ is already an isomorphism.
This isomorphism and its inverse are given in formulas by
$$
 Φ( x ⊗ y ) = (x, 0) ⋅ (0, y) \,,
 \quad
 Φ^{-1}( (x, y) ) = x ⊗ 1 + 1 ⊗ y \,,
$$
for all $x ∈ $, $y ∈ $.
The isomorphism $Φ$ is natural in both $$ and $$.
For every Lie algebra $$ we have the diagonal map
$$
 δ \colon  \longrightarrow  ⊕  \,, \quad x \longmapsto (x, x) \,.
$$
This map is a homomorphism of Lie algebras, and therefore induces a homomorphism of algebras
$$
 Δ
 \colon
 \mathrm{U}()
 \xrightarrow{\enspace \mathrm{U}(δ) \enspace}
 \mathrm{U}( ⊕ )
 \xrightarrow{\enspace Φ \enspace}
 \mathrm{U}() ⊗ \mathrm{U}() \,.
$$
The homomorphism $Δ$ is given by
$$
 Δ()(x) = x ⊗ 1 + 1 ⊗ x
$$
for all $x ∈ $.
It can now be shown that $Δ$ is comultiplicative, that it admits a counit $ε$, and that the resulting bialgebra structure on $\mathrm{U}()$ is already a Hopf algebra structure.
The counit $ε$ and antipode $S$ of this Hopf algebra structure are explicitly given by$ε(x) = 0$ and $S(x) = 0$ for all $x ∈ $.¹
Let now $I$ be the augmentation ideal of $\mathrm{U}()$ with respect to $ε$.
It follows from the PBW-theorem that the associated graded algebra $\operatorname{gr}_I \mathrm{U}()$ is the symmetric algebra $\mathrm{S}(\mathfrak{g})$.
The Hopf algebra structure on $\mathrm{S}()$ is given by the comultiplication $Δ(x) = x ⊗ 1 + 1 ⊗ x$ for all $x ∈ $.
In other words, all the elements of $$ are primitive in $\mathrm{S}()$.
The algebra $\mathrm{S}()$ is generated by $$, and therefore generated by primitive elements.

¹ Both $ε$ and $S$ also come from homomorphisms of Lie algebras:
the zero homomorphism $ \to 0$ induces the counit $\mathrm{U}() \to \mathrm{U}(0) = $, and the isomorphism $ \to ^{\mathrm{op}}$ induces the antipode $\mathrm{U}() \to \mathrm{U}(^{\mathrm{op}}) ≅ \mathrm{U}()^{\mathrm{op}}$.
