About the structure of a Hopf algebra on universal enveloping algebras of Lie algebras

Question

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.

Answer 1

For every two Lie algebras $𝔤$ and $𝔥$, their direct sum $𝔤 ⊕ 𝔥$ 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 $$ i \colon 𝔤 \longrightarrow 𝔤 ⊕ 𝔥 \,, \quad j \colon 𝔥 \longrightarrow 𝔤 ⊕ 𝔥 $$ given by $i(x) = (x, 0)$ and $j(y) = (0, y)$ are homomorphisms of Lie algebras. The induced homomorphisms of algebras $$ \mathrm{U}(i) \colon \mathrm{U}(𝔤) \longrightarrow \mathrm{U}(𝔤 ⊕ 𝔥) \,, \quad \mathrm{U}(j) \colon \mathrm{U}(𝔥) \longrightarrow \mathrm{U}(𝔤 ⊕ 𝔥) $$ can be combined into a single homomorphism of algebras $$ Φ \colon \mathrm{U}(𝔤) ⊗ \mathrm{U}(𝔥) \longrightarrow \mathrm{U}(𝔤 ⊕ 𝔥) \,. $$ 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 $x ∈ 𝔤$, $y ∈ 𝔥$. The isomorphism $Φ$ is natural in both $𝔤$ and $𝔥$.

For every Lie algebra $𝔤$ we have the diagonal map $$ δ \colon 𝔤 \longrightarrow 𝔤 ⊕ 𝔤 \,, \quad x \longmapsto (x, x) \,. $$ This map is a homomorphism of Lie algebras, and therefore induces a homomorphism of algebras $$ Δ \colon \mathrm{U}(𝔤) \xrightarrow{\enspace \mathrm{U}(δ) \enspace} \mathrm{U}(𝔤 ⊕ 𝔤) \xrightarrow{\enspace Φ \enspace} \mathrm{U}(𝔤) ⊗ \mathrm{U}(𝔤) \,. $$ The homomorphism $Δ$ is given by $$ Δ(𝔤)(x) = x ⊗ 1 + 1 ⊗ x $$ for all $x ∈ 𝔤$.

It can now be shown that $Δ$ is comultiplicative, that it admits a counit $ε$, and that the resulting bialgebra structure on $\mathrm{U}(𝔤)$ 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 $x ∈ 𝔤$.¹

Let now $I$ be the augmentation ideal of $\mathrm{U}(𝔤)$ with respect to $ε$. It follows from the PBW-theorem that the associated graded algebra $\operatorname{gr}_I \mathrm{U}(𝔤)$ is the symmetric algebra $\mathrm{S}(\mathfrak{g})$. The Hopf algebra structure on $\mathrm{S}(𝔤)$ is given by the comultiplication $Δ(x) = x ⊗ 1 + 1 ⊗ x$ for all $x ∈ 𝔤$. In other words, all the elements of $𝔤$ are primitive in $\mathrm{S}(𝔤)$. The algebra $\mathrm{S}(𝔤)$ is generated by $𝔤$, and therefore generated by primitive elements.

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