Exercise 25, § 7

Let $\mathfrak{a}$ be a Lie algebra and U its enveloping algebra. Apply the definitions and results of Exerc. 24 to the algebra U. In particular, $U'\subset U^*$ and the dual of $U'$ can be identified with the algebra $\widehat{U} =$ lim$_{\leftarrow -}U/\mathfrak{m}$; the canonical map $U\rightarrow \widehat{U}$ is injective (Chap. I, §7, Exerc. 3).

a) The coalgebra structure of U (Chap. II, §1, no. 4) defines an algebra structure on the dual $U^*$ (cf. Chap. II, §1, no. 5, Prop. 10, and Algebra, Chap. III, §11, no. 2). If $E,F$ are finite dimensional $\mathfrak{a}$-modules,

$\theta_{a,b}\circ \theta_{c,d}=\theta_{a\otimes c,b\otimes d}$ for $a\in E, b\in E^*, c\in F, d\in F^*$.

Deduce that $U'$ is a subalgebra of $U^*$. The coalgebra and algebra structures of $U'$ make it a commutative bigebra (Algebra, Chap. III, §11, no. 4).

b) Let $x$ be an element of $\widehat{U}$; identify $x$ with a linear form $U'\rightarrow k$. Prove the equivalence of the following properties:

(i) $x$ is a homomorphism of algebras from $U'$ to $k$.

(ii) $x_{E\otimes F}=x_E\otimes x_F$ for all finite dimensional $\mathfrak{a}$-modules E and F.

(Prove first that (ii) is equivalent to

(ii$'$)$x(\theta_{a\otimes c,b\otimes d}) =x(\theta_{a,b})x(\theta_{c,d})$ if $a\in E, b\in E^*, c\in F, d\in F^*$,

and use the fact that the $\theta_{a,b}$ generate $U'.)$

c) Let $x$ be an element of $\widehat{U}$ satisfying conditions (i) and (ii) of b). Show the equivalence of the following conditions:

(iii) $x$ takes the unit element of $U'$ to the unit element of $k$.

(iv) $x\not= 0$.

(v) If $k$ is given the trivial $\mathfrak{a}$-module structure, we have $x_k=$ Id.

(vi) $x_E$ is invertible for all E.

(The equivalence (iii) $\Leftarrow \Rightarrow$ (iv) follows from the fact that $x$ is a homomorphism of algebras. On the other hand, $x_k=\lambda$ Id, with $\lambda \in k$. Using the $\mathfrak{a}$-isomorphism $k\otimes E\rightarrow E$, deduce that $\lambda x_E=x_E$ for all E, and, in particular, that $\lambda^2=\lambda$ by taking $E =k$. The case $\lambda = 1$ corresponds to $x\not= 0$, hence (iv) $\Leftarrow \Rightarrow$ (v), and (vi) $=\Rightarrow$ (v). To prove that (v) $=\Rightarrow$ (vi), show that, if F is the dual of E, then $^tx_F\circ x_E=\lambda$ Id$_E.$)

d) Let G be the set of elements of $\widehat{U}$ satisfying conditions (i) to (vi) above. Show that G is a subgroup of the group of invertible elements of $\widehat{U}$.

Let $x\in G$. If E is a finite dimensional $\mathfrak{a}$-module, then $x_E\in \mathbf{G}\mathbf{L}(E)$. This applies in particular to $E =\mathfrak{a}$, equipped with the adjoint representation; this gives an element $x_{\mathfrak{a}}\in \mathbf{G}\mathbf{L}(\mathfrak{a})$. Show that $x_{\mathfrak{a}}$ is an automorphism of $\mathfrak{a}$ (use the $\mathfrak{a}$-homomorphism $\mathfrak{a}\otimes \mathfrak{a}\rightarrow \mathfrak{a}$ given by the bracket). This gives a homomorphism $v: G\rightarrow$ Aut($\mathfrak{a}$). If E is a finite dimensional $\mathfrak{a}$-module,

$x_E(y.e) =v(x)(y).x_E(e)$ if $y\in \mathfrak{a}, e\in E$

(use the $\mathfrak{a}$-homomorphism $\mathfrak{a}\otimes E\rightarrow E$ given by the operation of $\mathfrak{a}$ on E).

e) The principal anti-automorphism $\sigma$ of U extends by continuity to $\widehat{U}$. Its transpose leaves $U'$ stable and induces an inversion on $U'($Algebra, Chap. III, §11, Exerc. 4). If $x\in G$, then $\sigma (x) =x^{-1}$.

f) Assume that $\mathfrak{a}$ is semi-simple$^{16}$. Let $n$ be a nilpotent element of $\mathfrak{a}$. Then there exists a unique element $e^n$ of G such that $(e^n)_E=$ exp($n_E$) for every finite dimensional $\mathfrak{a}$-module E. We have $v(e^n) =$ exp(ad $n$)$\in$ Aut($\mathfrak{a}$), so Aut$_e(\mathfrak{a})\subset v(G)$.

Show that, if $\mathfrak{b}$ is a subalgebra of $\mathfrak{a}$ consisting of nilpotent elements, then

$e^n.e^m=e^{H(n,m)}$ for $n, m\in \mathfrak{b}$,

where H denotes the Hausdorff series (Chap. II, §6).