Exercise 8, § 1

Assume that $k$ is of characteristic $p >0$. Let $\mathfrak{g}$ be a Lie algebra over $k$ with basis $(e_1, . . . , e_n)$. Denote by U the enveloping algebra of $\mathfrak{g}$ and C the centre of U. For $i= 1, . . . , n$, choose a non-zero $p$-polynomial $f_i$, of degree $d_i$, such that $f_i$(ad $e_i$) $= 0$; then $f_i(e_i)\in C$, cf. Chap. I, §7, Exerc. 5. Put $z_i=f_i(e_i)$.

a) Show that $z_1, . . . , z_n$ are algebraically independent. If $A =k[z_1, . . . , z_n]$, show that U is a free A-module with basis the monomials $e^{\alpha_1}_1. . . e^{\alpha_n}_n$, where $0\leq \alpha_i\leq d_i$. (Use the Poincaré-Birkhoff-Witt theorem.) The rank [U : A] of U over A is equal to $d_1. . . d_n$; it is a power of $p$. Deduce that C is an A-module of finite type, hence a $k$-algebra of finite type and of dimension $n$ (Commutative Algebra, Chap. VIII).

b) Let K be the field of fractions of A, and let

Then $U_{(K)}\supset C_{(K)}\supset K$. Show that $U_{(K)}$ is a field with centre $C_{(K)}$, and that this is the quotient field (both left and right) of U, cf. Chap. I, §2, Exerc. 10. Deduce that $[U_{(K)}: C_{(K)}]$ is of the from $q^2$, where $q$ is a power of $p$; we have $[C_{(K)}: K] =q_C$, where $q_C$ is a power of $p$, and $[U : A] =q_Cq^2$.

c) Let $d$ be a non-zero element of A, and let $\Lambda$ be a subring of $U_{(K)}$ such that $U\subset \Lambda \subset d^{-1}U$. Show that $\Lambda = U$. [If $x=b/a,a\in A$ -- $\{0\}$, is an element of $\Lambda$, show by induction on $m$ that the relation $b\in Ua+ U_m$ implies that $b\in Ua+ U_{m-1}$, where $\{U_m\}$ is the canonical filtration of U. (For this, use the fact that gr U is integrally closed, and argue as in Prop. 15 of Commutative Algebra, Chap. V, §1, no. 4.) For $m= 0$, this gives $b\in Ua$, i.e. $x\in U.]$

Deduce that C is integrally closed.

d) Assume that $k$ is algebraically closed. Let $\rho :\mathfrak{g}\rightarrow \mathfrak{g}\mathfrak{l}(V)$ be an irreducible linear representation of $\mathfrak{g}$ and $\rho_U$ the corresponding representation of U. The restriction of $\rho_U$ to C is a homomorphism $\gamma_{\rho}$ from C to $k$ (identified with the homotheties of V); let $\alpha_{\rho}$ be its restriction to A. Show that for any homomorphism $\alpha$ (resp. $\gamma$ ) from the $k$-algebra A (resp. C) to $k$, there exists at least one irreducible representation $\rho$ of $\mathfrak{g}$ such that $\alpha_{\rho}=\alpha$ (resp. $\gamma_{\rho}=\gamma$ ) and that there are only finitely-many such representations (up to equivalence). Show that dim $V\leq q$, with the notation of $b).^5$

$^4$ For more details, cf. J. DIXMIER, Cohomologie des algèbres de Lie nilpotentes,

Acta Sci. Math. Szeged, Vol. XVI (1955), pp. 246-250.

$^5$ For more details, cf. H. ZASSENHAUS, The representations of Lie algebras of

prime characteristic, Proc. Glasgow Math. Assoc., Vol. II (1954), pp. 1-36.