Exercise 10, § 1

Assume that $k$ is of characteristic $p >0$. Denote by $\mathfrak{g}$ a Lie algebra with basis $\{e_1, e_2, e_3\}$, with $[e_1, e_2] =e_3,[e_1, e_3] = [e_2, e_3] = 0$.

a) Show that the centre of $U\mathfrak{g}$ is $k[e^p_1, e^p_2, e_3]$.

b) Assume that $k$ is algebraically closed. Show that, for all $(\lambda_1, \lambda_2, \lambda_3)\in k^3$, there exists (up to equivalence) a unique irreducible representation $\rho$ of $\mathfrak{g}$

$^6$ For more details, cf. H. ZASSENHAUS, Über Liesche Ringe mit Primzahlcharakteristik, Hamb. Abh., Vol. XIII (1939), pp. 1-100, and Darstellungstheorie nilpotenter Lie-Ringe bei Charakteristik $p >0$, Crelle’s J., Vol. CLXXXII (1940), pp. 150-155. such that $\lambda_i$ is the unique eigenvalue of $\rho (e_i) (i= 1,2,3)$; the degree of $\rho$ is $p$ if $\lambda_3\not= 0$, and is 1 if $\lambda_3= 0$. (Apply Exerc. 8 and 9, or argue directly.)