On the number of simple modules of a supersolvable restricted Lie algebra
A Lie algebra \(L\) is called supersolvable if there is a chain of ideals \(L_i\) \[ 0=L_n\subset\cdots\subset L_1\subset L_0=L \] such that \(\dim L_i/L_{i+1}=1\). Let \(L\) be a supersolvable Lie \(p\)-algebra over an algebraically closed field \(F\) of characteristic \(p\), \(u(L,\chi)\) be a reduced enveloping algebra of \(L\) corresponding to a \(p\)-character \(\chi\in L^*\). The aim of the paper is to obtain the number of isomorphism classes of simple \(u(L,\chi)\)-modules. For this purpose the simple \(u(L,\chi)\)-modules with a fixed eigenvalue function are considered. Let \(T_p(L)\) be the maximal torus in the center of \(L\). Given a simple \(u(L,\chi)\)-module \(S\), there exists \(\sigma\in T_p(L)^*\) such that \((t)_S=\sigma(t)\cdot 1_S\) for any \(t\in T_p(L)\). Obviously, \[ \sigma(t)^p-\sigma(t^{[p]})=\chi(t)^p. \] \(\sigma\) is called the eigenvalue function of \(S\). The set of all eigenvalue functions of simple \(u(L,\chi)\)-modules is denoted by \(\Pi(L,\chi)\) and the set of isomorphism classes of simple \(u(L,\chi)\)-modules with a fixed eigenvalue function \(\tau\) is denoted by \(\text{Irr}_{\tau}(L,\chi)\). Let \(G_0^L\) be the set of all \(\gamma\in L^*\) such that \(\gamma([L,L])=\gamma(T_p(L))=0\), \(\gamma(x^{[p]})=\gamma(x)^p\) for any \(x\in L\). It is shown that if \(M\) and \(N\) are simple \(u(L,\chi)\)-modules with the same eigenvalue function, then \(N=F_\gamma\otimes M\) for some \(\gamma\in G_0^L\) (Theorem 1). Here \(F_\gamma\) is one-dimensional \(L\)-module corresponding to \(\gamma\), i.e. \(x\cdot 1_\gamma=\gamma(x)1_\gamma\), \(x\in L\). It means that \(G_0^L\) acts transitively on \(\text{Irr}_\tau(L,\chi)\) for any \(\tau\in\Pi(L,\chi)\). Therefore, \(|\text{Irr}_\tau(L,\chi)|\) is a \(p\)-power (Proposition 1). For \(\tau\in\Pi(L,\chi)\) let \[ u_\tau(L,\chi)=u(L,\chi)/u(L,\chi)\{t-\tau(t)\cdot 1\mid t\in T_p(L)\}. \] For an arbitrary Lie \(p\)-algebra \(L\) it is shown that \(u(L,\chi)=\oplus_{\tau\in\Pi(L,\chi)}u_\tau(L,\chi)\). The dimensions and some properties of \(u_\tau(L,\chi)\) are established (Theorem 2). In particular, \(u_\tau(L,\chi)\) is proved to be a Frobenius algebra. Some corollaries on block ideals of \(u(L,\chi)\) for nilpotent and supersolvable restricted Lie algebras are obtained. Using the technique of projective covers the decomposition of the induced module \(\text{Ind}_{T_p(L)}^L(F_{\sigma},\chi)\) is found when \(L\) is a supersolvable Lie \(p\)-algebra (Theorem 3). This decomposition is used to compute \(|\text{Irr}_\tau(L,\chi)|\) in terms of \(\chi\)-admissible polarizations and then to derive lower and upper bounds for \(|\text{Irr}(L,\chi)|\), \[ p^{\dim T_p(L)}\leq|\text{Irr}(L,\chi)|\leq p^{\dim T_{\max}(L)} \] where \(T_{\max}(L)\) is a maximal torus of \(L\) (Theorem 4). When \(L\) is a strongly solvable Lie algebra (i.e. \([L,L]\) is \(p\)-nilpotent), it is shown that \[ |\text{Irr}(L,\chi)|=p^{\dim T_{\max}(K)}, \] for some \(\chi\)-admissible polarization \(K\) and for any maximal torus \(T_{\max}(K)\) of \(K\). The example of the three-dimensional supersolvable Lie algebra is considered to show that in general \(|\text{Irr}(L,\chi)|\) is not \(p\)-power.
scientific article; zbMATH DE number 1512971
scientific article