Modules for certain Lie algebras of maximal class
Let \(L\) be a finite-dimensional Lie algebra over a field \(F\) of characteristic zero and let \(\mu (L)= \min \{\dim M\), \(M\) is a faithful \(L\)-module\}. For several classes of Lie algebras, \(\mu (L)\leq 1+\dim L\). Let \(L= a(r, s, t)\) be the algebra with generators \(e_1, e_2, \dots\) and relations \([e_1, e_i ]= e_{i+1}\), \([e_2, e_3 ]= e_5\) and \([e_2, e_5 ]= re_7+ se_8+ te_9\) where \(r, s, t\in F\) and \(i= 1, 2, \dots\). If \(r\neq 0, {9\over 10}, 1, 2, 3\), then there exists a faithful \(L\)-module \(M\) of dimension 22 and \(M\) has no faithful submodule or quotient. Also, for \(s\neq 0\) and certain restrictions on \(r\), there exists no 12-dimensional faithful \(L\)-module.
scientific article; zbMATH DE number 838656