Lie powers of the natural module for \(\text{GL}(2)\)
Let \(L\) be a free Lie algebra of finite rank \(r\) over a field \(F\). Let \(L^n\) denote the degree \(n\) homogeneous component of \(L\) for each positive integer \(n\). When \(\operatorname {char}F=0\), \(L^n\) are semisimple. The authors study the case where \(F\) has a prime characteristic \(p\), in which case very few of \(L^n\) are semisimple. They consider the case \(r=2\) and \(F=F_p\), the field of \(p\) elements. They give the Krull-Schmidt multiplicities when either \(p=2\) or \(p=3\), or if \(n\) is not divisible by \(p\). When \(p=3\) they show that \(L^n\), as a module for a Sylow 3-subgroup of \(\text{GL}(2,3)\), has no one-dimensional direct summand. The authors also consider a challenging question concerning the homogeneous components of the free Lie algebra of rank \(r\) over \(\mathbb{Z}\), and they extend a previously known result to the maximal finite subgroups of \(\text{GL}(2,\mathbb{Z})\).
scientific article; zbMATH DE number 1502159
scientific article