Representative functions on discrete solvable groups (Q1821868)

The results here are mainly generalizations of work of Mostow concerning polycyclic groups to the class of solvable groups of finite torsion-free rank (A\({}_ 1\)-groups). In discussing the finite dimensional representations of a group \(\Gamma\) over a field k, one can consider the ''continuous dual'' \(k[\Gamma]^ 0\) of the group algebra, where the topology on k[\(\Gamma\) ] has as a fundamental system of neighbourhoods of zero the kernels of the finite- dimensional representations of \(\Gamma\) over k. \(k[\Gamma]^ 0\) has the structure of a coalgebra, and the locally finite-dimensional k[\(\Gamma\) ]-modules correspond to the \(k[\Gamma]^ 0\)-comodules. Any finite- dimensional \(k[\Gamma]^ 0\)-comodule is a submodule of the direct sum of finitely many copies of \(k[\Gamma]^ 0\), and, when k is algebraically closed, \(k[\Gamma]^ 0\) is the polynomial algebra of a pro-affine algebraic group \(G_ k(\Gamma)\) whose rational representations correspond exactly to the locally finite-dimensional representations of \(\Gamma\) over k. In Section 1, the author examines the unipotent radical \(U_ k(\Gamma)\) of \(G_ k(\Gamma)\), and proves that, on the category of \(A_ 1\)-groups, the assignment of \(U_ k(\Gamma)\) to \(\Gamma\) is an exact functor to the category of unipotent affine algebraic groups. In Section 2, he considers a certain homomorphic image \(B_ k(\Gamma)\) of \(G_ k(\Gamma)\), the ''lowest'' homomorphic image of \(G_ k(\Gamma)\) to ''preserve'' the unipotent radical \(U_ k(\Gamma)\), shows that the assignment of \(B_ k(\Gamma)\) is functorial on a subcategory of the category of solvable groups and determines the kernel of the canonical map from \(\Gamma\) to \(B_ k(\Gamma)\). Finally, in Section 3, he finds the ''unipotent radical'' of an \(A_ 1\)-group \(\Gamma\) and the kernel of the canonical map from \(\Gamma\) to \(G_ k(\Gamma)\). This gives a necessary and sufficient condition for an \(A_ 1\)-group to have a faithful locally finite- dimensional representation over a field of characteristic zero.