an:06133882
Zbl 1280.22013
Kasuya, Hisashi
Algebraic hulls of solvable groups and exponential iterated integrals on solvmanifolds
EN
Geom. Dedicata 162, 263-270 (2013).
00314115
2013
j
22E25 14H30 55P62 20F99
exponential iterated integral; algebraic hull; solvmanifold
Let \(G\) be a simply connected Lie group with Lie algebra \(\mathfrak{g}\), and \(\Gamma\) a cocompact discrete subgroup of \(G\). If \(G\) is nilpotent, then Chen's (closed) iterated integrals induced from \(\bigwedge \mathfrak{g}_\mathbb{C}^\ast\) represent the coordinate ring of the Malcev completion of \(\Gamma\). In this paper, a generalized case is discussed where \(G\) is a solvable Lie group and (hence) \(\Gamma\) is a torsion-free polycyclic group. The author applies the theory of \textit{C. Miller} [Topology 44, No. 2, 351--373 (2005; Zbl 1149.57315)] and shows that the coordinate ring of the algebraic hull of \(\Gamma\) is represented by Miller's (closed) exponential iterated integrals induced from a \(\mathbb{Z}\)-lattice of \(\mathfrak{g}_\mathbb{C}^\ast\) and \(\bigwedge \mathfrak{g}_\mathbb{C}^\ast\).
Hiroaki Nakamura (Osaka)
Zbl 1149.57315