Large sets of generating tuples for Lie groups
Alexandru Chirvasitu

TL;DR
This paper investigates the properties of generating pairs and subgroups in connected, semisimple linear Lie groups, showing their generative sets are dense and well-behaved in various topologies.
Contribution
It establishes that spaces of generating pairs and conjugates of circle subgroups are Zariski-open, Euclidean-open, and dense, providing a comprehensive understanding of generative structures in Lie groups.
Findings
Generating pairs form Zariski-open sets in the compact case.
Generating pairs are Euclidean-open in general.
Spaces of conjugates of generic circle subgroups are dense and Zariski-open.
Abstract
We prove that for a connected, semisimple linear Lie group the spaces of generating pairs of elements or subgroups are well-behaved in a number of ways: the set of pairs of elements generating a dense subgroup is Zariski-open in the compact case, Euclidean-open in general, and always dense. Similarly, for sufficiently generic circle subgroups , of , the space of conjugates of that generate a dense subgroup is always Zariski-open and dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra .
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Topics in Algebra · Homotopy and Cohomology in Algebraic Topology
