Large sets of generating tuples for Lie groups

2021/06/22 by Alexandru Chirvăsitu, Chirvasitu, Alexandru · 1 citation
Mathematics · #14L15 #17B20 #20G20 #22C05 #22E46 #22E60 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2106.11955

openalex publication_date 2021/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that for a connected, semisimple linear Lie group G 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 H_i, i=1,2 of G, the space of conjugates of H_i that generate a dense subgroup is always Zariski-open and dense. Similar statements hold for pairs of Lie subalgebras of the Lie algebra Lie(G).

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