On rational and real elements in a class of Lie groups

2025/08/25 by Mandal, Arunava, Singh, Shashank Vikram
#20E45 #20G20 #FOS: Mathematics #Group Theory (math.GR) #Primary 20H20 #Secondary 20D10

paper · doi:10.48550/arxiv.2508.18218

Abstract

For a class of groups G over a field \mathbbF, including certain Lie groups, Algebraic groups and finite groups, we develop a general method to determine rational and real elements, thereby unifying earlier group-specific results into a wider framework. As an application, we classify all real and rational elements in the semidirect product \rm SL(2,ℝ) \ltimes Symⁿ(ℝ²). Furthermore, for affine groups of the form \rm GL(n,ℝ) \ltimes ℝⁿ, we show that if x ∈ \rm GL(n,ℝ) is rational, then (x,v) is rational for every v ∈ ℝⁿ.

Citations

• Reversibility of Affine Transformations
• On the surjectivity of the power maps of a class of solvable groups

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