2022/04/07 by Krishnendu Gongopadhyay, Gongopadhyay, Krishnendu, Chandan Maity +1
Mathematics · #20G20 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Primary 20E45 #Secondary: 22E25
paper · pdf · doi:10.48550/arxiv.2204.03623
openalex publication_date 2022/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a linear Lie group that acts on it's Lie algebra \mathfrakg by the adjoint action: Ad(g)X=gXg-1. An element X∈ \mathfrak g is called AdG-real if -X = Ad(g)X for some g∈ G. An AdG-real element X is called strongly AdG -real if -X = Ad(τ) X for some involution τ∈ G. Let K=ℝ, ℂ or ℍ. Let Un(K) be the group of unipotent upper-triangular matrices over K. Let \mathfrakun (K) be the Lie algebra of Un(K) that consists of n × n upper triangular matrices with 0 in all the diagonal entries. In this paper, we consider the Ad-reality of the Lie algebra \mathfrakun(K) that comes from the adjoint action of the Lie group Un(K) on \mathfrakun(K). We prove that there is no non-trivial Ad Un(K)-real element in \mathfrakun (K). We also consider the adjoint action of the extended group Un^±(K) that consists of all upper triangular matrices over K having diagonal elements as 1 or -1, and construct a large class of Ad Un^±( K) -real elements. As applications of these results, we recover related results concerning classical reality in these groups.