A Note on Reversibility of Unipotent Matrices

Krishnendu Gongopadhyay () and Chandan Maity ()
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Krishnendu Gongopadhyay: Indian Institute of Science Education and Research (IISER) Mohali
Chandan Maity: Indian Institute of Science Education and Research (IISER) Berhampur

Chapter Chapter 11 in Essays on Geometry, 2025, pp 197-201 from Springer

Abstract: Abstract Let 𝔻 = ℝ $$\mathbb {D}=\mathbb {R}$$ , ℂ $$\mathbb {C}$$ or ℍ $$\mathbb {H}$$ . Let U n ( 𝔻 ) $$\mathrm {U}_n(\mathbb {D})$$ be the group of unipotent upper triangular matrices over 𝔻 $$\mathbb {D}$$ . Let 𝔲 n ( 𝔻 ) $$\mathfrak {u}_n (\mathbb {D})$$ be the Lie algebra of U n ( 𝔻 ) $$\mathrm {U}_n(\mathbb {D})$$ that consists of n × n $$n \times n$$ upper triangular matrices with 0 in all the diagonal entries. In this chapter, we consider the adjoint action of the extended group U n ± 1 ( 𝔻 ) $${\mathrm {U}_n^{\pm 1}}(\mathbb {D})$$ that consists of all upper triangular matrices over 𝔻 $$\mathbb {D}$$ having diagonal elements 1 or −1 $$-1$$ , and construct a large class of strongly Ad U n ± 1 ( 𝔻 ) $$ _{\mathrm {U}_n^{\pm 1}( \mathbb {D})} $$ -real elements in 𝔲 n ( 𝔻 ) $$\mathfrak {u}_n (\mathbb {D})$$ .

Date: 2025
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DOI: 10.1007/978-3-031-76257-4_11

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