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

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A Note on Reversibility of Unipotent Matrices

  • 章节

  • 发表日期：

    2025年

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  • Krishnendu Gongopadhyay,
  • Chandan Maity

摘要

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