论文标题
单位子组的真实伴随轨道
Real adjoint orbits of the unipotent subgroup
论文作者
论文摘要
让$ g $是一个线性谎言组，由eapaniaper动作进行作用于Lie代数$ \ Mathfrak {G} $：$ \ Mathrm {ad}（g）x = gxg^{ - 1} $。一个元素$ x \ in \ mathfrak {g} $称为$ \ mathrm {ad} _g $ -real，如果$ -x = \ mathrm {ad}（ad}（g）x $ for g $ in G $。 $ \ mathrm {ad} _g $ -real元素$ x $被称为$ \ mathrm {ad} _g $ -real，如果$ -x = \ mthrm {ad}（ad}（τ）x $用于某些互动$τ\ in G $。 令$ k = \ mathbb {r} $，$ \ mathbb {c} $或$ \ mathbb {h} $。令$ \ mathrm {u} _n（k）$为$ k $上的单位上三角矩阵的组。令$ \ mathfrak {u} _n（k）$为$ \ mathrm {u} _n（k）$的Lie代数，由$ n \ times n $ times n $上三角矩阵组成，在所有对角条目中，$ 0 $。在本文中，我们考虑$ \ mathrm {ad} $ - lie代数$ \ mathfrak {u} _n（k）$的现实，它来自lie group $ \ mathrm {u} _n（k）$ on $ \ mathfrak {u} _n（u} _n（k）$。我们证明，没有$ \ mathrm {u} _n（k）} $ - $ \ mathfrak {u} _n（k）$中的真实元素。我们还考虑了扩展组$ \ mathrm {u} _n^\ pm（k）$的伴随动作，该$由所有上层三角矩阵组成，其对角线元素为$ 1 $或$ -1 $，并构建一大类$ \ \ \ \ \ \ m m interrm {ad} _ {ad} _ {作为这些结果的应用，我们恢复了有关这些组中经典现实的相关结果。
Let $G$ be a linear Lie group that acts on it's Lie algebra $\mathfrak{g}$ by the adjoint action: $\mathrm{Ad}(g)X=gXg^{-1}$. An element $X\in \mathfrak {g}$ is called $\mathrm{Ad}_G$-real if $-X = \mathrm{Ad}(g)X $ for some $g\in G$. An $\mathrm{Ad}_G$-real element $X$ is called strongly $\mathrm{Ad}_G $-real if $-X = \mathrm{Ad}(τ) X $ for some involution $τ\in G$. Let $K=\mathbb{R}$, $\mathbb{C}$ or $\mathbb{H}$. Let $\mathrm{U}_n(K)$ be the group of unipotent upper-triangular matrices over $K$. Let $\mathfrak{u}_n (K)$ be the Lie algebra of $\mathrm{U}_n(K)$ that consists of $n \times n$ upper triangular matrices with $0$ in all the diagonal entries. In this paper, we consider the $\mathrm{Ad}$-reality of the Lie algebra $ \mathfrak{u}_n(K) $ that comes from the adjoint action of the Lie group $\mathrm{U}_n(K)$ on $ \mathfrak{u}_n(K)$. We prove that there is no non-trivial $\mathrm{Ad}_{\mathrm{ U}_n(K)}$-real element in $\mathfrak{u}_n (K)$. We also consider the adjoint action of the extended group $\mathrm{U}_n^\pm(K)$ that consists of all upper triangular matrices over $K$ having diagonal elements as $1$ or $-1$, and construct a large class of $\mathrm{Ad} _{\mathrm{ U}_n^\pm( K)} $-real elements. As applications of these results, we recover related results concerning classical reality in these groups.
