发表机构
Universidad Politécnica de Madrid（马德里理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究复单李代数的交换子界，证明Cartan 3-形式的谱范数复化后不增大，得到Dynkin指数为ℓ的表示像上Frobenius范数的最优常数2/ℓ，改进了相关不等式并揭示了紧致单李群的曲率性质。
AI 中文摘要
对于带有紧致共轭τ、Killing型B及埃尔米特型H(x,y)=-B(x,τy)的复单李代数𝔤，本文证明H([x,y],[x,y])≤(h∨)⁻¹H(x,x)H(y,y)，其中h∨为对偶Coxeter数，当且仅当在同构于𝔰𝔩(2,ℂ)的τ-稳定长根子代数中H-正交对时等号成立。等价地，Cartan 3-形式关于H的上余范数等于其在紧致实形式上的上余范数：Cartan 3-形式的谱范数在复化下不增大。对于Dynkin指数为ℓ的表示像上的Frobenius范数，最优常数为2/ℓ；该结果包含Böttcher–Wenzel与Bloch–Iserles不等式，将后者推广至复斜对称矩阵，并对F₄、E₆、E₇、E₈型例外李代数大幅改进了这些不等式。证明使用𝔤的Nahm代数及由τ诱导的共轭：相关函数的临界点对应τ-扭曲幂等元，其Hessian由τ-扭曲乘法算子控制，该算子与乘i的反交换性将极大值处的单侧二阶条件转化为双侧谱界。Dynkin指数为j的同构于𝔰𝔩(2,ℂ)的子代数给出临界值为1/(jh∨)的临界点，计算了其Morse指数，且极大点集为非退化临界流形。几何上，紧致单李群的极大复截面曲率等于其极大截面曲率。
英文摘要
For a complex simple Lie algebra $\mathfrak{g}$ with compact conjugation $τ$, Killing form $B$, and Hermitian form $H(x, y) = -B(x, τy)$, it is shown that $H([x,y],[x,y]) \leq (h^{\vee})^{-1}H(x,x)H(y,y)$, where $h^{\vee}$ is the dual Coxeter number, with equality exactly for $H$-orthogonal pairs in $τ$-stable long-root subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$. Equivalently, the comass of the Cartan $3$-form with respect to $H$ equals its comass on the compact real form: the spectral norm of the Cartan $3$-form does not increase under complexification. For the Frobenius norm on the image of a representation of Dynkin index $\ell$ the optimal constant is $2/\ell$; this contains the Böttcher--Wenzel and Bloch--Iserles inequalities, extends the latter to complex skew-symmetric matrices, and improves them substantially for the exceptional Lie algebras of types $F_{4}$, $E_{6}$, $E_{7}$, and $E_{8}$. The proof uses the Nahm algebra of $\mathfrak{g}$ with the conjugation induced by $τ$: critical points of the relevant function correspond to $τ$-twisted idempotents, and its Hessian is governed by $τ$-twisted multiplication operators, whose anticommutation with multiplication by $i$ turns the one-sided second order condition at a maximum into a two-sided spectral bound. Subalgebras isomorphic to $\mathfrak{sl}(2, \mathbb{C})$ of Dynkin index $j$ give critical points with critical value $1/(jh^{\vee})$, whose Morse indices are computed, and the set of maximum points is a nondegenerate critical manifold. Geometrically, the maximal complex sectional curvature of a compact simple Lie group equals its maximal sectional curvature.
Comments31 pages
