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Pluriclosed metrics on compact semisimple Lie groups

Jorge Lauret, Facundo Montedoro

Abstract

Given a compact semisimple Lie group G and a maximal torus T of G, we give an explicit description of all left and Ad(T)-invariant pluriclosed Hermitian structures on G in terms of the corresponding root system. They depend on 2d+1 parameters in the irreducible case, where dim(T)=2d. As applications, we obtain that the only left and Ad(T)-invariant pluriclosed metrics which are also CYT are bi-invariant metrics (i.e., Bismut flat) and study the pluriclosed flow as a neat ODE system.

Pluriclosed metrics on compact semisimple Lie groups

Abstract

Given a compact semisimple Lie group G and a maximal torus T of G, we give an explicit description of all left and Ad(T)-invariant pluriclosed Hermitian structures on G in terms of the corresponding root system. They depend on 2d+1 parameters in the irreducible case, where dim(T)=2d. As applications, we obtain that the only left and Ad(T)-invariant pluriclosed metrics which are also CYT are bi-invariant metrics (i.e., Bismut flat) and study the pluriclosed flow as a neat ODE system.

Paper Structure

This paper contains 6 sections, 9 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Hermitian geometry of Lie groups
  3. Pluriclosed metrics
  4. CYT metrics
  5. Pluriclosed flow
  6. Appendix: Roots

Key Result

Theorem 1.2

On a compact semisimple Lie group $G$ endowed with a Samelson complex structure $J$, any left and $\operatorname{Ad}(T)$-invariant pluriclosed metric is given as follows: $g_\mathfrak{t} (\mathfrak{t} _i,\mathfrak{t} _j)=0$ for all $i\ne j$ and on each simple factor, up to scaling, $g_\mathfrak{t} = $\Pi=\{\alpha_1,\dots,\alpha_n\}\subset\Delta^+$ is the set of simple roots and $x_i:=x_{\alpha_i}$

Theorems & Definitions (26)

  • Remark 1.1
  • Theorem 1.2
  • Example 2.1
  • Lemma 2.2
  • Remark 2.3
  • proof
  • Lemma 2.4
  • Remark 2.5
  • proof
  • Lemma 3.1
  • ...and 16 more