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Balanced Hermitian structures on twisted cartesian products

M. W. Mansouri, A. Oufkou

Abstract

We study Hermitian structures on twisted cartesian products $(\mathfrak{g}_{(ρ_{1},ρ_{2})},\mathrm{J},\cal{K})$ of two Hermitian Lie algebras according to two representations $ρ_{1}$ and $ρ_{2}$. We give the conditions on $(\mathfrak{g}_{(ρ_{1},ρ_{2})},\mathrm{J},\cal{K})$ to be balanced and locally conformally balanced. As an application we classify six-dimensional balanced Hermitian twisted cartesian products Lie algebras.

Balanced Hermitian structures on twisted cartesian products

Abstract

We study Hermitian structures on twisted cartesian products of two Hermitian Lie algebras according to two representations and . We give the conditions on to be balanced and locally conformally balanced. As an application we classify six-dimensional balanced Hermitian twisted cartesian products Lie algebras.
Paper Structure (5 sections, 8 theorems, 22 equations)

This paper contains 5 sections, 8 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Hermitian twisted cartesian products
  3. Applications in dimension six
  4. $\mathbf{\mathbb{R}^2\Join{\mathfrak{g}}}$ with $\dim({\mathfrak{g}})=4$
  5. $\mathrm{aff(\mathbb{R})\Join{\mathfrak{g}}_2}$ with $\dim({\mathfrak{g}})=4$

Key Result

Proposition 2.1

Let $({\mathfrak{g}}_1,{\mathrm{J}}_1,{\mathrm{k}}_1)$ and $({\mathfrak{g}}_2,{\mathrm{J}}_2,{\mathrm{k}}_2)$ be two Hermitian Lie algebras and let $(\rho_1,\rho_2)$ a representation compatible couple. Then $({\mathfrak{g}}_{(\rho_1,\rho_2)},\omega,{\mathrm{J}})$ is a Hermitian Lie algebra if and on

Theorems & Definitions (19)

  • Remark 1
  • Proposition 2.1
  • proof
  • Remark 2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.1
  • proof
  • ...and 9 more