# On Formality of Some Homogeneous Spaces

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Theorem**

**2.**

## 2. Preliminaries

#### 2.1. Presentation and Notation

#### 2.2. Formality

**Definition**

**1.**

- 1.
- $\mathcal{A}$ is the free algebra $\bigwedge V$ over a graded vector space $V\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{\u2a01}_{i}{V}^{i}$, and
- 2.
- there is a family of generators ${\left\{{a}_{\tau}\right\}}_{\tau \in I}$ indexed by some well-ordered set I, such that $|{a}_{\mu}|\phantom{\rule{0.166667em}{0ex}}\le \phantom{\rule{0.166667em}{0ex}}|{a}_{\tau}|$ if $\mu \phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}\tau $ and each $d{a}_{\tau}$ is expressed in terms of preceding ${a}_{\mu}$, $\mu \phantom{\rule{0.166667em}{0ex}}<\phantom{\rule{0.166667em}{0ex}}\tau $. Thus, $d{a}_{\tau}$ does not have a linear part.

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.3. Quaternionic-Kaehler and 3-Sasakian Manifolds

**Theorem**

**3.**

## 3. Proof of Theorem 1

#### 3.1. A Theorem on Formality of Homogeneous Spaces

**Theorem**

**4**

**.**Let $G/H$ be a homogeneous space of a compact semisimple Lie group G and let ${T}_{H}$ be a maximal torus in H. Then $G/H$ is formal if and only $G/{T}_{H}$ is formal.

#### 3.2. Cartan Algebras

**Theorem**

**5.**

#### 3.3. Formality of $G/{T}_{\beta}$

**Proposition**

**1.**

**Proof.**

**Case**

**1**

**(${C}_{n}$).**

**Case**

**2**

**(${B}_{n}$).**

**Case**

**3**

**(${D}_{n}$).**

**Case**

**4**

**(${A}_{n}$).**

#### 3.4. Completion of Proof of Theorem 1

## 4. Application: Formality of 3-Sasakian Homogeneous Manifolds of Classical Type

#### 4.1. Quaternionic-Kaehler Symmetric Spaces (Wolf Spaces)

**Theorem**

**6**

**Theorem**

**7**

**.**Let $G/K=G/{L}_{1}\xb7{A}_{1}$ be the quaternionic symmetric space. Then the homogeneous space $G/{L}_{1}$ is 3-Sasakian. All compact homogeneous Sasakian manifolds are obtained in this way.

**Remark**

**1.**

#### 4.2. Proof of Theorem 2

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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Tralle, A.
On Formality of Some Homogeneous Spaces. *Symmetry* **2019**, *11*, 1011.
https://doi.org/10.3390/sym11081011

**AMA Style**

Tralle A.
On Formality of Some Homogeneous Spaces. *Symmetry*. 2019; 11(8):1011.
https://doi.org/10.3390/sym11081011

**Chicago/Turabian Style**

Tralle, Aleksy.
2019. "On Formality of Some Homogeneous Spaces" *Symmetry* 11, no. 8: 1011.
https://doi.org/10.3390/sym11081011