# Non-Degeneracy of 2-Forms and Pfaffian

## Abstract

**:**

## 1. Introduction

## 2. Symplectic Vector Space and Pfaffian

#### Pfaffian and 2-Forms

**pf**$\left(A\right)={\left(-1\right)}^{n}\sqrt{detA}$ and in particular

**pf**$\left(J\right)={\left(-1\right)}^{n}$.

## 3. Rigidity of Symplectic Forms

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**pf**$\left({\widehat{A}}_{ii}\right)=0,$ because ${\widehat{A}}_{ii}$ is a $\left(2n-1\right)\times \left(2n-1\right)$ skew symmetric matrix and its determinant vanishes.

**Lemma**

**2.**

**Proof.**

**pf**$\left(A\right)$ each term in

**pf**$\left(A\right)$ containing ${A}_{ki}$. In general, for $l<m$

**pf**$\left({\widehat{A}}_{lm}\right)$, by the previous lemma

**Lemma**

**3.**

**Proof.**

**Corollary**

**1.**

**Theorem**

**1.**

**Proof.**

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Lee, J.-H.
Non-Degeneracy of 2-Forms and Pfaffian. *Symmetry* **2020**, *12*, 280.
https://doi.org/10.3390/sym12020280

**AMA Style**

Lee J-H.
Non-Degeneracy of 2-Forms and Pfaffian. *Symmetry*. 2020; 12(2):280.
https://doi.org/10.3390/sym12020280

**Chicago/Turabian Style**

Lee, Jae-Hyouk.
2020. "Non-Degeneracy of 2-Forms and Pfaffian" *Symmetry* 12, no. 2: 280.
https://doi.org/10.3390/sym12020280