# S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. The Lie Superalgebra of (Contact) Vector Fields on ${S}^{1\mid 1}$

#### 2.2. Detailed Overview of the Super-Space $\mathcal{S}\psi \mathcal{D}\mathcal{O}$

#### 2.3. Fuzzy Lie Algebras

**Note:**Condition (2) implies that $\mu (-x)\ge \mu (x)$ and $\mu (0)\ge \mu (x)$ for all x in V.

- $\mu $ is a fuzzy subspace of V;
- Each nonempty $U(\mu ,t)=\{x\in L|\mu (x)\ge t\}$ is a subspace of V.

**Note****:**- This theorem establishes the equivalence between fuzzy subspaces and conventional subspaces within the context of fuzzy sets.

**Example**

**1.**

#### 2.4. The Structure of $\mathcal{S}\varrho $ as a $\overrightarrow{(}{S}^{1|1})$-Module

## 3. Computations of the Space ${\mathit{H}}^{\mathbf{1}}(\overrightarrow{(}{\mathit{S}}^{\mathbf{1}|\mathbf{1}}),\mathcal{S}\mathit{\psi}\mathcal{D}\odot ))$

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Remark**

**1.**

## 4. Deformations, Cohomology, and Integrability of Infinitesimal Deformations

#### 4.1. Deformations

**polynomial**if it takes the form:

**Remark**

**2.**

**infinitesimal deformation (inf deformation)**.

#### 4.2. Integrability of Infinitesimal Deformation

**Lemma**

**4.**

**Proof.**

**Theorem**

**2.**

**Lemma**

**5.**

**Proof.**

#### 4.2.1. Exploring Integrable Infinitesimal Deformations in Fuzzy Lie Algebras

**Lemma**

**6.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**2.**

**Proof.**

#### 4.2.2. A Variation of the Central Charge

**Remark**

**4.**

**Corollary**

**3.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Pogudin, G.; Razmyslov, Y.P. Prime Lie algebras satisfying the standard Lie identity of degree 5. J. Algebra
**2016**, 468, 182–192. [Google Scholar] [CrossRef] - Bahturin, Y. Identical Relations in Lie Algebras; Walter de Gruyter GmbH & Co KG: Berlin, Germany, 2021; Volume 68. [Google Scholar]
- Kanel-Belov, A.; Rowen, L.H. Computational Aspects of Polynomial Identities; AK Peters/CRC Press: Boca Raton, FL, USA, 2005. [Google Scholar]
- Davies, J.M. Elliptic cohomology is unique up to homotopy. J. Aust. Math. Soc.
**2023**, 115, 99–118. [Google Scholar] [CrossRef] - Baklouti, A. Quadratic Hom-Lie triple systems. J. Geom. Phys.
**2017**, 121, 166–175. [Google Scholar] [CrossRef] - Baklouti, A.; Benayadi, S. Symplectic Jacobi-Jordan algebras. Linear Multilinear Algebra
**2021**, 69, 1557–1578. [Google Scholar] [CrossRef] - Roger, C.; Ovsienko, V. Deforming the Lie algebra of vector fields on S
^{1}inside the Lie algebra of pseudodifferential symbols on S^{1}. arXiv**1998**, arXiv:math/9812074. [Google Scholar] - Ovsienko, V.; Roger, C. Deforming the Lie algebra of vector fields on S1 inside the Lie algebra of pseudodifferential symbols on S1, Differential topology, infinite-dimensional Lie algebras, and applications. Am. Math. Soc. Transl.
**1999**, 194, 211–226. [Google Scholar] - Creutzig, T.; Linshaw, A. The super W
_{(1∞)}algebra with integral central charge. Trans. Am. Math. Soc.**2015**, 367, 5521–5551. [Google Scholar] [CrossRef] - Cheng, S.J.; Wang, W. Lie subalgebras of differential operators on the super circle. Publ. Res. Inst. Math. Sci.
**2003**, 39, 545–600. [Google Scholar] [CrossRef] - García, J.I.; Liberati, J.I. Quasifinite Representations of Classical Subalgebras of the Lie Superalgebra of Quantum Pseudodifferential Operators. Int. Sch. Res. Not.
**2013**, 2013, 672872. [Google Scholar] [CrossRef] - Yehia, S.E.B. The adjoint representation of fuzzy Lie algebras. Fuzzy Sets Syst.
**2001**, 119, 409–417. [Google Scholar] [CrossRef] - Assiry, A.; Baklouti, A. Exploring Roughness in Left Almost Semigroups and Its Connections to Fuzzy Lie Algebras. Symmetry
**2023**, 15, 1717. [Google Scholar] [CrossRef] - Baklouti, A. Multiple-Attribute Decision Making Based on the Probabilistic Dominance Relationship with Fuzzy Algebras. Symmetry
**2023**, 15, 1188. [Google Scholar] [CrossRef] - Radul, A.O. Non-trivial central extensions of Lie algebras of differential operators in two and higher dimensions. Phys. Lett. B
**1991**, 265, 86–91. [Google Scholar] [CrossRef] - Agrebaoui, B.; Ben Fraj, N. On the cohomology of the Lie superalgebra of contact vector fields on S1|1. Belletin Soc. R. Sci. Liege
**2004**, 72, 365–375. [Google Scholar] [CrossRef] - Manin, Y.I.; Radul, A.O. A supersymmetric extension of the Kadomtsev-Petviashvili hierarchy. Commun. Math. Phys.
**1985**, 98, 65–77. [Google Scholar] [CrossRef] - Ali, A.; Alali, A.S.; Zishan, A. Applications of Fuzzy Semiprimary Ideals under Group Action. Axioms
**2023**, 12, 606. [Google Scholar] [CrossRef] - Altassan, A.; Mateen, M.H.; Pamucar, D. On Fundamental Theorems of Fuzzy Isomorphism of Fuzzy Subrings over a Certain Algebraic Product. Symmetry
**2021**, 13, 998. [Google Scholar] [CrossRef] - Shaqaqha, S. Fuzzy Hom–Lie Ideals of Hom–Lie Algebras. Axioms
**2023**, 12, 630. [Google Scholar] [CrossRef] - Agrebaoui, B.; Ben Fraj, N.; Omri, S. On the cohomology of the Lie superalgebra of contact vector fields on S1|2. J. Nonlinear Math. Phys.
**2006**, 13, 523–534. [Google Scholar] [CrossRef] - Fuks, D.B. Cohomology of Infinite-Dimensional Lie Algebras; Springer Science & Business Media: New York, NY, USA, 2012. [Google Scholar]
- Poletaeva, E. The analogs of Riemann and Penrose tensors on supermanifolds. arXiv
**2005**, arXiv:math/0510165. [Google Scholar] - Nijenhuis, A.; Richardson, R.W., Jr. Deformations of homomorphisms of Lie groups and Lie algebras. Bull. Am. Math. Soc.
**1967**, 73, 175–179. [Google Scholar] [CrossRef] - Agrebaoui, B.; Ammar, F.; Lecomte, P.; Ovsienko, V. Multi-parameter deformations of the module of symbols of differential operators. Int. Math. Res. Not.
**2002**, 2002, 847–869. [Google Scholar] [CrossRef] - Fialowski, A.; Fuchs, D. Construction of miniversal deformations of Lie algebras. J. Funct. Anal.
**1999**, 161, 76–110. [Google Scholar] [CrossRef] - Fraj, N.B.; Omri, S. Deforming the Lie superalgebra of contact vector fields on S1|1 inside the Lie superalgebra of superpseudodifferential operators on S1|1. J. Nonlinear Math. Phys.
**2006**, 13, 19–33. [Google Scholar] [CrossRef] - Grozman, P.; Leites, D.; Shchepochkina, I. Lie superalgebras of string theories. arXiv
**1997**, arXiv:hep-th/9702120. [Google Scholar] - Radul, A.O. Superstring schwartz derivative and the bott cocycle. In Integrable and Superintegrable Systems; World Scientic: Singapore, 1990. [Google Scholar]

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**MDPI and ACS Style**

Assiry, A.; Mansour, S.; Baklouti, A.
S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras. *Axioms* **2024**, *13*, 2.
https://doi.org/10.3390/axioms13010002

**AMA Style**

Assiry A, Mansour S, Baklouti A.
S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras. *Axioms*. 2024; 13(1):2.
https://doi.org/10.3390/axioms13010002

**Chicago/Turabian Style**

Assiry, Abdullah, Sabeur Mansour, and Amir Baklouti.
2024. "S-Embedding of Lie Superalgebras and Its Implications for Fuzzy Lie Algebras" *Axioms* 13, no. 1: 2.
https://doi.org/10.3390/axioms13010002