### Abstract

For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we explicitly give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and - in the arXiv version of the paper - in other characteristics for completeness of the picture). In the modular and super cases, we define notions of Chevalley generators and Cartan matrix, and an auxiliary notion of the Dynkin diagram. The relations of simple Lie algebras of the A, D, E types are not only Serre ones. These non-Serre relations are same for Lie superalgebras with the same Cartan matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan matrix (indigenous for characteristic 2) are also given.

Original language | English (US) |
---|---|

Pages (from-to) | 237-278 |

Number of pages | 42 |

Journal | Homology, Homotopy and Applications |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2010 |

### Fingerprint

### Keywords

- Defining relation
- Divided power cohomology
- Modular Lie superalgebra

### ASJC Scopus subject areas

- Mathematics (miscellaneous)

### Cite this

*Homology, Homotopy and Applications*,

*12*(1), 237-278. https://doi.org/10.4310/HHA.2010.v12.n1.a13

**Divided power (co)homology. presentations of simple finite dimensional modular lie superalgebras with cartan matrix.** / Bouarroudj, Sofiane; Grozman, Pavel; Lebedev, Alexei; Leites, Dimitry.

Research output: Contribution to journal › Article

*Homology, Homotopy and Applications*, vol. 12, no. 1, pp. 237-278. https://doi.org/10.4310/HHA.2010.v12.n1.a13

}

TY - JOUR

T1 - Divided power (co)homology. presentations of simple finite dimensional modular lie superalgebras with cartan matrix

AU - Bouarroudj, Sofiane

AU - Grozman, Pavel

AU - Lebedev, Alexei

AU - Leites, Dimitry

PY - 2010/1/1

Y1 - 2010/1/1

N2 - For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we explicitly give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and - in the arXiv version of the paper - in other characteristics for completeness of the picture). In the modular and super cases, we define notions of Chevalley generators and Cartan matrix, and an auxiliary notion of the Dynkin diagram. The relations of simple Lie algebras of the A, D, E types are not only Serre ones. These non-Serre relations are same for Lie superalgebras with the same Cartan matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan matrix (indigenous for characteristic 2) are also given.

AB - For modular Lie superalgebras, new notions are introduced: Divided power homology and divided power cohomology. For illustration, we explicitly give presentations (in terms of analogs of Chevalley generators) of finite dimensional Lie (super)algebras with indecomposable Cartan matrix in characteristic 2 (and - in the arXiv version of the paper - in other characteristics for completeness of the picture). In the modular and super cases, we define notions of Chevalley generators and Cartan matrix, and an auxiliary notion of the Dynkin diagram. The relations of simple Lie algebras of the A, D, E types are not only Serre ones. These non-Serre relations are same for Lie superalgebras with the same Cartan matrix and any distribution of parities of the generators. Presentations of simple orthogonal Lie algebras having no Cartan matrix (indigenous for characteristic 2) are also given.

KW - Defining relation

KW - Divided power cohomology

KW - Modular Lie superalgebra

UR - http://www.scopus.com/inward/record.url?scp=77954626103&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77954626103&partnerID=8YFLogxK

U2 - 10.4310/HHA.2010.v12.n1.a13

DO - 10.4310/HHA.2010.v12.n1.a13

M3 - Article

VL - 12

SP - 237

EP - 278

JO - Homology, Homotopy and Applications

JF - Homology, Homotopy and Applications

SN - 1532-0073

IS - 1

ER -