### Abstract

All finite-dimensional simple modular Lie algebras with Cartan matrix fail to have deformations, even infinitesimal ones, if the characteristic p of the ground field is equal to 0 or exceeds 3. If p = 3, then the orthogonal Lie algebra o(5) is one of two simple modular Lie algebras with Cartan matrix that do have deformations (the Brown algebras br(2; α) appear in this family of deformations of the 10-dimensional Lie algebras, and therefore are not listed separately); moreover, the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra which has a Cartan matrix and admits a deformation. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of an algebraic group O(5) of automorphisms of the Lie algebra o(5) on the space H^{2}(o(5); o(5)) of infinitesimal deformations and presented representatives of the isomorphism classes. We give here an explicit description of the global deformations of the Lie algebra o(5) and describe the deformations of a simple analog of this orthogonal algebra in characteristic 2. In characteristic 3, we have found the representatives of the isomorphism classes of the deformed algebras that linearly depend on the parameter.

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

Pages (from-to) | 777-791 |

Number of pages | 15 |

Journal | Mathematical Notes |

Volume | 89 |

Issue number | 5 |

DOIs | |

State | Published - Jun 1 2011 |

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### Keywords

- Brown algebra
- Cartan matrix
- Chevalley basis
- finite-dimensional simple modular Lie algebra
- global deformation
- infinitesimal deformation
- Jacobi identity
- Massey bracket
- Maurer-Cartan equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Mathematical Notes*,

*89*(5), 777-791. https://doi.org/10.1134/S0001434611050191

**Deformations of the Lie algebra o(5) in characteristics 3 and 2.** / Bouarroudj, Sofiane; Lebedev, A. V.; Wagemann, F.

Research output: Contribution to journal › Article

*Mathematical Notes*, vol. 89, no. 5, pp. 777-791. https://doi.org/10.1134/S0001434611050191

}

TY - JOUR

T1 - Deformations of the Lie algebra o(5) in characteristics 3 and 2

AU - Bouarroudj, Sofiane

AU - Lebedev, A. V.

AU - Wagemann, F.

PY - 2011/6/1

Y1 - 2011/6/1

N2 - All finite-dimensional simple modular Lie algebras with Cartan matrix fail to have deformations, even infinitesimal ones, if the characteristic p of the ground field is equal to 0 or exceeds 3. If p = 3, then the orthogonal Lie algebra o(5) is one of two simple modular Lie algebras with Cartan matrix that do have deformations (the Brown algebras br(2; α) appear in this family of deformations of the 10-dimensional Lie algebras, and therefore are not listed separately); moreover, the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra which has a Cartan matrix and admits a deformation. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of an algebraic group O(5) of automorphisms of the Lie algebra o(5) on the space H2(o(5); o(5)) of infinitesimal deformations and presented representatives of the isomorphism classes. We give here an explicit description of the global deformations of the Lie algebra o(5) and describe the deformations of a simple analog of this orthogonal algebra in characteristic 2. In characteristic 3, we have found the representatives of the isomorphism classes of the deformed algebras that linearly depend on the parameter.

AB - All finite-dimensional simple modular Lie algebras with Cartan matrix fail to have deformations, even infinitesimal ones, if the characteristic p of the ground field is equal to 0 or exceeds 3. If p = 3, then the orthogonal Lie algebra o(5) is one of two simple modular Lie algebras with Cartan matrix that do have deformations (the Brown algebras br(2; α) appear in this family of deformations of the 10-dimensional Lie algebras, and therefore are not listed separately); moreover, the 29-dimensional Brown algebra br(3) is the only other simple Lie algebra which has a Cartan matrix and admits a deformation. Kostrikin and Kuznetsov described the orbits (isomorphism classes) under the action of an algebraic group O(5) of automorphisms of the Lie algebra o(5) on the space H2(o(5); o(5)) of infinitesimal deformations and presented representatives of the isomorphism classes. We give here an explicit description of the global deformations of the Lie algebra o(5) and describe the deformations of a simple analog of this orthogonal algebra in characteristic 2. In characteristic 3, we have found the representatives of the isomorphism classes of the deformed algebras that linearly depend on the parameter.

KW - Brown algebra

KW - Cartan matrix

KW - Chevalley basis

KW - finite-dimensional simple modular Lie algebra

KW - global deformation

KW - infinitesimal deformation

KW - Jacobi identity

KW - Massey bracket

KW - Maurer-Cartan equation

UR - http://www.scopus.com/inward/record.url?scp=79959647002&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=79959647002&partnerID=8YFLogxK

U2 - 10.1134/S0001434611050191

DO - 10.1134/S0001434611050191

M3 - Article

AN - SCOPUS:79959647002

VL - 89

SP - 777

EP - 791

JO - Mathematical Notes

JF - Mathematical Notes

SN - 0001-4346

IS - 5

ER -