## DEFORMATIONS OF RESTRICTED SIMPLE LIE ALGEBRAS II (2007)

Citations: | 6 - 5 self |

### BibTeX

@MISC{Viviani07deformationsof,

author = {Filippo Viviani},

title = {DEFORMATIONS OF RESTRICTED SIMPLE LIE ALGEBRAS II},

year = {2007}

}

### OpenURL

### Abstract

Abstract. We compute the infinitesimal deformations of two families of restricted simple modular Lie algebras of Cartan-type: the Contact and the Hamiltonian Lie algebras. 1.

### Citations

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Citation Context ...rd facts of deformation theory, the infinitesimal deformations of a Lie algebra are parametrized by the second cohomology of the Lie algebra with values in the adjoint representation (see for example =-=[GER64]-=-). It is a classical result (see [HS97]) that for a simple Lie algebra g over a field of characteristic 0 it holds that H i (g, g) = 0 for every i ≥ 0, which implies in particular that such Lie algebr... |

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Citation Context ...nitesimal deformations of a Lie algebra are parametrized by the second cohomology of the Lie algebra with values in the adjoint representation (see for example [GER64]). It is a classical result (see =-=[HS97]-=-) that for a simple Lie algebra g over a field of characteristic 0 it holds that H i (g, g) = 0 for every i ≥ 0, which implies in particular that such Lie algebras are rigid. The proof of this fact re... |

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Citation Context ...r for the second page of the first spectral sequence 2.5, we have the equality (2.7) E p,0 2 = Hp (g, h; M), where H ∗ (g, h; M) are the relative cohomology groups defined (by Chevalley and Eilenberg =-=[CE48]-=-) from the sub-complex C p (g, h; M) ⊂ C p (g, M) consisting of4 FILIPPO VIVIANI cochains orthogonal to h, that is cochains satisfying the two conditions: (2.8) (2.9) f |h = 0, df |h = 0 or equivalen... |

68 | Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson - Demazure, Gabriel - 1970 |

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Citation Context ...the ones which are restricted. These can be characterized as those modular Lie algebras such that the p-power of an inner derivation (which in characteristic p is a derivation) is still inner (see FS =-=[FS88]-=- or STR [STR04]). Important examples of restricted Lie algebras are the ones coming from groups schemes (see DG [DG70, Chap. 2]). This paper is devoted to the study of the infinitesimal deformations o... |

53 |
Cohomology of Lie algebras
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Citation Context ...s results on the cohomology of Lie algebras 2.1. Review of general theory. In this subsection we review, in order to fix notations, the classical theory of cohomology of Lie algebras (see for example =-=[HS53]-=-). If g is a Lie algebra over a field F and M is a g-module, then the cohomology groups H∗ (g, M) can be computed from the complex of n-dimensional cochains Cn (g, M) (n ≥ 0), that are alternating n-l... |

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Citation Context ... are restricted. These can be characterized as those modular Lie algebras such that the p-power of an inner derivation (which in characteristic p is a derivation) is still inner (see FS [FS88] or STR =-=[STR04]-=-). Important examples of restricted Lie algebras are the ones coming from groups schemes (see DG [DG70, Chap. 2]). This paper is devoted to the study of the infinitesimal deformations of the restricte... |

21 |
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Citation Context ...ling form may degenerate. Indeed the classification problem remained open for a long time until it was recently solved, if the characteristic of the base field is greater than 3, by Wilson-Block (see =-=[BW88]-=-), Strade (see [STR89], [STR92], [STR91], [STR93], [STR94], [STR98]) and Premet-Strade (see [PS97], [PS99], [PS01]). The classification remains still open in characteristic 2 and 3 (see [STR04, page 2... |

17 | Simple Lie algebras of small characteristic. I. Sandwich elements. J. Algebra 189
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Citation Context ... was recently solved, if the characteristic of the base field is greater than 3, by Wilson-Block (see [BW88]), Strade (see [STR89], [STR92], [STR91], [STR93], [STR94], [STR98]) and Premet-Strade (see =-=[PS97]-=-, [PS99], [PS01]). The classification remains still open in characteristic 2 and 3 (see [STR04, page 209]). According to this classification, simple modular (that is over a field of positive character... |

16 |
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Citation Context ...algebras. The algebras of classical-type are obtained by the simple Lie algebras in characteristic zero by first taking a model over the integers (via Chevalley bases) and then reducing modulo p (see =-=[SEL67]-=-). The algebras of Cartan-type were constructed by Kostrikin-Shafarevich in 1966 (see [KS66]) as finite-dimensional analogues of the infinite-dimensional complex simple Lie algebras, which occurred in... |

14 | Classification of Simple Lie Algebras over Algebraically Closed Fields of Prime Characteristic - Strade, Wilson - 1991 |

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Citation Context ...that such Lie algebras are rigid if the characteristic of the base field is greater than or equal to 5 while in characteristic 2 and 3 there are non-rigid classical Lie algebras (see [CHE05], [CK00], =-=[CKK00]-=-). The purpose of this article is to compute the infinitesimal deformations of the first two families of restricted simple Lie algebras of Cartan type: the Witt-Jacobson algebras W(n) and the Special ... |

11 |
A course in homological algebra, Second edition
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Citation Context ...estricted Lie algebras, Cartan-type simple Lie algebras. The author was partially supported by a grant from the Mittag-Leffler Institute (Stockholm). 12 FILIPPO VIVIANI It is a classical result (see =-=[HS97]-=-) that for a simple Lie algebra g over a field of characteristic 0 it holds that H i (g, g) = 0 for every i ≥ 0, which implies in particular that such Lie algebras are rigid. The proof of this fact re... |

10 |
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Citation Context ... showed that such Lie algebras are rigid if the characteristic of the base field is greater than or equal to 5 while in characteristic 2 and 3 there are non-rigid classical Lie algebras (see [CHE05], =-=[CK00]-=-, [CKK00]). The purpose of this article is to compute the infinitesimal deformations of the first two families of restricted simple Lie algebras of Cartan type: the Witt-Jacobson algebras W(n) and the... |

10 |
The classification of the simple modular Lie algebras. VI. Solving the final case
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Citation Context ...ed open for a long time until it was recently solved, if the characteristic of the base field is greater than 3, by Wilson-Block (see [BW88]), Strade (see [STR89], [STR92], [STR91], [STR93], [STR94], =-=[STR98]-=-) and Premet-Strade (see [PS97], [PS99], [PS01]). The classification remains still open in characteristic 2 and 3 (see [STR04, page 209]). According to this classification, simple modular (that is ove... |

10 |
Shapiro’s lemma and its consequences in the cohomology theory of modular Lie algebras
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Citation Context ...d here constitute part of my doctoral thesis. I thank my advisor prof. Schoof for useful advices and constant encouragement. I thank the referee for the suggestion of using the results of [FAR86] and =-=[FS91]-=- in order to simply some of the proofs of this paper. 2. Contact algebra 2.1. Definition and Basic properties. We first introduce some notations about the set N n of n-tuple of natural numbers. We con... |

7 |
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Citation Context ...istic zero by first taking a model over the integers (via Chevalley bases) and then reducing modulo p (see [SEL67]). The algebras of Cartan-type were constructed by Kostrikin-Shafarevich in 1966 (see =-=[KS66]-=-) as finite-dimensional analogues of the infinite-dimensional complex simple Lie algebras, which occurred in Cartan’s classification of Lie pseudogroups, and are divided into four families, called Wit... |

7 |
Melikian: Simple Lie algebras of characteristic 5
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Citation Context ...ing a volume form, a Hamiltonian form and a contact form, respectively. Moreover in characteristic 5 there is one exceptional simple modular Lie algebra called the Melikian algebra (introduced in MEL =-=[MEL80]-=-). A particular important class of simple modular Lie algebras which are the ones which are restricted. These can be characterized as those modular Lie algebras such that the p-power of an inner deriv... |

7 |
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Citation Context ...of the infinitesimal deformations of the restricted simple Lie algebras. The simple Lie algebras of classical type are known to be rigid over a field of characteristic different from 2 and 3 (see RUD =-=[RUD71]-=-), in analogy of what happens in characteristic zero. In the paper Viv [VIV], the author computed the infinitesimal deformations of the Witt-Jacobson and Special Lie algebras. In this paper, we comput... |

6 |
The Classification of the Simple Modular Lie Algebras: I. Determination of the two-sections
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Citation Context ...ate. Indeed the classification problem remained open for a long time until it was recently solved, if the characteristic of the base field is greater than 3, by Wilson-Block (see [BW88]), Strade (see =-=[STR89]-=-, [STR92], [STR91], [STR93], [STR94], [STR98]) and Premet-Strade (see [PS97], [PS99], [PS01]). The classification remains still open in characteristic 2 and 3 (see [STR04, page 209]). According to thi... |

6 |
The Classification of the Simple Modular Lie Algebras: III. Solution of the Classical Case
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Citation Context ...ed the classification problem remained open for a long time until it was recently solved, if the characteristic of the base field is greater than 3, by Wilson-Block (see [BW88]), Strade (see [STR89], =-=[STR92]-=-, [STR91], [STR93], [STR94], [STR98]) and Premet-Strade (see [PS97], [PS99], [PS01]). The classification remains still open in characteristic 2 and 3 (see [STR04, page 209]). According to this classif... |

6 |
The Classification of the Simple Modular Lie Algebras: V. Algebras with Hamiltonian Two-sections
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- 1994
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Citation Context ...em remained open for a long time until it was recently solved, if the characteristic of the base field is greater than 3, by Wilson-Block (see [BW88]), Strade (see [STR89], [STR92], [STR91], [STR93], =-=[STR94]-=-, [STR98]) and Premet-Strade (see [PS97], [PS99], [PS01]). The classification remains still open in characteristic 2 and 3 (see [STR04, page 209]). According to this classification, simple modular (th... |

6 |
Seligman: Modular Lie algebras. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40
- B
- 1967
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Citation Context ...bras. The algebras of classical-type are obtained by the simple Lie algebras in characteristic zero by first taking a model over the integers (via Chevalley bases) and then reducing modulo p (see SEL =-=[SEL67]-=-). The algebras of Cartan-type were constructed by Kostrikin-Shafarevich in 1966 (see KS [KS66]) as finite-dimensional analogues of the infinite-dimensional complex simple Lie algebras, which occurred... |

6 |
Chebochko: Deformations of classical Lie algebras with a homogeneous root system in characteristic two
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Citation Context ... (see [RUD71]) showed that such Lie algebras are rigid if the characteristic of the base field is greater or equal to 5 while in characteristic 2 and 3 there are non-rigid classical Lie algebras (see =-=[CHE05]-=-, [CK00], [CKK00]). The purpose of this article is to compute the infinitesimal deformations of the first two families of restricted simple Lie algebras of Cartan type: the Witt-Jacobson algebras W(n)... |

5 |
Deformations of classical Lie algebras with a homogeneous root system in characteristic two
- CHEBOCHKO
- 2005
(Show Context)
Citation Context ... [RUD71]) showed that such Lie algebras are rigid if the characteristic of the base field is greater than or equal to 5 while in characteristic 2 and 3 there are non-rigid classical Lie algebras (see =-=[CHE05]-=-, [CK00], [CKK00]). The purpose of this article is to compute the infinitesimal deformations of the first two families of restricted simple Lie algebras of Cartan type: the Witt-Jacobson algebras W(n)... |

5 |
Simple Lie algebras of characteristic 5
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- 1980
(Show Context)
Citation Context ... fixing a volume form, a Hamiltonian form and a contact form, respectively. Moreover in characteristic 5 there is one exceptional simple modular Lie algebra called the Melikian algebra (introduced in =-=[MEL80]-=-). We are interested in a particular class of modular Lie algebras called restricted. These can be characterized as those modular Lie algebras such that the p-power of an inner derivation (which in ch... |

5 |
Deformations of simple Lie algebras
- RUDAKOV
- 1971
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Citation Context ... also for the simple modular Lie algebras of classical type over a field of characteristic not dividing the determinant of the Killing form and the dimension of the Lie algebra. Actually Rudakov (see =-=[RUD71]-=-) showed that such Lie algebras are rigid if the characteristic of the base field is greater than or equal to 5 while in characteristic 2 and 3 there are non-rigid classical Lie algebras (see [CHE05],... |

5 |
The Classification of the Simple Modular Lie Algebras
- Strade
- 1992
(Show Context)
Citation Context ...ed the classification problem remained open for a long time until it was recently solved, if the characteristic of the base field is greater than 3, by Wilson-Block (see [BW88]), Strade (see [STR89], =-=[STR92]-=-, [STR91], [STR93], [STR94], [STR98]) and Premet-Strade (see [PS97], [PS99], [PS01]). The classification remains still open in characteristic 2 and 3 (see [STR04, page 209]). According to this classif... |

4 |
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Citation Context ...aev although we took from him the idea to consider relative cohomology with respect to the subalgebra of negative degree elements. As a byproduct of our proof, we recover the results of Celousov (see =-=[CEL70]-=-) on the first cohomology group of the adjoint representation (Theorems 3.3 and 4.5). The results presented here constitute part of my doctoral thesis. I thank my advisor prof. R. Schoof for useful ad... |

4 |
Deformations of the restricted Melikian algebra
- VIVIANI
(Show Context)
Citation Context ...e Lie algebras of classical type are known to be rigid over a field of characteristic different from 2 and 3 (see [RUD71]), in analogy of what happens in characteristic zero. In the papers [VIV1] and =-=[VIV2]-=-, the author computed the infinitesimal deformations of the Witt-Jacobson, Special Date: 02 May 2008. 1991 Mathematics Subject Classification. Primary 17B50; Secondary 17B20, 17B56. Key words and phra... |

4 |
Central extensions and invariant forms of graded Lie algebras
- Farnsteiner
- 1986
(Show Context)
Citation Context ...ult presented here constitute part of my doctoral thesis. I thank my advisor prof. Schoof for useful advices and constant encouragement. I thank the referee for the suggestion of using the results of =-=[FAR86]-=- and [FS91] in order to simply some of the proofs of this paper. 2. Contact algebra 2.1. Definition and Basic properties. We first introduce some notations about the set N n of n-tuple of natural numb... |

4 |
Dˇzumadildaev: Deformations of general Lie algebras of Cartan type (Russian
- S
- 1980
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Citation Context ...on that the infinitesimal deformations of simple Lie algebras of Cartan-type (in the general non-restricted case) has been considered already by Dˇzumadildaev-Kostrikin in [DK78] and Dˇzumadildaev in =-=[DZU80]-=-, [DZU81] and [DZU89] but a complete picture as well as detailed proofs were missing. More precisely: in [DK78] the authors compute the infinitesimal deformations of the Jacobson-Witt algebras of rank... |

4 |
Dˇzumadildaev: Relative cohomology and deformations of the Lie algebras of Cartan types (Russian). Dokl. Akad. Nauk SSSR 257
- S
- 1981
(Show Context)
Citation Context ...he infinitesimal deformations of simple Lie algebras of Cartan-type (in the general non-restricted case) has been considered already by Dˇzumadildaev-Kostrikin in [DK78] and Dˇzumadildaev in [DZU80], =-=[DZU81]-=- and [DZU89] but a complete picture as well as detailed proofs were missing. More precisely: in [DK78] the authors compute the infinitesimal deformations of the Jacobson-Witt algebras of rank 1, in [D... |

3 |
Kostrikin: Deformations of the Lie algebra W1(m) (Russian). Algebra, number theory and their applications. Trudy
- Dˇzumadildaev, I
- 1978
(Show Context)
Citation Context ...nitesimal deformations of simple Lie algebras of Cartan-type (in the general non-restricted case) have been considered already by Dˇzumadildaev in [DZU80, DZU81, DZU89] and Dˇzumadildaev-Kostrikin in =-=[DK78]-=- but a complete picture as well as detailed proofs were missing. More precisely: in [DK78] the authors compute the infinitesimal deformations of the Jacobson-Witt algebras of rank 1, in [DZU80, Theore... |

3 |
Dˇzumadildaev: Deformations of the Lie algebras Wn(m
- S
- 1989
(Show Context)
Citation Context ...mal deformation of the Jacobson-Witt algebras of any rank but without a proof, in [DZU81] a general strategy for the Jacobson-Witt and Hamiltonian algebras is outlined (without proofs) and finally in =-=[DZU89]-=- the author clarifies this strategy and then applies it to the Jacobson-Witt algebras but with a half-page sketch of the proof. i=1INFINITESIMAL DEFORMATIONS OF RESTRICTED SIMPLE LIE ALGEBRAS I 3 Our... |

3 |
Deformations of restricted simple Lie algebras I. math.RA/0612861
- Viviani
(Show Context)
Citation Context ...imple Lie algebras of classical type are known to be rigid over a field of characteristic different from 2 and 3 (see RUD [RUD71]), in analogy of what happens in characteristic zero. In the paper Viv =-=[VIV]-=-, the author computed the infinitesimal deformations of the Witt-Jacobson and Special Lie algebras. In this paper, we compute the infinitesimal defromations of the Contact algebras K(n) and the Hamilt... |

1 | Deformations of general Lie algebras of Cartan type - Dˇzumadildaev - 1980 |

1 |
Relative cohomology and deformations of the Lie algebras of Cartan types
- Dˇzumadildaev
- 1981
(Show Context)
Citation Context ...esimal deformations of the Jacobson-Witt algebras of rank 1, in [DZU80, Theorem 4] the author describes the infinitesimal deformation of the Jacobson-Witt algebras of any rank but without a proof, in =-=[DZU81]-=- a general strategy for the Jacobson-Witt and Hamiltonian algebras is outlined (without proofs) and finally in [DZU89] the author clarifies this strategy and then applies it to the Jacobson-Witt algeb... |

1 |
Deformations of simple finite group schemes, submitted. (available at arXiv:0705.0821
- Viviani
(Show Context)
Citation Context ...te the infinitesimal deformations of the remaining restricted simple Lie algebras of Cartan-type, namely the Hamiltonian, the Contact and the exceptional Melikian algebras. Moreover, in another paper =-=[VIV4]-=-, we apply these results to the study of the infinitesimal deformations of the simple finite group schemes corresponding to the restricted simple Lie algebras of Cartan type. Let us mention that the i... |