## Simple finite group schemes and their infinitesimal deformations (811)

### BibTeX

@MISC{811simplefinite,

author = {},

title = {Simple finite group schemes and their infinitesimal deformations},

year = {811}

}

### OpenURL

### Abstract

We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.

### Citations

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Citation Context ...bra of odd rank, sp(2n) is the symplectic algebra and so(2n) is the special orthogonal algebra of even rank. For the simple Lie algebras corresponding to the exceptional Dynkin diagrams, see the book =-=[19]-=- or the nice account in [3]. These simple Lie algebras admit a model over the integers via the (so-called) Chevalley bases. Therefore, via reduction modulo a prime p, one obtains a restricted Lie alge... |

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(Show Context)
Citation Context ...of a Dynkin diagram and an automorphism of a finite field. The resulting groups are called: 2 B2(2 2n+1 ), 3 D4(q 3 ), E6(q), 2 E6(q 2 ), E7(q), E8(q), F4(q), 2 F4(2 2n+1 ), G2(q), 2 G2(3 2n+1 ) (see =-=[7]-=-).Simple finite group schemes and their infinitesimal deformations 5 4 Classification of simple restricted Lie algebras First of all, we show that there is a bijection between simple restricted Lie a... |

323 |
Representations of Algebraic Groups
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Citation Context ...es in the adjoint representation (see [16]). The restricted cohomology is related to the ordinary cohomology by two spectral sequences { p,q E1 = HomFr(S p g, H q−p (g, M)) ⇒ H p+q ∗ (g, M) if p ̸= 2 =-=[20]-=-, E p,q 2 = HomFr(Λ q g, H p ∗ (g, M)) ⇒ Hp+q (g, M) [12], where S p g and Λ q g denote, respectively, the p-th symmetric power and the q-th alternating power, and HomFr denotes the homomorphisms whic... |

125 | The octonions
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Citation Context ...he symplectic algebra and so(2n) is the special orthogonal algebra of even rank. For the simple Lie algebras corresponding to the exceptional Dynkin diagrams, see the book [19] or the nice account in =-=[3]-=-. These simple Lie algebras admit a model over the integers via the (so-called) Chevalley bases. Therefore, via reduction modulo a prime p, one obtains a restricted Lie algebra over Fp, which is simpl... |

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Citation Context ...], [2] for a nice historical account). Simple Lie algebras over an algebraically closed field F of characteristic p ̸= 2, 3 have recently been classified by Block-Wilson-Premet-Strade (see [5], [29], =-=[28]-=-). The classification says that for p ≥ 7 the simple Lie algebras can be of two types: of classical type and of generalized Cartan type. The algebras of classical type are obtained by considering the ... |

34 |
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Citation Context ... examples can be constructed as finite-dimensional analogues of the four classes of infinitedimensional complex simple Lie algebras, which occurred in Cartan’s classification of Lie pseudogroups (see =-=[6]-=-). Denote with O(m) the divided power k-algebra in m-variables. It is the commutative and associative algebra with unit defined by the generators xα for α ∈ Nm satisfying the relations x α · x β = ( α... |

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(Show Context)
Citation Context ...]/(X p )) over a field k of characteristic p > 3 is simple with a degenerate Killing form. In the succeeding three decades, many more non-classical restricted simple Lie algebras have been found (see =-=[18]-=-, [14], [1], [15]). The first comprehensive conceptual approach to constructing these non-classical restricted simple Lie algebras was proposed by Kostrikin-Shafarevich and Kac (see [22], [23], [21]).... |

22 |
Graded Lie algebras of finite characteristic
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Citation Context ...nd (see [18], [14], [1], [15]). The first comprehensive conceptual approach to constructing these non-classical restricted simple Lie algebras was proposed by Kostrikin-Shafarevich and Kac (see [22], =-=[23]-=-, [21]). They showed that all the known examples can be constructed as finite-dimensional analogues of the four classes of infinitedimensional complex simple Lie algebras, which occurred in Cartan’s c... |

18 | The status of the classification of the finite simple groups
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Citation Context ...s (the simple Lie algebras have been classified only for p ̸= 2, 3). Simple finite groups were classified during the years 1955-1985 thanks to the contribution of many mathematicians (see [26], [27], =-=[2]-=- for a nice historical account). Simple Lie algebras over an algebraically closed field F of characteristic p ̸= 2, 3 have recently been classified by Block-Wilson-Premet-Strade (see [5], [29], [28]).... |

18 |
Cohomology of restricted Lie algebras
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Citation Context ...ted Lie algebra (g, [p]), the cohomology Hi (G; g) of G with values in the adjoint representation is equal to the restricted cohomology Hi ∗ (g, g) of g with values in the adjoint representation (see =-=[16]-=-). The restricted cohomology is related to the ordinary cohomology by two spectral sequences { p,q E1 = HomFr(S p g, H q−p (g, M)) ⇒ H p+q ∗ (g, M) if p ̸= 2 [20], E p,q 2 = HomFr(Λ q g, H p ∗ (g, M))... |

16 | A brief history of the classification of the finite simple groups
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(Show Context)
Citation Context ...objects (the simple Lie algebras have been classified only for p ̸= 2, 3). Simple finite groups were classified during the years 1955-1985 thanks to the contribution of many mathematicians (see [26], =-=[27]-=-, [2] for a nice historical account). Simple Lie algebras over an algebraically closed field F of characteristic p ̸= 2, 3 have recently been classified by Block-Wilson-Premet-Strade (see [5], [29], [... |

16 |
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Citation Context ...], [27], [2] for a nice historical account). Simple Lie algebras over an algebraically closed field F of characteristic p ̸= 2, 3 have recently been classified by Block-Wilson-Premet-Strade (see [5], =-=[29]-=-, [28]). The classification says that for p ≥ 7 the simple Lie algebras can be of two types: of classical type and of generalized Cartan type. The algebras of classical type are obtained by considerin... |

13 |
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Citation Context ...s 9 Theorem 5.2 (Rudakov). If k is a field of char(k) = p ≥ 5 and g is a simple Lie algebra of classical type, then g is rigid. Note, however, that the preceding result is false if p = 2, 3 (see [9], =-=[10]-=-, [8]). 5.3 Deformations of Lie algebras of Cartan type The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in [34] (building upo... |

12 |
Deformations of classical Lie algebras
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Citation Context ...ations 9 Theorem 5.2 (Rudakov). If k is a field of char(k) = p ≥ 5 and g is a simple Lie algebra of classical type, then g is rigid. Note, however, that the preceding result is false if p = 2, 3 (see =-=[9]-=-, [10], [8]). 5.3 Deformations of Lie algebras of Cartan type The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in [34] (buildi... |

11 |
On Finite Simple Groups and Their Classification
- Solomon
- 1995
(Show Context)
Citation Context ...imple objects (the simple Lie algebras have been classified only for p ̸= 2, 3). Simple finite groups were classified during the years 1955-1985 thanks to the contribution of many mathematicians (see =-=[26]-=-, [27], [2] for a nice historical account). Simple Lie algebras over an algebraically closed field F of characteristic p ̸= 2, 3 have recently been classified by Block-Wilson-Premet-Strade (see [5], [... |

8 |
Simple Lie algebras of characteristic p
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(Show Context)
Citation Context ...er a field k of characteristic p > 3 is simple with a degenerate Killing form. In the succeeding three decades, many more non-classical restricted simple Lie algebras have been found (see [18], [14], =-=[1]-=-, [15]). The first comprehensive conceptual approach to constructing these non-classical restricted simple Lie algebras was proposed by Kostrikin-Shafarevich and Kac (see [22], [23], [21]). They showe... |

7 |
On the classification of simple Lie algebras over a field of nonzero characteristic
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(Show Context)
Citation Context ...e [18], [14], [1], [15]). The first comprehensive conceptual approach to constructing these non-classical restricted simple Lie algebras was proposed by Kostrikin-Shafarevich and Kac (see [22], [23], =-=[21]-=-). They showed that all the known examples can be constructed as finite-dimensional analogues of the four classes of infinitedimensional complex simple Lie algebras, which occurred in Cartan’s classif... |

7 |
Shafarevich: Cartan’s pseudogroups and the p-algebras of Lie (Russian). Dokl. Akad. Nauk SSSR 168
- Kostrikin, R
- 1966
(Show Context)
Citation Context ...en found (see [18], [14], [1], [15]). The first comprehensive conceptual approach to constructing these non-classical restricted simple Lie algebras was proposed by Kostrikin-Shafarevich and Kac (see =-=[22]-=-, [23], [21]). They showed that all the known examples can be constructed as finite-dimensional analogues of the four classes of infinitedimensional complex simple Lie algebras, which occurred in Cart... |

6 |
The restricted simple Lie algebras are of classical or Cartan type
- Block, Wilson
- 1984
(Show Context)
Citation Context ...e [26], [27], [2] for a nice historical account). Simple Lie algebras over an algebraically closed field F of characteristic p ̸= 2, 3 have recently been classified by Block-Wilson-Premet-Strade (see =-=[5]-=-, [29], [28]). The classification says that for p ≥ 7 the simple Lie algebras can be of two types: of classical type and of generalized Cartan type. The algebras of classical type are obtained by cons... |

6 |
Deformations of classical Lie algebras with a homogeneous root system in characteristic two
- Chebochko
(Show Context)
Citation Context ...eorem 5.2 (Rudakov). If k is a field of char(k) = p ≥ 5 and g is a simple Lie algebra of classical type, then g is rigid. Note, however, that the preceding result is false if p = 2, 3 (see [9], [10], =-=[8]-=-). 5.3 Deformations of Lie algebras of Cartan type The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in [34] (building upon [30... |

6 |
Cohomology groups of reduced enveloping algebras
- Farnsteiner
- 1991
(Show Context)
Citation Context ...ed cohomology is related to the ordinary cohomology by two spectral sequences { p,q E1 = HomFr(S p g, H q−p (g, M)) ⇒ H p+q ∗ (g, M) if p ̸= 2 [20], E p,q 2 = HomFr(Λ q g, H p ∗ (g, M)) ⇒ Hp+q (g, M) =-=[12]-=-, where S p g and Λ q g denote, respectively, the p-th symmetric power and the q-th alternating power, and HomFr denotes the homomorphisms which are Frobenius linear. In the case of the adjoint repres... |

6 |
Two new classes of simple Lie algebras
- Frank
- 1964
(Show Context)
Citation Context ...field k of characteristic p > 3 is simple with a degenerate Killing form. In the succeeding three decades, many more non-classical restricted simple Lie algebras have been found (see [18], [14], [1], =-=[15]-=-). The first comprehensive conceptual approach to constructing these non-classical restricted simple Lie algebras was proposed by Kostrikin-Shafarevich and Kac (see [22], [23], [21]). They showed that... |

6 | Infinitesimal deformations of restricted simple Lie algebras II
- VIVIANI
(Show Context)
Citation Context ...5.3 Deformations of Lie algebras of Cartan type The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in [34] (building upon [30], =-=[31]-=- and [32]) the infinitesimal deformations of the simple finite group schemes associated to simple and restricted Lie algebras of Cartan type. Theorem 5.3 ([34]). ⎧⎪ h ⎨ 2 ∗(W(n; 1), W(n; 1)) = n, h 2 ... |

5 |
Trace forms on Lie algebras
- Block
- 1962
(Show Context)
Citation Context ...on-degenerate except at a finite number of primes. Moreover, they can be characterized as those restricted simple Lie algebras admitting a projective representation with nondegenerate trace form (see =-=[4]-=-). 4.2 Lie algebras of Cartan type However, there are restricted simple Lie algebras which have no analogous in characteristic zero and therefore are called non-classical. The first example of a non-c... |

5 | Simple Lie algebras of characteristic 5 - MELIKIAN - 1980 |

5 |
Deformations of simple Lie algebras
- RUDAKOV
- 1971
(Show Context)
Citation Context ...e Killing form. If char(k) = p does not divide the discriminant of the Killing form, then the same proof gives the rigidity of simple Lie algebras of classical type in charactestic p. Indeed Rudakov (=-=[25]-=-) has shown the followingSimple finite group schemes and their infinitesimal deformations 9 Theorem 5.2 (Rudakov). If k is a field of char(k) = p ≥ 5 and g is a simple Lie algebra of classical type, ... |

4 |
Deformations of the restricted Melikian algebra
- VIVIANI
(Show Context)
Citation Context ...mations of Lie algebras of Cartan type The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in [34] (building upon [30], [31] and =-=[32]-=-) the infinitesimal deformations of the simple finite group schemes associated to simple and restricted Lie algebras of Cartan type. Theorem 5.3 ([34]). ⎧⎪ h ⎨ 2 ∗(W(n; 1), W(n; 1)) = n, h 2 ∗(S(n; 1)... |

4 | Restricted simple Lie algebras and their infinitesimal deformations, Proceedings Conference ”From Lie Algebras to Quantum Groups”, Centro Internacional de Matematica de Coimbra no. 28, 2006 (available at arXiv:math.RA/0702755 - Viviani |

2 |
Infinitesimal deformations of restricted simple Lie algebras I
- VIVANI
(Show Context)
Citation Context ...[8]). 5.3 Deformations of Lie algebras of Cartan type The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in [34] (building upon =-=[30]-=-, [31] and [32]) the infinitesimal deformations of the simple finite group schemes associated to simple and restricted Lie algebras of Cartan type. Theorem 5.3 ([34]). ⎧⎪ h ⎨ 2 ∗(W(n; 1), W(n; 1)) = n... |

1 |
A new class of simple Lie algebras,Proc
- Frank
- 1954
(Show Context)
Citation Context ... )) over a field k of characteristic p > 3 is simple with a degenerate Killing form. In the succeeding three decades, many more non-classical restricted simple Lie algebras have been found (see [18], =-=[14]-=-, [1], [15]). The first comprehensive conceptual approach to constructing these non-classical restricted simple Lie algebras was proposed by Kostrikin-Shafarevich and Kac (see [22], [23], [21]). They ... |

1 |
Abstract derivations and Lie algebras,Trans
- Jacobson
- 1937
(Show Context)
Citation Context ...be the finite group schemes of height one, we need to recall the definition of the restricted Lie algebras (sometimes called p-Lie algebras) over a field k of positive characteristic. Definition 2.2 (=-=[17]-=-). A Lie algebra L over a field k of characteristic p > 0 is said to be restricted (or a p-Lie algebra) if it is endowed with a map (called p-map) [p] : L → L, x ↦→ x [p] , which verifies the followin... |

1 |
Deformations of simple finite group schemes. Preprint available at arXiv:0705.0821
- Viviani
(Show Context)
Citation Context ..., 3 (see [9], [10], [8]). 5.3 Deformations of Lie algebras of Cartan type The Lie algebras of Cartan type do have infinitesimal deformations, unlike the Lie algebras of classical type. We computed in =-=[34]-=- (building upon [30], [31] and [32]) the infinitesimal deformations of the simple finite group schemes associated to simple and restricted Lie algebras of Cartan type. Theorem 5.3 ([34]). ⎧⎪ h ⎨ 2 ∗(W... |

1 | Introduction to Affine Group Schemes - Watherhouse - 1979 |