#### DMCA

## Modules for certain Lie algebras of maximal class (1995)

Venue: | J. Pure Appl. Algebra |

Citations: | 13 - 7 self |

### Citations

197 |
Representation of nilpotent Lie groups and their applications. Part 1: Basic theory and examples
- Corwin
- 1990
(Show Context)
Citation Context ...a over k . Then there is asnite-dimensional vectorspace V together with a faithful representation % : g ! gl(V ) , such that %(X) is nilpotent for all X 2 g: 14 The construction goes as follows (see =-=[6]-=-): Let g be k -step nilpotent, g(1) = g and g(i+1) = [g; g(i)]: Choose a basis X1; : : : ; Xn of g such that thesrst n1 elements span g (k) , thesrst n2 elements span g (k1) and so on. We will constr... |

85 |
et algèbres de Lie, Ch
- Bourbaki
- 1972
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Citation Context ...the connected nilpotent Lie groups corresponding to the a(r; s; t) of Theorem A do not admit such an action. It should be noted, that the results are contradictory to the articles of Boyom and Nisse (=-=[3]-=- , [11]). 2 1. PRELIMINARIES Let k be aseld of characteristic zero and a(r; s; t) as dened in the introduction. In the following we consider thesliform algebras a(r; s; t) for r 2 A2 . They are gener... |

18 |
The structure of complete left-symmetric algebras
- Segal
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Citation Context ...ts c1; c2 > 0 . On the other hand there are classes of Lie algebras g for which one has the much better bound (g) dim g + 1: This holds, for instance, for all nilpotent Lie algebras of class 3 ( =-=[14]-=- ) or in low dimensions, for Z - graded Lie algebras, or for those which posses a nonsingular derivation. Accordingly it is quite dicult tosnd a nilpotent Lie algebra g with (g) > dim g + 1: Thesrst... |

14 |
Projective Geometry on Manifolds, Lecture Notes
- Goldman
- 1988
(Show Context)
Citation Context ...d anely on Rn . This includes the problem, pointed out by Milnor and Auslander ( [12] , [1] ), whether G always admits a complete left-invariant locallysat ane structure or not (see [5], [10], [7], =-=[8]-=-, [14], [13], [15] ). It is well known that once G has such an action then the Lie algebra g of G has a faithful module of dimension dim g+1 . More precisely, g then admits an ane structure, i.e. a f... |

8 | Radical d’une algèbre symétrique à gauche, Annales de l’Institut Fourier - Helmstetter - 1979 |

6 | Left-symmetric structures on simple modular Lie algebras
- Burde
- 1994
(Show Context)
Citation Context ... transitively and anely on Rn . This includes the problem, pointed out by Milnor and Auslander ( [12] , [1] ), whether G always admits a complete left-invariant locallysat ane structure or not (see =-=[5]-=-, [10], [7], [8], [14], [13], [15] ). It is well known that once G has such an action then the Lie algebra g of G has a faithful module of dimension dim g+1 . More precisely, g then admits an ane str... |

4 |
On fundamental groups of complete anely manifolds
- Milnor
- 1977
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Citation Context ...nnected nilpotent Lie groups corresponding to the a(r; s; t) of Theorem A do not admit such an action. It should be noted, that the results are contradictory to the articles of Boyom and Nisse ([3] , =-=[11]-=-). 2 1. PRELIMINARIES Let k be aseld of characteristic zero and a(r; s; t) as dened in the introduction. In the following we consider thesliform algebras a(r; s; t) for r 2 A2 . They are generated by... |

3 |
Simply transitive groups of ane motions
- AUSLANDER
- 1977
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Citation Context ...potent Lie groups G on ane space Rn . The problem here is to determine which such G act simply transitively and anely on Rn . This includes the problem, pointed out by Milnor and Auslander ( [12] , =-=[1]-=- ), whether G always admits a complete left-invariant locallysat ane structure or not (see [5], [10], [7], [8], [14], [13], [15] ). It is well known that once G has such an action then the Lie algebr... |

3 |
Three-dimensional ane crystallographic Groups
- Fried, Goldman
- 1983
(Show Context)
Citation Context ...ly and anely on Rn . This includes the problem, pointed out by Milnor and Auslander ( [12] , [1] ), whether G always admits a complete left-invariant locallysat ane structure or not (see [5], [10], =-=[7]-=-, [8], [14], [13], [15] ). It is well known that once G has such an action then the Lie algebra g of G has a faithful module of dimension dim g+1 . More precisely, g then admits an ane structure, i.e... |

3 |
Complete left-invariant ane structures on nilpotent Lie groups
- KIM
- 1986
(Show Context)
Citation Context ...sitively and anely on Rn . This includes the problem, pointed out by Milnor and Auslander ( [12] , [1] ), whether G always admits a complete left-invariant locallysat ane structure or not (see [5], =-=[10]-=-, [7], [8], [14], [13], [15] ). It is well known that once G has such an action then the Lie algebra g of G has a faithful module of dimension dim g+1 . More precisely, g then admits an ane structure... |

2 |
Une nilvariete non ane
- Benoist
- 1995
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Citation Context ...r those which posses a nonsingular derivation. Accordingly it is quite dicult tosnd a nilpotent Lie algebra g with (g) > dim g + 1: Thesrst example of this phenomenon was discovered by Y. Benoist ( =-=[2]-=- ) : Let a(r; s; t) be the Lie algebra given by the vector space generators e1; e2; e3; : : : and the relations [e1; ei] = ei+1 [e2; e3] = e5(1) [e2; e5] = re7 + se8 + te9 for i = 1; 2; 3; : : : and r... |

1 |
les structures anes homotopes a zero des groupes de
- BOYOM, Sur
- 1990
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Citation Context ... . We shall reduce this list now LEMMA 3.2 If a(r; s; t) has a - module M then we may assume that the type of M is one of the following: 6 (1) ; (2) fig i = 6; :::; 11 (3) fi; i + 1g i = 6; :::; 10 =-=(4)-=- fi; i + 1 j 11 + ig i = 6; :::; 10 (5) fi; i + 1; j j 11 + ig i = 6; 7; 8 j > i + 2 (6) fi; j; j + 1 j 11 + jg j = 6; :::; 10 i < j 1 Proof: If the module M has type fi; 11 j : : :g for i = 1; :::;... |

1 |
Structure ane des infranilvarietes et des infrasolvarietes
- NISSE
(Show Context)
Citation Context ...ted nilpotent Lie groups G on ane space Rn . The problem here is to determine which such G act simply transitively and anely on Rn . This includes the problem, pointed out by Milnor and Auslander ( =-=[12]-=- , [1] ), whether G always admits a complete left-invariant locallysat ane structure or not (see [5], [10], [7], [8], [14], [13], [15] ). It is well known that once G has such an action then the Lie ... |

1 |
Examples of locally ane spaces
- SCHEUNEMAN
- 1971
(Show Context)
Citation Context ... Rn . This includes the problem, pointed out by Milnor and Auslander ( [12] , [1] ), whether G always admits a complete left-invariant locallysat ane structure or not (see [5], [10], [7], [8], [14], =-=[13]-=-, [15] ). It is well known that once G has such an action then the Lie algebra g of G has a faithful module of dimension dim g+1 . More precisely, g then admits an ane structure, i.e. a faithful line... |

1 |
Derivations and ane structures of some nilpotent Lie algebras, Mem
- YAMAGUCHI
- 1980
(Show Context)
Citation Context ...This includes the problem, pointed out by Milnor and Auslander ( [12] , [1] ), whether G always admits a complete left-invariant locallysat ane structure or not (see [5], [10], [7], [8], [14], [13], =-=[15]-=- ). It is well known that once G has such an action then the Lie algebra g of G has a faithful module of dimension dim g+1 . More precisely, g then admits an ane structure, i.e. a faithful linear rep... |