Geodesic
Lie Algebras Attached to Clifford Modules and Simple Graded Lie Algebras
Kenro Furutani1 ; Mauricio Godoy Molina2 ; Irina Markina3 ; Tohru Morimoto4 ; Alexander Vasil'ev3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
1Dept. of Mathematics, Science University of Tokyo, Japan
2Dep. de Matemática y Estadística, Universidad de la Frontera, Chile
3Dept. of Mathematics, University of Bergen, Norway
4Seki Kowa Inst. of Mathematics, Yokkaichi University, Inst. K. Oka de Mathématiques, Nara Women's University, Japan
5We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo H-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type B
6with |2|-grading do not contain non-Heisenberg pseudo H-type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types A
7, C
8and D
9provide such a possibility. Among exceptional algebras only F
10and E
11contain non-Heisenberg pseudo H-type Lie algebras as their negative part of |2|-grading. An analogous question addressed to real simple graded Lie algebras is more delicate, and we give results revealing the main differences with the complex situation.
12Keywords: Simple Lie algebras, root system, Dynkin diagram, graded Lie algebras, parabolic subalgebras, H-type algebra, Clifford algebra, non-degenerate bilinear form.
13MSC: 17B10, 17B22, 17B25, 22E46.
14[
Journal of Lie Theory, Tome 28 (2018) no. 3, pp. 843-864

Voir la notice de l'article provenant de la source Heldermann Verlag

We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo H-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type Bn with |2|-grading do not contain non-Heisenberg pseudo H-type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types An, Cn and Dn provide such a possibility. Among exceptional algebras only F4 and E6 contain non-Heisenberg pseudo H-type Lie algebras as their negative part of |2|-grading. An analogous question addressed to real simple graded Lie algebras is more delicate, and we give results revealing the main differences with the complex situation.
Classification : 17B10, 17B22, 17B25, 22E46
Mots-clés : Simple Lie algebras, root system, Dynkin diagram, graded Lie algebras, parabolic subalgebras, H-type algebra, Clifford algebra, non-degenerate bilinear form

Kenro Furutani  1   ; Mauricio Godoy Molina  2   ; Irina Markina  3   ; Tohru Morimoto  4   ; Alexander Vasil'ev  3 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14

1 Dept. of Mathematics, Science University of Tokyo, Japan
2 Dep. de Matemática y Estadística, Universidad de la Frontera, Chile
3 Dept. of Mathematics, University of Bergen, Norway
4 Seki Kowa Inst. of Mathematics, Yokkaichi University, Inst. K. Oka de Mathématiques, Nara Women's University, Japan
5 We study possible cases of complex simple graded Lie algebras of depth 2, which are the Tanaka prolongations of pseudo H-type Lie algebras arising through representation of Clifford algebras. We show that the complex simple Lie algebras of type B
6 with |2|-grading do not contain non-Heisenberg pseudo H-type Lie algebras as their negative nilpotent part, while the complex simple Lie algebras of types A
7 , C
8 and D
9 provide such a possibility. Among exceptional algebras only F
10 and E
11 contain non-Heisenberg pseudo H-type Lie algebras as their negative part of |2|-grading. An analogous question addressed to real simple graded Lie algebras is more delicate, and we give results revealing the main differences with the complex situation.
12 Keywords: Simple Lie algebras, root system, Dynkin diagram, graded Lie algebras, parabolic subalgebras, H-type algebra, Clifford algebra, non-degenerate bilinear form.
13 MSC: 17B10, 17B22, 17B25, 22E46.
14 [ 
Kenro Furutani; Mauricio Godoy Molina; Irina Markina; Tohru Morimoto; Alexander Vasil'ev. Lie Algebras Attached to Clifford Modules and Simple Graded Lie Algebras. Journal of Lie Theory, Tome 28 (2018) no. 3, pp. 843-864. http://geodesic.mathdoc.fr/item/JOLT_2018_28_3_a13/
@article{JOLT_2018_28_3_a13,
     author = {Kenro Furutani and Mauricio Godoy Molina and Irina Markina and Tohru Morimoto and Alexander Vasil'ev},
     title = {Lie {Algebras} {Attached} to {Clifford} {Modules} and {Simple} {Graded} {Lie} {Algebras}},
     journal = {Journal of Lie Theory},
     pages = {843--864},
     year = {2018},
     volume = {28},
     number = {3},
     url = {http://geodesic.mathdoc.fr/item/JOLT_2018_28_3_a13/}
}
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