On constructions of Lie (super) algebras and (𝜀,δ)-Freudenthal–Kantor triple systems defined by bilinear forms

2019 ◽  
Vol 19 (11) ◽  
pp. 2050223
Author(s):  
Noriaki Kamiya ◽  
Daniel Mondoc
Keyword(s):  
Lie Algebras ◽  
Complex Structure ◽  
Bilinear Forms ◽  
Simple Lie Algebras ◽  
Triple Systems ◽  
Dynkin Diagrams

In this work, we discuss a classification of [Formula: see text]-Freudenthal–Kantor triple systems defined by bilinear forms and give all examples of such triple systems. From these results, we may see a construction of some simple Lie algebras or superalgebras associated with their Freudenthal–Kantor triple systems. We also show that we can associate a complex structure into these ([Formula: see text]-Freudenthal–Kantor triple systems. Further, we introduce the concept of Dynkin diagrams associated to such [Formula: see text]-Freudenthal–Kantor triple systems and the corresponding Lie (super) algebra construction.

scholarly journals Classification of the restricted simple Lie algebras

Journal of Algebra ◽  
1988 ◽  
Vol 114 (1) ◽  
pp. 115-259 ◽  
Author(s):  
Richard E Block ◽  
Robert Lee Wilson
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras

Classification of forms of restricted simple lie algebras of cartan type

1991 ◽  
Vol 19 (6) ◽  
pp. 1603-1628 ◽  
Author(s):  
Shirlei Serconek ◽  
Robert Lee Wilson
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras ◽  
Cartan Type

scholarly journals Classification of simple Lie algebras on a lattice

10.1112/plms/pds042 ◽  
2012 ◽  
Vol 106 (3) ◽  
pp. 508-564 ◽  
Author(s):  
Kenji Iohara ◽  
Olivier Mathieu
Keyword(s):  
Lie Algebras ◽  
Simple Lie Algebras

scholarly journals Coclosed G2-structures inducing nilsolitons

Forum Mathematicum ◽  
2018 ◽  
Vol 30 (1) ◽  
pp. 109-128 ◽  
Author(s):  
Leonardo Bagaglini ◽  
Marisa Fernández ◽  
Anna Fino
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Complex Structure ◽  
Almost Complex Structure ◽  
Nilpotent Lie Algebras ◽  
Almost Complex ◽  
Metric Structures ◽  
Trivial Center ◽  
Nilsoliton Metric

Abstract We show obstructions to the existence of a coclosed {\mathrm{G}_{2}} -structure on a Lie algebra {\mathfrak{g}} of dimension seven with non-trivial center. In particular, we prove that if there exists a Lie algebra epimorphism from {\mathfrak{g}} to a six-dimensional Lie algebra {\mathfrak{h}} , with the kernel contained in the center of {\mathfrak{g}} , then any coclosed {\mathrm{G}_{2}} -structure on {\mathfrak{g}} induces a closed and stable three form on {\mathfrak{h}} that defines an almost complex structure on {\mathfrak{h}} . As a consequence, we obtain a classification of the 2-step nilpotent Lie algebras which carry coclosed {\mathrm{G}_{2}} -structures. We also prove that each one of these Lie algebras has a coclosed {\mathrm{G}_{2}} -structure inducing a nilsoliton metric, but this is not true for 3-step nilpotent Lie algebras with coclosed {\mathrm{G}_{2}} -structures. The existence of contact metric structures is also studied.

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