Lie algebras with complex structures having nilpotent eigenspaces
Let (g,[·,·]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of gC isomorphic to the algebra (g,[*]J) with bracket [X * Y]J = 1/2 ([X,Y] â [JX, JY]). We consider here the case where these subalgebras are nilpotent and prove that the orig...
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| Autores principales: | , |
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| Lenguaje: | English |
| Publicado: |
Universidad Católica del Norte, Departamento de Matemáticas
2011
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| Materias: | |
| Acceso en línea: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0716-09172011000200008 |
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| Sumario: | Let (g,[·,·]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of gC isomorphic to the algebra (g,[*]J) with bracket [X * Y]J = 1/2 ([X,Y] â [JX, JY]). We consider here the case where these subalgebras are nilpotent and prove that the original (g,[·,·]) Lie algebra must be solvable. We consider also the 6-dimensional case and determine explicitly the possible nilpotent Lie algebras (g,[*]J). Finally we produce several examples illustrating different situations, in particular we show that for each given s there exists g with complex structure J such that (g,[*]J) is s-step nilpotent. Similar examples of hypercomplex structures are also built. |
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