arXiv subjects
Publications and source records attributed to Louis Magnin.
We consider the class of Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals, of finite dimensional semi-simple Lie algebras. A list of Kac-Moody types for indecomposable nilpotent Lie algebras of dimension at most 7 is given.
We compare the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones by a very elementary approach. The comparison gives some conditions, which are easy to verify for a given Lie algebra, for deciding whether it has more Leibniz deformations than just the Lie ones. We also give the complete description of a Leibniz (and Lie) versal deformation of the 4-dimensional diamond Lie algebra, and study the case of its 5-dimensional analogue.
We compute the torsion-free linear maps from the Lie algebra su(2) into itself, deduce a new determination of the integrable complex structures and their equivalence classes under the action of the automorphism group for u(2) and su(2)xsu(2), and prove that in both cases the set of complex structures is a differentiable manifold. u(2)x u(2), su(2)^N and u(2)^N are also considered. Extensions of complex structures from u(2) to su(2)xsu(2) are studied, local holomorphic charts given, and attention is paid to what representations of u(2) we can get from a substitute to the regular representation on a space of holomorphic functions for the complex structure.
We compare by a very elementary approach the second adjoint and trivial Leibniz cohomology spaces of a Lie algebra to the usual ones. Examples are given of coupled cocycles. Some properties are deduced as to Leibniz deformations. We also consider the class of Lie algebras for which the Koszul 3-form is zero, and prove that it contains all quotients of Borel subalgebras, or of their nilradicals, of finite dimensional semisimple Lie algebras. Finally, a list of Kac-Moody types for indecomposable nilpotent Lie algebras of dimension $\leqslant 7$ is given.