CR-structures on a real Lie algebra

✍ Scribed by Giuliana Gigante; Giuseppe Tomassini

• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 737 KB
• Volume: 94
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1016/0001-8708(92)90033-h

✦ Synopsis

Levi-fat and then the Lie group G, corresponding to go is a Levi-flat CR-manifold. Moreover if q is an ideal, then p (with the complex structure J) is a complex subalgebra. In particular Go contains a complex Lie subgroup of positive dimension.

In this paper we are interested in CR-structures of real codimension 1 and of "k-torsion 0" (Section 2) on a reductive Lie algebra of the first category [S, 71. This situation appears as the natural generalization of the complex case when the torsion is vanishing [6].

The main result we prove is that such CR-structures are determined by CR-structures on a compact Cartan subalgebra ho of go and by a direct sum of root spaces. As a consequence we obtain a complete description of the "Moduli space" (Section 4).

The main tool in our proof is provided by the results of Wolf and Malysev [ 10, 51 on decompositions of g of the form g,, + 9, 9' a Lie subalgebra. The above results can be generalized to the real codimension r > 1 under the additional hypothesis [V, V] c "Y-, [V", q] c q.