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Extending invariant complex structures. (English) Zbl 1352.17007

Authors’ abstract: We study the problem of extending a complex structure to a given Lie algebra \(\mathfrak{g}\), which is firstly defined on an ideal \(\mathfrak{h}\subset\mathfrak{g}\). We consider the next situations: \(\mathfrak{h}\) is either complex or it is totally real. The next question is to equip \(\mathfrak{g}\) with an additional structure, such as a (non)-definite metric or a symplectic structure and to ask either \(\mathfrak{h}\) is non-degenerate, isotropic, etc. with respect to this structure, by imposing a compatibility assumption. We show that this implies certain constraints on the algebraic structure of \(\mathfrak{g}\). Constructive examples illustrating this situation are shown, in particular computations in dimension six are given.

MSC:

17B05 Structure theory for Lie algebras and superalgebras
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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