A new invariant of quadratic Lie algebras. (English) Zbl 1271.17004
In this nice paper, the authors define and study in detail a new invariant of quadratic Lie algebras. This invariant, called the dup-number, essentially measures the decomposability of the 3-form \(I\) defined by \(I(X,Y,Z)=B([X,Y],Z)\), where \(B\) denotes the bilinear form associated to a non-Abelian quadratic Lie algebra. It is shown that the range of the dup-number is \(0,1,3\). With the help of this invariant, the authors classify those quadratic Lie algebras for which the invariant does not vanish. In addition, some relevant questions concerning double extensions and the orbit method are also analyzed.
Reviewer: Rutwig Campoamor-Stursberg (Madrid)
MSC:
| 17B05 | Structure theory for Lie algebras and superalgebras |
| 17B20 | Simple, semisimple, reductive (super)algebras |
Keywords:
quadratic Lie algebras; invariants; double extensions; adjoint orbits; solvable Lie algebrasReferences:
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