On the strong rigidity of solvable Lie algebrasOn the strong rigidity of solvable Lie algebras
Faculty of Sciences. Mathematics and Computer Science

Department of Mathematics - Computer Sciences

Fundamental Mathematics

conferenceObject

2006New York, 2006

Mathematics

Lecture notes in pure and applied mathematics. - New York

NOG Workshop of the International Algebra Congress, SEP 17-23, 2002, St Petersburg, RUSSIA

243(2006), p. 162-174

0075-8469

0-8247-2349-X

000236608600008

E

English (eng)

University of Antwerp

We call a finite-dimensional complex Lie algebra g strongly rigid if its universal enveloping algebra Ug is rigid as an associative algebra, i.e. every formal associative deformation is equivalent to the trivial deformation. The aim of this paper is to study the strong rigidity properties of solvable Lie algebras. First, we show that a strongly rigid Lie algebra has to be rigid as Lie algebra, this restricts the research to rigid Lie algebras. In addition the second scalar cohomology group has to vanish. Therefore the nilpotent Lie algebras of dimension greater or equal than two are not strongly rigid and the torus's dimension of strongly rigid solvable Lie algebra has to be one. Moreover, the Kontsevitch's theory of deformation quantization helps to see that every polynomial deformation of the linear Poisson structure on g* which induces a nonzero cohomology class of g leads to a nontrivial deformation of Ug. Since the rigidity is intimately related to cohomology, the cohomology groups are characterized. At last, we classify the n-dimensional strongly rigid solvable Lie algebras where n <= 6 and give some remarks on linearizability of their corresponding Poisson structure.

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