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Journal of Lie Theory 20 (2010), No. 3, 525--541
Copyright Heldermann Verlag 2010

Graded Nilpotent Lie Algebras of Infinite Type

Boris Doubrov
Belarussian State University, Nezavisimosti av. 4, 220030 Minsk, Belarus

Olga Radko
Institute of Mathematics, Surganova 11, 220072 Minsk, Belarus

The paper gives the complete characterization of all graded nilpotent Lie algebras with infinite-dimensional Tanaka prolongation as extensions of graded nilpotent Lie algebras of lower dimension by means of a commutative ideal. We introduce a notion of weak characteristics of a vector distribution and prove that if a bracket-generating distribution of constant type does not have non-zero complex weak characteristics, then its symmetry algebra is necessarily finite-dimensional. The paper also contains a number of illustrative algebraic and geometric examples including the proof that any metabelian Lie algebra with a 2-dimensional center always has an infinite-dimensional Tanaka prolongation.

Keywords: Graded nilpotent Lie algebras, Tanaka prolongation, metabelian Lie algebras, Lie algebra cohomology.

MSC: 17B70, 53C30, 58A17

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