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
doubrov@islc.org
Olga Radko
Institute of Mathematics, Surganova 11, 220072 Minsk, Belarus
radko@islc.org
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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