Journal of Lie Theory 24 (2014), No. 1, 001--027
Copyright Heldermann Verlag 2014
Applications of Index Sets and Nikolayevsky Derivations to Positive Rank Nilpotent Lie Algebras
Tracy L. Payne
Dept. of Mathematics, Idaho State University, 921 S. 8th Ave., Pocatello, ID 83209-8085, U.S.A.
payntrac@isu.edu
We consider real nilpotent Lie algebras of positive rank. We fix a set Λ indexing the nonzero structure constants for a Lie algebra g with respect to a basis of eigenvectors for an R-split torus in the derivation algebra of g. We give criteria for when two Lie algebras with the same index set are isomorphic. We present a criterion for when there is a nilsoliton metric Lie algebra having a given index set, and we determine which nilsoliton metric Lie algebras have a given index set, up to isometric isomorphism and rescaling, in some common situations. We study the Nikolayevsky derivation, showing that it commutes with automorphisms that preserve certain inner products, and we find conditions on the Nikolayevsky derivation that insure that the isometry group of a metric Lie algebra is finite. We give examples showing that index sets and the Nikolayevsky derivation are useful invariants for nilpotent Lie algebras.
Keywords: Nilpotent Lie algebra, nilsoliton, soliton metric, soliton inner product, pre-Einstein derivation, Nikolayevsky derivation, isometry group.
MSC: 22E25, 17B30, 53C25
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