Construction of Milnor’s Generalized Bases for Some Four-dimensional Metric Lie Algebras
Abstract
The Milnor bases are orthonormal bases with structure constants being dependent on a small number of parameters. These bases are convenient for calculations of left invariant tensor fields, and they are well known for three-dimensional Lie groups with a left invariant Riemannian metrics. However, a special technique is required for construction of similar bases for metric Lie algebras of a dimension higher than 3. In this paper, a method of Milnor generalized bases construction for four-dimensional metric Lie algebras with structural algebra constants being dependent on a small number of parameters is considered. This method is based on studying the space of orbits of Lie groups left invariant Riemannian metrics. The proposed method can be used for the basis construction in finite-dimensional metric Lie algebras. The G.M. Mubarakzyanov’s classification of real four-dimensional metric Lie algebras is adopted for the process of Milnor generalized bases construction. Also, well-known facts about the automorphism groups of these algebras are used. The constructed bases are useful for calculating and studying of invariant tensor fields and signatures of curvature operators on Lie groups with leftinvariant Riemannian metric.
DOI 10.14258/izvasu(2015)1.1-13
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References
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