Abstract
With the exception of the three step real free Lie algebra on two generators, all real free Lie algebras of step at least three are shown to have trivial Tanaka prolongation. This result, together with the known results concerning the step two real free Lie algebras and the step three real free Lie algebra on two generators, gives a complete list of Tanaka prolongations for real free Lie algebras.
Similar content being viewed by others
References
Gardner, R.B.: The method of equivalence and its applications, volume 58 of CBMS-NSF Regional Conference Series in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (1989)
Hall M. Jr. (1950). A basis for free Lie rings and higher commutators in free groups. Proc. Am. Math. Soc. 1: 575–581
Korányi A. and Reimann H.M. (1985). Quasiconformal mappings on the Heisenberg group. Invent. Math. 80(2): 309–338
Reutenauer C.: Free Lie algebras, volume 7 of London Mathematical Society Monographs. New Series. The Clarendon Press Oxford University Press. Oxford Science Publications, New York (1993)
Tanaka N. (1967). On generalized graded Lie algebras and geometric structures. I. J. Math. Soc. Japan 19: 215–254
Tanaka N. (1970). On differential systems, graded Lie algebras and pseudogroups. J. Math. Kyoto Univ. 10: 1–82
Yamaguchi, K.: Differential systems associated with simple graded Lie algebras. In Progress in differential geometry, volume 22 of Adv. Stud. Pure Math., pp. 413–494. Math. Soc. Japan, Tokyo (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Warhurst, B. Tanaka prolongation of free Lie algebras. Geom Dedicata 130, 59–69 (2007). https://doi.org/10.1007/s10711-007-9205-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-007-9205-1