Malcev Lie algebra

In mathematics, a Malcev Lie algebra, or Mal'tsev Lie algebra, is a generalization of a rational nilpotent Lie algebra, and Malcev groups are similar. Both were introduced by Quillen (1969, Appendix A3), based on the work of (Mal'cev 1949).

According to Papadima & Suciu (2004) a Malcev Lie algebra is a rational Lie algebra $L$ together with a complete, descending ${\mathbb {Q} }$ -vector space filtration $\{F_{r}L\}_{r\geq 1}$ , such that:

$F_{1}L=L$
$[F_{r}L,F_{s}L]\subset F_{r+s}L$
the associated graded Lie algebra $\oplus _{r\geq 1}F_{r}L/F_{r+1}L$ is generated by elements of degree one.

Malcev Lie algebras come up, e.g., in the theory of mixed Hodge structures.

References

Mal'cev, A. I. (1949), "Nilpotent torsion-free groups", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 13: 201–212, ISSN 0373-2436, MR 0028843
Papadima, Stefan; Suciu, Alexander I. (2004), "Chen Lie algebras", International Mathematics Research Notices (21): 1057–1086, doi:10.1155/S1073792804132017, ISSN 1073-7928, MR 2037049
Quillen, Daniel (1969), "Rational homotopy theory", Annals of Mathematics. Second Series, 90: 205–295, ISSN 0003-486X, JSTOR 1970725, MR 0258031

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