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 $\{F_r L\}_{r\ge 1}$, such that:

• $F_1 L = L$
• $[F_rL, F_sL]\subset F_{r+s}L$
• the associated graded Lie algebra $\oplus_{r\ge 1} F_rL/F_{r+1}L$ 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