Malcev Lie algebra

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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

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