nLab
n-Lie algebra

-Lie algebras

This entry is about the concept that became famous with the BLG model while it is NOT about what in homotopy theory are known as Lie n-algebras (homotopy-theoretic Lie algebras), NOR n-algebras (categorifies associative algebras).


Context

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

nn-Lie algebras

Idea

An nn-Lie algebra is defined to be an algebraic structure which

  • looks formally like the special case of an L L_\infty-algebra for which only the nn-ary bracket D nD_n is non-vanishing (see there);

  • but without necessarily the grading underlying an L L_\infty-algebra, and in particular without the requirement that D nD_n be of homogeneous degree 1-1 in any grading.

Therefore, any “nn-Lie algebras” that appear in the literature are not examples of Lie n-algebras, hence of L-∞ algebras. (So in particular nn-Lie algebras in this sense in general don’t integrate to Lie infinity-groupoids via the usual Lie theory. )

Instead, at least certain “3-Lie algebras” can be understood as encoding structure in Lie 2-algebras equipped with a binary invariant polynomial (Saemann-Ritter 13, section 2.5).

References

Original

A discussion of nn-Lie algebras (without the L L_\infty-grading) is in

  • v. T. Filippov, nn-Lie algebras, Sib. Math. Zh. No 6 126–140 (195)

  • v. T. Filippov, On the nn-Lie algebra of Jacobians, Sibirsk. Mat. Zh., 1998, Volume 39, Number 3, Pages 660–669 (English translation)

  • A. S. Dzhumadil’daev, Wronskians as nn-Lie multiplications (arXiv:math/0202043)

Similar (but different) discussion is in

  • P. Hanlon, M. Wachs, On Lie k-Algebras, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236

  • José de Azcárraga, J. C. Perez Bueno, Higher-order simple Lie algebras, (arXiv:hep-th/9605213)

Re-inventions

The notion of nn-Lie algebras, for n=3n=3, was re-invented, in the form of the M-brane 3-algebra by string physicists in the BLG model

which sparked a tremendous amount of activity.

See the blog entry

for further details and links. And see this blog discussion

for discussion about the relation to proper L L_\infty-algebraic formalism.

Further re-inventions of the concept of nn-Lie algebras in this context are appearing. For instance in

  • Tamar Friedman, Orbifold singularities, the LATKe, and Yang-Mills with Matter (arXiv)

Relation to representations of metric Lie algebras

The full generalized axioms on the M2-brane 3-algebra and first insights into their relation to Lie algebra representations of metric Lie algebras is due to

The full identification of M2-brane 3-algebras with dualizable Lie algebra representations over metric Lie algebras is due to

See also:

Last revised on December 6, 2019 at 10:09:23. See the history of this page for a list of all contributions to it.