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). Some authors also say Filippov algebra.
∞-Lie theory (higher geometry)
Background
Smooth structure
Higher groupoids
Lie theory
∞-Lie groupoids
∞-Lie algebroids
Formal Lie groupoids
Cohomology
Homotopy
Related topics
Examples
-Lie groupoids
-Lie groups
-Lie algebroids
-Lie algebras
An -Lie algebra is defined to be an algebraic structure which
looks formally like the special case of an -algebra for which only the -ary bracket is non-vanishing (see there);
but without necessarily the grading underlying an -algebra, and in particular without the requirement that be of homogeneous degree in any grading.
Therefore, any “-Lie algebras” that appear in the literature are not examples of Lie n-algebras, hence of L-∞ algebras. (So in particular -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).
A discussion of -Lie algebras (without the -grading) is in
V. T. Filippov, -Lie algebras, Sib. Math. Zh. No 6 126–140 (195)
V. T. Filippov, On the -Lie algebra of Jacobians, Sibirsk. Mat. Zh., 1998, Volume 39, Number 3, Pages 660–669 (English translation)
A. S. Dzhumadil’daev, Wronskians as -Lie multiplications (arXiv:math/0202043)
Similar (but different) discussion is in
Phil Hanlon, Michelle Wachs, On Lie -Algebras, Advances in Mathematics Volume 113, Issue 2, July 1995, Pages 206–236 (doi:10.1006/aima.1995.1038)
José de Azcárraga, J. C. Perez Bueno, Higher-order simple Lie algebras, Commun. Math. Phys. 184 (1997) 669-681 [arXiv:hep-th/9605213, doi:10.1007/s002200050079]
José de Azcárraga, José M. Izquierdo, -Ary algebras: a review with applications, J. Phys. A: Math. Theor. 43 (2010) 293001 [doi:10.1088/1751-8113/43/29/293001]
The notion of -Lie algebras, for , 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 -algebraic formalism.
Further re-inventions of the concept of -Lie algebras in this context are appearing. For instance in
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:
Sam Palmer, Christian Saemann, section 2 of M-brane Models from Non-Abelian Gerbes, JHEP 1207:010, 2012 (arXiv:1203.5757)
Patricia Ritter, Christian Saemann, section 2.5 of Lie 2-algebra models, JHEP 04 (2014) 066 (arXiv:1308.4892)
Christian Saemann, appendix A of Lectures on Higher Structures in M-Theory (arXiv:1609.09815)
José Figueroa-O'Farrill, Triple systems and Lie superalgebras in M2-branes, ADE and Lie superalgebras, talk at IPMU 2009 (pdf, pdf)
Last revised on June 19, 2023 at 14:55:29. See the history of this page for a list of all contributions to it.