∞-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
A Lie algebra is nilpotent if there exists a uniform such that acting via the Lie bracket on any one of its elements with other elements times yields zero. In other words, such that for any , , the composition .
The lower central series or (descending central series) of a Lie algebra is a sequence of nested ideals defined inductively by , . The Lie algebra is said to be nilpotent if for some .
In other words, a Lie algebra is nilpotent if and only the improper ideal is a nilpotent element in the ideal lattice with respect to the ideal product .
A Lie algebra is locally nilpotent if any finite-dimensional subalgebra is nilpotent.
A Lie algebra is ad-nilpotent if for any , its endomorphism is nilpotent. In general, this is a weaker property.
Every finite-dimensional matrix Lie algebra over a field is nilpotent iff it is ad-nilpotent.
Trivially: Every abelian Lie algebra is nilpotent.
The Lie algebra of even-degree elements of the tensor product of a super translation Lie algebra (super Minkowski super Lie algebra) with a superpoint is nilpotent.
A finite-dimensional Lie algebra is nilpotent precisely if its Chevalley-Eilenberg algebra is a Sullivan algebra (necessarily minimal). See also at rational homotopy theory for more on this.
Jean-Pierre Serre: Nilpotent and Solvable Lie Algebras, Chapter V in: Lie Algebras and Lie Groups – 1964 Lectures given at Harvard University, Lecture Notes in Mathematics 1500, Springer (1992) [doi:10.1007/978-3-540-70634-2]
Arthur A. Sagle, Ralph E. Walde: Nilpotent Lie Groups and Algebras, Chapter 11 in: Introduction to Lie Groups and Lie Algebras, Pure and Applied Mathematics 51, Elsevier (1973) 215-227 [doi:10.1016/S0079-8169(08)61671-2]
James Milne, pp. 260 in: Algebraic Groups – The theory of group schemes of finite type over a field, Cambridge University Press (2017) [doi:10.1017/9781316711736, webpage, pdf]
Laurence Corwin, Frederick P. Greenleaf: Section 1.2 of: Representations of Nilpotent Lie Groups and their Applications – Volume 1 Part 1. Basic Theory and Examples, Cambridge University Press (2004) [ISBN:9780521604956]
Victor Kac (notes by Marc Doss), Nilpotent and Solvable Lie algebras (2010) [pdf, pdf]
Daniel Huybrechts: Nilpotent Lie Algebras, Section 1 of Stability Structures on Lie Algebras, after Kontsevich and Soibelman (2011) [pdf, pdf]
Veronique Fischer, Michael Ruzhansky, Section 1.6 in: Quantization on Nilpotent Lie Groups, Birkhäuser (2016) [doi:10.1007/978-3-319-29558-9]
Nilpotent Lie algebras [pdf]
See also:
Wikipedia, Nilpotent Lie algebra
Wikipedia: Engel’s theorem
GroupProps: Lie correspondence between nilpotent Lie algebras and unipotent algebraic groups
On the Sullivan models which are Chevalley-Eilenberg algebra of nilpotent Lie algebras:
On nilpotent super Lie algebras:
Manfred Scheunert, §III.2 in: The theory of Lie superalgebras. An introduction, Lect. Notes Math. 716 (1979) [doi:10.1007/BFb0070929]
Luc Frappat, Antonio Sciarrino, Paul Sorba: §26 in: Dictionary on Lie Superalgebras, Academic Press (2000) [arXiv:hep-th/9607161, ISBN:978-0122653407]
Last revised on September 3, 2024 at 13:03:52. See the history of this page for a list of all contributions to it.