nLab nilpotent Lie algebra

Contents

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A Lie algebra is nilpotent if there exists a uniform nn such that acting via the Lie bracket on any one of its elements with other elements nn times yields zero. In other words, nN\exists n\in\mathbf{N} such that for any x ix_i, i=1,,ni = 1,\ldots, n, the composition adx 1adx 2adx n=0ad x_1 ad x_2 \ldots ad x_n = 0.

Definition

The lower central series or (descending central series) of a Lie algebra 𝔤\mathfrak{g} is a sequence of nested ideals 𝔤 k+1𝔤 k\mathfrak{g}^{k+1} \trianglelefteq \mathfrak{g}^{k} defined inductively by 𝔤 1𝔤\mathfrak{g}^1 \coloneqq \mathfrak{g}, 𝔤 k+1[𝔤,𝔤 k]\mathfrak{g}^{k+1} \coloneqq [\mathfrak{g}, \mathfrak{g}^k]. The Lie algebra is said to be nilpotent if 𝔤 k=0\mathfrak{g}^{k} = 0 for some kk \in \mathbb{N}.

In other words, a Lie algebra 𝔤\mathfrak{g} is nilpotent if and only the improper ideal 𝔤\mathfrak{g} 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 xx, its endomorphism adxad x is nilpotent. In general, this is a weaker property.

Engel’s theorem

Every finite-dimensional matrix Lie algebra Lgl(m)L\subset gl(m) over a field is nilpotent iff it is ad-nilpotent.

Examples

Properties

  • Every nilpotent Lie algebra is solvable?.

Relation to Sullivan models

A finite-dimensional Lie algebra 𝔤\mathfrak{g} is nilpotent precisely if its Chevalley-Eilenberg algebra CE(𝔤)CE(\mathfrak{g}) is a Sullivan algebra (necessarily minimal). See also at rational homotopy theory for more on this.

References

  • 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:

On the Sullivan models which are Chevalley-Eilenberg algebra of nilpotent Lie algebras:

On nilpotent super Lie algebras:

Last revised on September 3, 2024 at 13:03:52. See the history of this page for a list of all contributions to it.