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An Introduction to Lie Algebras: Nilpotency and Engel's Theorem

Liu, Xinyun LU (2026) In Bachelor's Thesis in Mathematical Sciences MATK11 20261
Centre for Mathematical Sciences
Mathematics (Faculty of Sciences)
Mathematics (Faculty of Engineering)
Abstract
In 1890, Friedrich Engel sketched a proof of a theorem relating the nilpotency of a Lie algebra to nilpotent linear transformations. A complete proof was later given by his student K.A. Umlauf. In this thesis, we introduce the theory of the finite-dimensional Lie algebras, with particular emphasis on linear Lie algebras and illustrative examples. We investigate the algebraic structure of Lie algebras through concepts such as centralizers, normalizers and ideals. We then introduce nilpotent endomorphisms and representations of Lie algebras, which provide the necessary background for studying Engel's Theorem. Finally, we prove Engel's Theorem and apply it to characterize the nilpotency of a finite-dimensional Lie algebra in terms of its... (More)
In 1890, Friedrich Engel sketched a proof of a theorem relating the nilpotency of a Lie algebra to nilpotent linear transformations. A complete proof was later given by his student K.A. Umlauf. In this thesis, we introduce the theory of the finite-dimensional Lie algebras, with particular emphasis on linear Lie algebras and illustrative examples. We investigate the algebraic structure of Lie algebras through concepts such as centralizers, normalizers and ideals. We then introduce nilpotent endomorphisms and representations of Lie algebras, which provide the necessary background for studying Engel's Theorem. Finally, we prove Engel's Theorem and apply it to characterize the nilpotency of a finite-dimensional Lie algebra in terms of its adjoint representation. (Less)
Popular Abstract
Lie algebras are an important area of abstract algebra, with applications in fields such as geometry, theoretical physics, and computer science. This thesis introduces the basic concepts and results to study finite-dimensional Lie algebras and presents several examples of linear Lie algebras, which can be represented using matrices. The central result of the thesis is Engel’s Theorem, which provides a criterion for determining when a Lie algebra is nilpotent, meaning that repeatedly taking Lie brackets eventually produces zero. The theorem also illustrates how linear transformations can be used to understand the structure of Lie algebras.
Please use this url to cite or link to this publication:
author
Liu, Xinyun LU
supervisor
organization
course
MATK11 20261
year
type
M2 - Bachelor Degree
subject
publication/series
Bachelor's Thesis in Mathematical Sciences
report number
LUNFMA-4197-2026
ISSN
1654-6229
other publication id
2026:K28
language
English
id
9250409
date added to LUP
2026-09-14 16:30:01
date last changed
2026-09-14 16:30:01
@misc{9250409,
  abstract     = {{In 1890, Friedrich Engel sketched a proof of a theorem relating the nilpotency of a Lie algebra to nilpotent linear transformations. A complete proof was later given by his student K.A. Umlauf. In this thesis, we introduce the theory of the finite-dimensional Lie algebras, with particular emphasis on linear Lie algebras and illustrative examples. We investigate the algebraic structure of Lie algebras through concepts such as centralizers, normalizers and ideals. We then introduce nilpotent endomorphisms and representations of Lie algebras, which provide the necessary background for studying Engel's Theorem. Finally, we prove Engel's Theorem and apply it to characterize the nilpotency of a finite-dimensional Lie algebra in terms of its adjoint representation.}},
  author       = {{Liu, Xinyun}},
  issn         = {{1654-6229}},
  language     = {{eng}},
  note         = {{Student Paper}},
  series       = {{Bachelor's Thesis in Mathematical Sciences}},
  title        = {{An Introduction to Lie Algebras: Nilpotency and Engel's Theorem}},
  year         = {{2026}},
}