# Lie group

A Lie group is a set endowed with compatible structures of a group and a smooth manifold: the group elements can be described by coordinates, and the operations of multiplication \( (g,h) \mapsto gh \) and inversion \( g \mapsto g^{-1} \) are smooth maps<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup><sup> • </sup><sup>[2](https://youyuanwu.github.io/misc-docs/math/01-group-theory/15-lie-groups.html)</sup>. The name honors [Sophus Lie](https://www.edgepedia.org/sophus-lie), the Norwegian mathematician who developed the theory of continuous transformation groups in the winter of 1873–74<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Lie/)</sup>. Lie groups are the mathematical setting for continuous symmetry: rotations, Lorentz transformations, and gauge transformations all form Lie groups<sup>[2](https://youyuanwu.github.io/misc-docs/math/01-group-theory/15-lie-groups.html)</sup>.

| Key fact | Detail |
|---|---|
| Definition | A group \( G \) that is a smooth manifold such that \( (x,y) \mapsto xy^{-1} \) is smooth (analytic in the original formulation)<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup> |
| Variants | Real, complex, or \( p \)-adic, according to the field of the manifold; every complex Lie group is naturally a real one<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup> |
| Matrix form | Closed subgroups of \( \mathrm{GL}(n,\mathbb{R}) \) are Lie groups (J. von Neumann, 1927)<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup> |
| Lie algebra | The tangent space at the identity, equivalently the left-invariant vector fields, with a bracket measuring noncommutation<sup>[4](https://books.physics.oregonstate.edu/GELG/lie.html)</sup><sup> • </sup><sup>[5](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)</sup> |
| Exponential map | A smooth map \( \exp: \mathfrak{g} \to G \) carrying \( X \) to the one-parameter subgroup \( \exp(tX) \); a diffeomorphism near \( 0 \)<sup>[5](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)</sup> |
| Surjectivity of exp | Always surjective for compact connected \( G \); fails for \( \mathrm{SL}(2,\mathbb{R}) \)<sup>[6](https://terrytao.wordpress.com/wp-content/uploads/2014/11/gsm-153.pdf)</sup> |
| Structure theorem | Every connected Lie group is diffeomorphic to \( K \times \mathbb{R}^n \) for a maximal compact subgroup \( K \), unique up to conjugation<sup>[7](https://math.ucr.edu/home/baez/lie_groups/)</sup> |
| Main applications | Particle physics (\( \mathrm{SU}(n) \), Lorentz and Poincaré groups) and robotics (\( \mathrm{SO}(3) \), \( \mathrm{SE}(3) \))<sup>[8](https://arxiv.org/abs/2012.00834)</sup><sup> • </sup><sup>[9](https://www.cis.upenn.edu/~cis6100/geombchap14.pdf)</sup> |

## Definition and first examples

The compatibility requirement is what distinguishes a Lie group from an arbitrary group that happens to carry a manifold structure. Requiring the single map \( (x,y) \mapsto xy^{-1} \) to be analytic, as the Encyclopedia of Mathematics formulation does, is equivalent to requiring both multiplication and inversion to be smooth<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup>. A Lie group is called real, complex, or \( p \)-adic depending on the field over which its manifold is considered; most of the theory concerns the real case<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup>.

**Matrix groups supply the standard examples.** The general linear group \( \mathrm{GL}(n,\mathbb{R}) \) of invertible real \( n \times n \) matrices is a Lie group, and so is any subgroup of it closed in the natural Euclidean topology, a result due to J. von Neumann in 1927<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup>. Familiar examples include \( \mathbb{R}^n \) under translation, the circle \( U(1) \), and the groups \( \mathrm{SO}(n) \), \( \mathrm{SU}(n) \), \( \mathrm{SL}(n,\mathbb{R}) \), \( \mathrm{SL}(n,\mathbb{C}) \), the [Lorentz group](https://www.edgepedia.org/lorentz-group), and the [Poincaré group](https://www.edgepedia.org/poincare-group)<sup>[2](https://youyuanwu.github.io/misc-docs/math/01-group-theory/15-lie-groups.html)</sup>. For matrix groups, smoothness simply means that the matrix entries depend smoothly on the parameters<sup>[2](https://youyuanwu.github.io/misc-docs/math/01-group-theory/15-lie-groups.html)</sup>.

The closed subgroup theorem of Cartan makes the matrix picture essentially general: a closed subgroup of a linear Lie group is itself a Lie group with the subspace topology<sup>[10](https://bpb-us-w2.wpmucdn.com/faculty.umaine.edu/dist/1/19/files/2022/08/liegroups.pdf)</sup>. In fact every finite-dimensional Lie group can be regarded locally as a matrix group<sup>[4](https://books.physics.oregonstate.edu/GELG/lie.html)</sup>.

## The Lie algebra

To each Lie group \( G \) is attached its [Lie algebra](https://www.edgepedia.org/lie-algebra) \( \mathfrak{g} \), most simply described as the tangent space at the identity element<sup>[4](https://books.physics.oregonstate.edu/GELG/lie.html)</sup>. Equivalently, \( \mathrm{Lie}(G) \) is the algebra of left-invariant vector fields on \( G \)<sup>[5](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)</sup>. The bracket operation \( [X,Y] \) on this vector space measures how the corresponding infinitesimal transformations fail to commute; Lie himself combined transformations in a way he called an "infinitesimal group", which is what is today called a Lie algebra<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Lie/)</sup>.

For matrix groups the Lie algebra is a concrete space of matrices. For \( U(n) \) it is \( \mathfrak{u}(n) \), the skew-Hermitian matrices; for \( \mathrm{SL}(n,\mathbb{R}) \) it is \( \mathfrak{sl}(n,\mathbb{R}) \), the trace-free matrices; for \( \mathrm{SU}(n) \) it is the trace-free skew-Hermitian matrices; and differentiating the condition \( R(t)^T R(t) = I \) shows that the Lie algebra of \( \mathrm{SO}(n) \) consists of the skew-symmetric matrices<sup>[11](https://www.homepages.ucl.ac.uk/~ucahjde/lg/lg-notes.pdf)</sup>.

The practical value of the Lie algebra is that it reduces the study of a complicated analytic object, the group, largely to the study of a purely algebraic object<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup>. A Lie group homomorphism \( F: G \to H \) induces a Lie algebra homomorphism \( F_*: \mathrm{Lie}(G) \to \mathrm{Lie}(H) \), and when \( G \) is simply connected every Lie algebra homomorphism lifts back to a Lie group homomorphism<sup>[5](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)</sup><sup> • </sup><sup>[10](https://bpb-us-w2.wpmucdn.com/faculty.umaine.edu/dist/1/19/files/2022/08/liegroups.pdf)</sup>.

## The exponential map

The bridge between the group and its algebra is the exponential map \( \exp: \mathrm{Lie}(G) \to G \), which sends an element \( X \) to the value at time 1 of the integral curve of \( X \) starting at the identity; the curve \( s \mapsto \exp(sX) \) is the one-parameter subgroup generated by \( X \)<sup>[5](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)</sup>.

The main properties, as summarized in standard lecture notes, are<sup>[5](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)</sup>:

- \( \exp \) is smooth, and its differential at the origin is the identity.
- \( \exp((s+t)X) = \exp(sX)\,\exp(tX) \), with \( (\exp X)^{-1} = \exp(-X) \) and \( (\exp X)^n = \exp(nX) \).
- \( \exp \) restricts to a diffeomorphism from some neighborhood of \( 0 \) in \( \mathfrak{g} \) onto a neighborhood of the identity in \( G \), giving exponential coordinates near \( e \)<sup>[5](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)</sup><sup> • </sup><sup>[11](https://www.homepages.ucl.ac.uk/~ucahjde/lg/lg-notes.pdf)</sup>.
- The map is natural: for a Lie group homomorphism \( F \), \( \exp \circ F_* = F \circ \exp \)<sup>[5](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)</sup>.

**Globally, the exponential map need not be a homeomorphism.** Injectivity fails for any group containing a circle subgroup, a class that includes all positive-dimensional compact Lie groups and all non-simply-connected Lie groups: on the circle group \( S^1 \), whose Lie algebra is \( \mathbb{R} \), the map \( t \mapsto e^{2\pi i t} \) is periodic and many-to-one<sup>[6](https://terrytao.wordpress.com/wp-content/uploads/2014/11/gsm-153.pdf)</sup>. Surjectivity fails for disconnected groups, since the image of a connected domain is connected, and it can fail even for connected groups: the matrix \( \begin{pmatrix} -4 & 0 \\ 0 & -1/4 \end{pmatrix} \) in \( \mathrm{SL}(2,\mathbb{R}) \) has no square root in that group, so it cannot be an exponential<sup>[6](https://terrytao.wordpress.com/wp-content/uploads/2014/11/gsm-153.pdf)</sup>. Surjectivity is recovered in important cases: for connected, simply connected, nilpotent Lie groups, and for every compact connected Lie group<sup>[6](https://terrytao.wordpress.com/wp-content/uploads/2014/11/gsm-153.pdf)</sup>.

## Structure and classification

A connected topological group has no proper open subgroups, a fact used to prove global properties of connected Lie groups from local ones<sup>[12](https://math.stanford.edu/~conrad/210CPage/handouts/lie_groups_notes.pdf)</sup>. Globally, every connected Lie group is diffeomorphic to the product of a maximal compact subgroup \( K \) with a [Euclidean space](https://www.edgepedia.org/euclidean-space) \( \mathbb{R}^n \), and \( K \) is unique up to conjugation<sup>[7](https://math.ucr.edu/home/baez/lie_groups/)</sup>.

Classification proceeds through the Lie algebra. The classification of real semisimple Lie groups is more involved than the complex case and depends on the classification of their real forms<sup>[1](https://encyclopediaofmath.org/wiki/Lie_group)</sup>. A continuous homomorphism between closed subgroups of general linear groups is automatically analytic, so the Lie structure is uniquely determined by the topology<sup>[13](https://www.math.ucla.edu/~vsv/liegroups2007/6.pdf)</sup>.

## Applications in physics

Symmetry underlies the theoretical study of particle physics, and Lie groups supply its language: the special unitary groups \( \mathrm{SU}(n) \) and the Lorentz and Poincaré groups are the central examples, treated through their Lie algebras and representations<sup>[8](https://arxiv.org/abs/2012.00834)</sup>. Gauge theories rest on unitary irreducible representations of groups such as \( \mathrm{SU}(N) \) and on the Poincaré group and its little groups<sup>[14](https://arxiv.org/abs/2602.01258)</sup>. Graduate courses in particle and field theory accordingly build on Lie groups, Lie algebras, and their representations, including explicit and spontaneous symmetry breaking and the [Higgs mechanism](https://www.edgepedia.org/higgs-mechanism)<sup>[15](https://www.drps.ed.ac.uk/26-27/dpt/cxpgph11097.htm)</sup>.

## Applications in robotics

The configuration space of a rigid body in [three-dimensional space](https://www.edgepedia.org/three-dimensional-space) is the special [Euclidean group](https://www.edgepedia.org/euclidean-group) \( \mathrm{SE}(3) = \mathbb{R}^3 \rtimes \mathrm{SO}(3) \), a semidirect product reflecting the coupling between rotation and translation<sup>[16](https://www.sciencedirect.com/science/article/pii/S0307904X26002386)</sup>. The surjectivity of the exponential maps \( \exp: \mathfrak{so}(3) \to \mathrm{SO}(3) \) and \( \exp: \mathfrak{se}(3) \to \mathrm{SE}(3) \) is important in robotics: joint motions in kinematics can be written as exponentials \( e^{\theta s} \), where \( \theta \) is a joint angle and \( s \in \mathfrak{se}(3) \) is the joint screw, in the tradition of Robert Ball's 1900 treatise on screw theory<sup>[9](https://www.cis.upenn.edu/~cis6100/geombchap14.pdf)</sup>.

Because the exponential map is surjective, a multivalued logarithm \( \log: \mathrm{SE}(3) \to \mathfrak{se}(3) \) exists, and motion interpolation can be performed by taking the linear interpolant \( (1-t)\log(B_1) + t\log(B_2) \) of two rigid motions in the algebra and mapping it back through \( \exp \)<sup>[9](https://www.cis.upenn.edu/~cis6100/geombchap14.pdf)</sup>. Lie-theoretic formulations also yield succinct block-triangular factorizations of the Jacobian, mass matrix, Coriolis, and gravity terms in the dynamics of cooperating robot systems<sup>[17](https://www.sciencedirect.com/science/article/pii/S0921889096008020)</sup>, and practitioners use \( \mathrm{SO}(3) \), \( \mathrm{SE}(2) \), \( \mathrm{SE}(3) \), and related groups directly in vision-based control, estimation, and optimization rather than working around singular parameterizations such as [Euler angles](https://www.edgepedia.org/euler-angles)<sup>[18](https://github.com/byu-magicc/lie_groups)</sup>.

## History

Sophus Lie (1842–1899) made major advances in the theory of continuous groups of transformations and differential equations<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Lie/)</sup>. His work on contact transformations, examined through Jacobi's process of generating new solutions of differential equations from a given one, led him to combine transformations in a way he called an infinitesimal group, the modern Lie algebra<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Lie/)</sup>. In the winter of 1873–74 he began developing systematically what became his theory of continuous transformation groups, later named Lie groups; [Wilhelm Killing](https://www.edgepedia.org/wilhelm-killing) subsequently examined the associated Lie algebras<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Lie/)</sup>. Lie's contemporaries were slow to accept the work, partly because his papers were difficult to follow and his geometric intuition exceeded that of other mathematicians of his time<sup>[3](https://mathshistory.st-andrews.ac.uk/Biographies/Lie/)</sup>.

## References

1. [Lie group, Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Lie_group)
2. [Lie Groups and the Exponential Map, youyuanwu](https://youyuanwu.github.io/misc-docs/math/01-group-theory/15-lie-groups.html)
3. [Sophus Lie (1842–1899), MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/Biographies/Lie/)
4. [Lie Groups and Lie Algebras, Oregon State University](https://books.physics.oregonstate.edu/GELG/lie.html)
5. [Introduction to Smooth Manifolds & Lie Groups, UCSD lecture notes (T. Kemp)](https://www.math.ucsd.edu/~tkemp/251/250-251.Notes.pdf)
6. [Two notes on Lie groups, T. Tao, Graduate Studies in Mathematics 153](https://terrytao.wordpress.com/wp-content/uploads/2014/11/gsm-153.pdf)
7. [Lie Groups (Spring 2021 course), J. Baez, UC Riverside](https://math.ucr.edu/home/baez/lie_groups/)
8. [Lie Groups and their applications to Particle Physics: A Tutorial for Undergraduate Physics Majors, arXiv:2012.00834](https://arxiv.org/abs/2012.00834)
9. [Applications of Lie Groups and Lie Algebras, Chapter 14, UPenn geometry notes](https://www.cis.upenn.edu/~cis6100/geombchap14.pdf)
10. [Lecture Notes in Lie Groups, University of Maine](https://bpb-us-w2.wpmucdn.com/faculty.umaine.edu/dist/1/19/files/2022/08/liegroups.pdf)
11. [Lie groups and Lie algebras, UCL lecture notes (J. Evans)](https://www.homepages.ucl.ac.uk/~ucahjde/lg/lg-notes.pdf)
12. [Compact Lie Groups, Stanford MATH 210C notes (B. Conrad)](https://math.stanford.edu/~conrad/210CPage/handouts/lie_groups_notes.pdf)
13. [Matrix Lie groups, UCLA notes (V. Sverak course materials)](https://www.math.ucla.edu/~vsv/liegroups2007/6.pdf)
14. [An introduction to gauge theories and group theory in particle physics, arXiv:2602.01258](https://arxiv.org/abs/2602.01258)
15. [Symmetries of Particles and Fields (PGPH11097), University of Edinburgh course catalogue](https://www.drps.ed.ac.uk/26-27/dpt/cxpgph11097.htm)
16. [Left-invariant geometric modelling and tracking control for fully actuated robotic manipulators, Applied Mathematical Modelling](https://www.sciencedirect.com/science/article/pii/S0307904X26002386)
17. [A Lie group formulation of the dynamics of cooperating robot systems, Robotics and Autonomous Systems](https://www.sciencedirect.com/science/article/pii/S0921889096008020)
18. [byu-magicc/lie_groups: Implementation-focused introduction to Lie groups for roboticists, GitHub](https://github.com/byu-magicc/lie_groups)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Lie theory › Lie algebra structure*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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