Lie Groups and Lie Algebras

Why do a robot arm and a quantum spin share the same mathematics? Because both live in Lie groups-geometric objects where symmetry meets calculus.

• **Robotics:** SE(3) is the configuration space of a rigid body. SLAM, motion planning, and pose interpolation all use it.
• **Computer Vision:** Camera calibration, the essential matrix in structure-from-motion - all via SO(3) and SE(3).
• **Particle Physics:** SU(2) and SU(3) are the symmetry groups of the Standard Model. Gluons and quarks are representations of Lie groups.

Предварительные знания

• Smooth Manifolds
• Connections and Covariant Derivative

Lie Groups: Group + Manifold

Every drone rotation is an element of SO(3), a Lie group with 3 degrees of freedom. ROS 2 represents robot pose as SE(3) elements. Unity and Unreal use quaternion multiplication - a Lie group operation - for interpolating 10M+ transforms per frame in AAA games.

**SO(3)**: rotation group of 3D space (3-dimensional manifold). **SE(3)**: rotations + translations (six degrees of freedom). **GL(n,ℝ)**: invertible matrices. **U(1)**: unit circle (phases).

Lie Algebra: Tangent Space at the Identity

The **Lie algebra** 𝔤 of a Lie group G is the tangent space T_e G at the identity element e, equipped with the **Lie bracket** [·,·]. The Lie bracket measures the 'non-commutativity' of infinitesimal group elements.

**so(3)**: skew-symmetric 3×3 matrices (correspond to angular velocities). **se(3)**-4×4 matrices of the form [ω, v; 0, 0]. The Lie bracket: [A,B] = AB − BA.

The Exponential Map exp: 𝔤 → G

The **exponential map** sends elements of the Lie algebra (infinitesimal motions) to finite group elements. For matrix groups this is literally the matrix exponential.

For a unit axis ω̂ and angle θ: exp(θ·ω̂×) = I + sin(θ)·ω̂× + (1−cos θ)·(ω̂×)². This allows any rotation to be parameterized by the vector ω = θ·ω̂.

The Adjoint Representation and the BCH Formula

The **adjoint representation** Ad: G → Aut(𝔤) is defined by conjugation: Ad_g(X) = gXg⁻¹. Its differential ad: 𝔤 → End(𝔤) is the Lie bracket: ad_X(Y) = [X,Y]. The **BCH formula** expresses log(exp(X)·exp(Y)) in terms of iterated brackets.

In robotics and computer vision, SE(3) represents rigid body poses. The exponential and logarithm allow pose interpolation, velocity computation, and optimization directly on Lie groups (Lie group optimization).

Key Ideas

• **Lie group**: group + smooth manifold (continuous symmetries)
• **Lie algebra 𝔤**: tangent space at the identity, with bracket [A,B] = AB−BA
• **exp: 𝔤 → G**: exponential map from algebra to group (Rodrigues' formula for SO(3))
• **ad_X(Y) = [X,Y]**: adjoint representation; BCH formula for products of exponentials

Related Topics

Lie groups permeate modern mathematics and physics:

• Connections and Covariant Derivative — A connection on a principal bundle is a 𝔤-valued form; parallel transport integrates over the group
• Differential Forms — The Maurer-Cartan form ω = g⁻¹dg is the canonical 𝔤-valued form on a Lie group
• Smooth Manifolds — A Lie group is a manifold with group structure; all manifold constructions apply

Вопросы для размышления

• Why is the Lie algebra of SO(3) skew-symmetric matrices? What does skew-symmetry mean physically?
• The BCH formula is an infinite series of brackets. In which groups does it terminate after finitely many terms?
• How does the exponential map relate to the trajectory of a one-parameter subgroup?

Связанные уроки

• aa-01-groups-intro