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Reading notebook: manifolds, Lie groups, and Lie algebras
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In [1]:
Markdown Overview

Notebook on manifolds, Lie groups, and Lie algebra.

Let M be a smooth n-manifold. Locally, every point p in M admits a chart (U, phi) with phi: U -> R^n. A Lie group G is then a smooth manifold equipped with smooth multiplication m: G x G -> G and smooth inversion i: G -> G. Its Lie algebra g is the tangent space T_e G at the identity, together with the bracket induced by left-invariant vector fields. Read the cells in order and use the output pane as the running geometric model for these definitions.

  • A smooth atlas gives compatible local coordinates on a curved space.
  • Smooth group operations turn that space into a geometry of symmetries.
  • The Lie algebra records first-order behavior at the identity element.
# notation M: smooth manifold G: Lie group e in G: identity element g = T_e G: Lie algebra exp: g -> G
In [2]:
Markdown Local model

1. Start with a manifold.

A smooth manifold is a Hausdorff, second-countable topological space equipped with an atlas of charts whose transition maps are smooth. The point is not that the space is globally flat, but that each neighborhood U can be compared with an open set in R^n through a chart phi: U -> R^n. Differential constructions are performed in coordinates and then transported back to M.

In the output, the moving patch represents a first-order approximation near a point p. The ambient surface curves, but the tangent patch behaves like the affine model for T_p M. This is the geometric reason local analysis on a manifold resembles ordinary multivariable calculus.

chart phi: U -> R^n transition phi_a o phi_b^{-1}: phi_b(U_a /\ U_b) -> phi_a(U_a /\ U_b)
Interpretation. The highlighted patch should be read as a visual proxy for T_p M together with local coordinates near p.
In [3]:
Markdown Smooth composition

2. Add a smooth group law.

A Lie group is a smooth manifold G with multiplication m(g, h) = gh and inversion i(g) = g^{-1}, both smooth. Consequently, each left translation L_g(h) = gh and right translation R_g(h) = hg is a diffeomorphism. Geometry and algebra are therefore fused: points of the manifold are also group elements that act by smooth symmetries.

The output can be read as a one-parameter family of rotations. Moving around the orbit is not merely tracing a curve in space; it is composition inside G. For the model example G = SO(2), the parameter theta labels a rotation R_theta and composition corresponds to angle addition modulo 2 pi.

m: G x G -> G i: G -> G L_g(h) = gh
Interpretation. The orbit of frames represents group elements organized as a smooth manifold rather than as a discrete list.
In [4]:
Markdown Linearization

3. Zoom into the identity.

The Lie algebra g of G is the tangent space T_e G at the identity together with a bilinear antisymmetric bracket satisfying the Jacobi identity. One way to define the bracket is through left-invariant vector fields; for matrix groups it is the commutator [A, B] = AB - BA. This linearization near e is what makes infinitesimal structure computable.

The output singles out one tangent direction X in T_e G and then suggests the corresponding flow t -> exp(tX). The isolated arrow is infinitesimal data; the surrounding field indicates how that local direction integrates into a one-parameter subgroup of G.

g = T_e G [A, B] = AB - BA (matrix groups) exp: g -> G
Interpretation. A vector X in g determines the initial velocity of a curve through e, and exp(tX) recovers the associated finite motion in the group.
In [5]:
Markdown Summary

4. Keep the ladder in view.

The hierarchy is now precise. A manifold provides local smooth coordinates. A Lie group is a manifold with smooth multiplication and inversion. Its Lie algebra is the tangent space at the identity, equipped with a bracket that captures infinitesimal noncommutativity. The exponential map then connects the linear object back to finite group elements.

Step 1

Manifold

A smooth atlas makes each neighborhood locally diffeomorphic to an open set in R^n.

Step 2

Lie group

The manifold also carries smooth maps m: G x G -> G and i: G -> G.

Step 3

Lie algebra

The space g = T_e G carries a bracket induced by left-invariant fields.

Step 4

Exponential map

Each X in g generates a curve t -> exp(tX) inside G.

# structural summary (M, atlas) (G, m, i) g = T_e G with bracket [.,.] X in g => t -> exp(tX) in G