Visualizing the Lie Bracket in 𝖘𝖚(2)

An interactive 3D notebook for visualizing the Lie bracket structure of 𝖘𝖚(2) using a geometric cross-product analogy. Developed as a demonstration for Lecture 3 (Matrix Lie Algebras) in the Lie Groups with Applications course at Quantum Formalism.

Key Goals

• Compute the Lie bracket $[X, Y] = XY - YX$ for elements in 𝖘𝖚(2)
• Map each element of 𝖘𝖚(2) to a vector in $\mathbb{R}^3$ using the Pauli basis
• Visualize $X$, $Y$, and $[X, Y]$ as 3D vectors on the unit sphere
• Demonstrate that the Lie bracket corresponds to the cross product in this representation

Tools & Concepts

• Basis derived from scaled Pauli matrices: $i\sigma_x/2, i\sigma_y/2, i\sigma_z/2$
• Lie algebra computations with NumPy
• 3D visualization via matplotlib and mplot3d
• Interactive control with ipywidgets

This notebook illustrates deep algebraic structure through concrete geometric visuals — an ideal companion to coursework on Lie algebras, cross products, and matrix groups.

Note: Repository will be made public soon.