This repository contains Jupyter notebooks that provide an interactive exploration of the representation theory of the Lie algebra sl(2,C). The notebooks implement symbolic computation to demonstrate how the standard basis elements (H, E, F) act on spaces of homogeneous polynomials, allowing users to visualize the weight space decomposition and ladder structure of these fundamental representations.
- Symbolic computation of the differential operators corresponding to the standard basis of sl(2,C)
- Interactive widgets allowing real-time application of Lie algebra operators to custom polynomials
- Visual representation of weight spaces and the ladder structure of irreducible representations
- Educational tools bridging abstract algebra with computational mathematics
- Python 3.8+
- Jupyter Notebook or JupyterLab
- SymPy
- IPython widgets (ipywidgets)
The primary notebook Lie_Groups_Lecture4.ipynb allows you to:
- View the explicit action of H, E, and F operators on homogeneous polynomial spaces
- Enter your own homogeneous polynomials and apply different operators
- Observe how the weight structure evolves as operators are applied
- Experiment with different degree polynomials to explore representations of various dimensions
This project explores the finite-dimensional irreducible representations of sl(2,C), which are parameterized by non-negative integers m. Each representation acts on the space V_m of homogeneous polynomials of degree m in two complex variables. The standard basis elements act as differential operators:
- H (Cartan element): -z1·∂/∂z1 + z2·∂/∂z2
- E (raising operator): -z2·∂/∂z1
- F (lowering operator): -z1·∂/∂z2
These representations form the mathematical foundation for quantum angular momentum, spherical harmonics, and numerous applications in physics and mathematics.
The representation theory of sl(2,C) has applications in:
- Quantum mechanics (angular momentum)
- Particle physics (classification of elementary particles)
- 3D computer graphics (spherical harmonics)
- Machine learning (equivariant neural networks)
- Quantum computing (representation of quantum gates)
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This repository is intended for educational purposes. Redistribution, modification, or commercial use of this material without prior written permission is prohibited.
Brian Hepler, PhD is a mathematician specializing in Lie theory, quantum computing, and data science. This material was developed as part of the "Lie Groups with Applications" course at Quantum Formalism Academy.