Visualizing Representations of 𝔰𝔩(2,ℂ)

An interactive notebook that illustrates representations of the Lie algebra $\mathfrak{sl}(2, \mathbb{C})$ by implementing the action of its generators on homogeneous polynomials. Designed for Lecture 4 (Representation Theory) of the Lie Groups with Applications course at Quantum Formalism.

Goal

To explore the action of the $\mathfrak{sl}(2, \mathbb{C})$ generators — the weight operator $H$, raising operator $E$, and lowering operator $F$ — on the space of homogeneous polynomials $V_m$ in two variables $(z_1, z_2)$, and to visualize the ladder structure of irreducible representations.

Key Activities

• Defined the representation space $V_m$, consisting of degree-$m $homogeneous polynomials in two variables
• Implemented $H, E$, and $ F$ as differential operators acting on symbolic expressions in sympy
• Built an interactive interface for applying these operators to a given polynomial using ipywidgets
• Demonstrated weight shifts and the full ladder structure for each irreducible $V_m$

Technologies & Concepts

• Python, SymPy (symbolic differentiation & polynomial manipulation)
• ipywidgets for interactivity
• Representation theory of $\mathfrak{sl}(2, \mathbb{C})$
• Differential operators on polynomial spaces

Note: This project is currently in development — the code repository will be made public soon.