Lie Groups and Lie Algebras: Winter-2023
Practice Exercises 5 : Due Friday 2/10

1. Two-dimensional root diagrams I S0 4611S

   The goal of this problem is to construct all possible root diagrams in two dimensions. For the purposes of this problem, a root \(\alpha\) is an element of a subset \(\Delta\subset\mathbb{R}^n\) with the following properties:

   • \(\alpha\ne0\);
   • \(\alpha\in\Delta \Longleftrightarrow -\alpha\in\Delta\);
   • The elements of \(\Delta\) span \(\mathbb{R}^n\);
   • If \(\alpha,\beta\in\Delta\), then \(2\frac{\alpha\cdot\beta}{\alpha\cdot\alpha}\in\mathbb{Z}\).
   (See Constrained Dot Products I for the consequences of the last assumption.)

   For the purposes of this problem, a root diagram is a graph showing all of the roots as well as the origin, connected by lines with the following properties.

   • Each root lies in an adjoint representation of \(\mathfrak{su}(2)\), that is, lies on a line through the origin with exactly three points.
   • Each root lies in a representation of each such copy of \(\mathfrak{su}(2)\), that is, lies on a line (possibly of zero length) parallel to the corresponding root vector that is centered appropriately (with midpoint on the perpendicular bisector of the line through the root vector).
   • There are no “missing” lines, that is, if two roots can be connected by a line parallel to some root vector, that line is present.

   Find all possible root diagrams in two dimensions.

2. Two-dimensional root diagrams II S0 4611S

   Any root diagram can be divided in half by a plane through the origin that contains no roots, with the roots on one side being labeled positive and the others negative. Simple roots are positive roots that can not be expressed as the sum of any combination of other positive roots.

   For each root diagram in two dimensions,

   • Choose a set of positive roots;
   • Find the simple roots;
   • Find the angle(s) between them;
   • Show explicitly that the remaining positive roots can be expressed as linear combinations of the simple roots with positive, integer coefficients.