Online Training Session 1, Part B

Defining groups and structure theory.

1. Questions from Part A

Welcome back. Questions from previous hour. Function arguments/outputs queries: whattype "function" ?, "function"@("input"), inputs/outputs for the + function, e.g. +@(mat,int)

2. Data Types, Root Datum

Data type: LieType, atlas functions: simply_connected, adjoint, simple_roots, simple_coroots. Root datum in atlas, isogenies; \(SL(2)\), \(GL(2)\), \(PSL(2)\)

3. More about Lie Groups

More Lie groups, The file, realform data type, GL(2,R), Sp(4,R), posroots, poscorrots, rho, GL(3,R), SL(3,R)

4. Fundamental Weight Coordinates, \(Sp(8,\mathbb{R})\)

The file, strings, type C4, fundamental weight coordinates, \(Sp(8,\mathbb{R})\)

5. Simple Roots of \(SL(5,\mathbb{R})\)

Simple roots of \(SL(5,\mathbb{R})\), atlas coordinates.

6. More on Coordinates

More on atlas’ choice of coordinates, help files.

7. The File

The file

8. Get Lie Groups Information

More Lie group information, Lie algebra, connectednes, root datum, split, quasisplit, adjoint; nr_of_Cartan_classes, print_cartan_info (for a real torus), imaginary, real and complex roots

9. Real Forms

real_forms, for loop to print real forms, cartan classes along realforms, sp(4)

10. Inner Class

real forms in an inner class. \(SL(5,\mathbb{R})\), distinguished involution, outer outomorphisms