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 groups.at, 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 lietypes.at, 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 atlas-functions.help File

The file atlas-functions.help

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