# 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