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,R)¶
The file lietypes.at, strings, type C4, fundamental weight coordinates, Sp(8,R)
5. Simple Roots of SL(5,R)¶
Simple roots of SL(5,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,R), distinguished involution, outer outomorphisms