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