.. _session_1B: 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)`` .. raw:: html 2. Data Types, Root Datum ---------------------------- Data type: LieType, atlas functions: ``simply_connected``, ``adjoint``, ``simple_roots``, ``simple_coroots``. Root datum in atlas, isogenies; :math:`SL(2)`, :math:`GL(2)`, :math:`PSL(2)` .. raw:: html 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)`` .. raw:: html 4. Fundamental Weight Coordinates, :math:`Sp(8,\mathbb{R})` ------------------------------------------------------------------ The file lietypes.at, strings, type C4, fundamental weight coordinates, :math:`Sp(8,\mathbb{R})` .. raw:: html 5. Simple Roots of :math:`SL(5,\mathbb{R})` --------------------------------------------------- Simple roots of :math:`SL(5,\mathbb{R})`, atlas coordinates. .. raw:: html 6. More on Coordinates --------------------------- More on atlas' choice of coordinates, help files. .. raw:: html 7. The atlas-functions.help File -------------------------------------- The file atlas-functions.help .. raw:: html 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 .. raw:: html 9. Real Forms ----------------- ``real_forms``, for loop to print real forms, cartan classes along realforms, sp(4) .. raw:: html 10. Inner Class -------------------- real forms in an inner class. :math:`SL(5,\mathbb{R})`, distinguished involution, outer outomorphisms .. raw:: html