Complex semisimple Lie algebras, 2020-2021.
Page updated : 21/12/2020.
BEWARE : the examination of "Algèbres de Lie I", cancelled in october due to the lockdown, will take place on december the 15th, 2020, from 13:30 to 16:30, Halle aux farines, room 12E.
Lecture Notes : Lecture notes on "Complex semisimple Lie algebras" (version of 07/12/2020).
Various Notes :
1) Tutorials : Explicit computation of Cartan-Chevalley decomposition of $sl_n(\k)$.
2) Tutorials : Explicit computation of Cartan-Chevalley decomposition of $sp_{2n}(\k)$.
3) Course complement : Tensor product and related constructions (version of 05/11/2020).
4) Course complement : Exterior algebra, exterior powers and representations.
5) Tutorials : Fundamental representations of $sl_n(\k)$.
6) Tutorials : Solution to Exercise V.2.20.
7) Course complement : Symmetric powers of representations.
8) Tutorials : Symmetric powers of the natural representation of $sl_2(\k)$.
9) Tutorials : Dual representation and weights, Tensor product and weights.
10) Tutorials : Clebsch-Gordan decomposition.
11) Tutorials : Weight diagrams for $sl_3(\k)$.
Week 1 -- (from 07/09/2020 to 13/09/2020)
Course : sections I.1 to I.3.
Tutorials : remainder on Linear Algebra over a field (including Jordan-Chevalley decomposition of an endomorphism).
Week 2 -- (from 14/09/2020 to 20/09/2020)
Course : sections I.4 to I.7.
Tutorials :Exercise I.7.3.
Week 3 -- (from 21/09/2020 to 27/09/2020)
Course : sections II.1 to II.5.
Tutorials : part of Section II.4 (finite dim. irreducible representations of $sl_2(k)$).
Week 4 -- (from 28/09/2020 to 04/10/2020)
Course : sections II.6 to III.4.
Tutorials : Exercises III.3.8 and III.6.2.
Week 5 -- (from 05/10/2020 to 11/10/2020)
Course : sections III.5 to III.8
Tutorials : Explicit computation of Cartan-Chevalley decomposition of $sl_n(\k)$ (see "Various Notes, 1" above).
Week 6 -- (from 12/10/2020 to 19/10/2020)
Course : sections III.10 and IV.1. (Note: section III.9 will be studied in the second part of the course.)
Tutorials : Explicit computation of Cartan-Chevalley decomposition of $sp_{2n}(\k)$ (see "Various Notes, 2" above).
Week 7 -- (from 02/11/2020 to 08/11/2020)
Course : section III.9, beginning of section IV.2
Tutorials : Course complement: "Tensor product and related constructions" (see "Various Notes, 3" above).
Week 8 -- (from 09/11/2020 to 15/11/2020)
Course : section IV.2, section IV.3 until Remark IV.3.7.
Tutorials :
Week 9 -- (from 16/11/2020 to 22/11/2020)
Course : section IV.3 and Part V until Theorem V.2.8. (See, also, the new Remark IV.2.11)
Tutorials :
Week 10 -- (from 23/11/2020 to 29/11/2020)
Course : section V.2 until Theorem V.2.16.
Tutorials : Exterior algebra, exterior powers and representations, Fundamental representations of $sl_n(\k)$ (see "Various Notes, 4, 5" above).
Week 11 -- (from 30/11/2020 to 06/12/2020)
Course : section V.2 until Theorem V.2.22.
Tutorials : Exercise V.2.20 of the Lecture Notes (see "Various Notes 6" above for a solution).
Week 12 -- (from 07/12/2020 to 13/12/2020)
Course : section V.2 until Theorem V.2.26.
Tutorials : Symmetric powers of representations, Symmetric powers of the natural representation of $sl_2(\k)$ (see "Various Notes 7, 8" above).
Tutorials : Dual representation and weights, Tensor product and weights, Clebsch-Gordan decomposition, Weight diagrams for $sl_3(\k)$ (see "Various Notes 9, 10, 11" above).