## Complex semisimple Lie algebras.

Page updated : 16/12/2021.
The page of the course of 2020-2021 is available : Complex semisimple Lie algebras, 2020-2021.
Here is the examination sheet of "Algèbres de Lie 1" for 2020-2021 : Examination, Algèbres de Lie I, 2020-2021.

Lecture Notes : Lecture notes on "Complex semisimple Lie algebras" (version of 12/12/2021).
Beware :
-- Section I.8, Proposition II.5.19 and Remark II.5.20 will be studied latter.

Various Notes :
1) Course complement : Tensor product of vector spaces (version of 19/03/2021).
2) Course complement : Tensor product (version of 05/11/2020).
3) Tutorials : Simplicity of \$sl_n(\k)\$ .
4) Tutorials : Cartan-Chevalley decomposition of \$sl_n(\k)\$ .
5) Course complement : Exterior algebra, exterior powers and representations.
6) Tutorials : Fundamental representations of \$sl_n(\k)\$ .
7) Course complement : Complements to the proof of Proposition V.3.6.
8) Tutorials : Tensor product and weights.
9) Tutorials : Dual representation and weights.
10) Tutorials : Weight diagrams for \$sl_3(\k)\$ .

Week 1 -- (from 13/09/2021 to 19/09/2021)
Course : sections I.1 to I.3.
Tutorials : remainder on Linear Algebra over a field (including Jordan-Chevalley decomposition of an endomorphism and tensor product of vector spaces).

Week 2 -- (from 20/09/2021 to 26/09/2021)
Course : sections I.4 to I.6 ; section I.7 up to Lemma I.7.18.
Tutorials : simplicity of \$sl_2(\k)\$.

Week 3 -- (from 27/09/2021 to 02/10/2021)
Course : section I.7 ; Part II up to Corollary II.5.8.
Tutorials : simplicity of \$sl_n(\k)\$.

Week 4 -- (from 04/10/2021 to 10/10/2021)
Course : section II.5, from Proposition II.5.9 to Exercise II.5.18 ; section II.6 ; part III, up to Remark III.2.12.
Tutorials :

Week 5 -- (from 11/10/2021 to 17/10/2021)
Course : end of section III.2 ; section III.3 to III.5.
Tutorials : Cartan-Chevalley decomposition of \$sl_n(\k)\$ (beginning).

Week 6 -- (from 18/10/2021 to 24/10/2021)
Course : sections III.6 to III.8.
Tutorials : Cartan-Chevalley decomposition of \$sl_n(\k)\$ (end).

Week 7 -- (from 08/11/2021 to 14/11/2021)
Course : section III.10 ; brief comments on the link between "maximal toral subalgebras" and "Cartan subalgebras" ; section IV.2 ; section IV.3 until Definition IV.3.7.
At that point, the only statements you are supposed to know about "Cartan subalgebras" are Definition I.8.7, Proposition II.5.19 and Theorem IV.1.5.

Tutorials

Week 8 -- (from 15/11/2021 to 21/11/2021)
Course : end of section IV.3, section IV.4 until Lemma IV.4.4.
Tutorials

Week 9 -- (from 22/11/2021 to 28/11/2021)
Course : section II.7, section III.9, end of section IV.4.
Tutorials

Week 10 -- (from 29/11/2021 to 05/12/2021)
Course : Part V, until Example V.2.15.
Tutorials : course complement on exterior algebras and applications to representation theory (see "Various Notes" 5, 6 above.)

Week 11 -- (from 06/12/2021 to 12/12/2021)
Course : Part V, until the end of section V.2.
Tutorials : fundamental representations of \$sl_n(\k)\$ (see "Various Notes" 6 above.)

Week 12 -- (from 13/12/2021 to 19/12/2021)
Course : Part V, until the end of section V.3.
Tutorials : Tensor products, duals and weights (see "Various Notes" 8, 9 above) ; weight diagrams for \$sl_3(\k)\$ (see "Various Notes" 10 above.)