Math 774 Lie algebras, Fall 2019

Location & time:  Phillips 301, TuTh 930--10:45am.
Text:  Humphrey's Introduction to Lie algebras and representation theory
OH:  Drop in or by appointment.
General plan:  Cover basic information on Lie algebras, Cartan subalgebras, Weyl groups, root systems etc., i.e. classification of simple Lie algebras over C. Some representation theory. For an approximate schedule see the syllabus.

Schedule Oct 28--Dec 3
Oct 29--31: S 14. Isomorphisms of Lie algebras and root systems
Nov 5: S 18. Lie algebras from root systems (Serre's Theorem)
Nov 7: S 17. Universal enveloping algebras & PBW, hopefully
Nov 12--14: S 16. Uniqueness of Borels and Cartans
Nov 19--21: S 13. Weights roots and representations
Nov 25: Simple representations. Dec 3: Weyl Char formula

Homework below
Problems labeled by section are from Humphrey's text

Due_Sept_6 Humphrey's I.1 # 4, 6, 8, 9 -- I.2 # 3, 5, 6 -- I.3 # 2, 3, 4, 6, 7, 8
Due Sept 20 S. 3 # 6 -- S. 4 # 1, 5, 7 -- S. 5 # 2, 3, 5, 6 -- S. 6 # 2, 3, 5, 6, 7
Due Oct 7 1. For V a rep over sl_2, define the character ch(V)=\sum_n dim(V_n)t^n. (a) Prove that V is classified up to isomorphism by its character. (b) Give an "easy formula" for ch(L(m)), like the formula for the sum \sum_{j=1}^M j. (c) Use the character to prove the Clebsch-Gordon formula for the tensor product L(m)\otimes L(n) of simple sl_2 reps. (d) Observe that the category rep(sl_2) is "generated" by the 2-dimensional simple L(1), aka the standard representation. (What must we mean by "generated" here?)

S. 7 # 7 -- S. 8 # 3, 10 -- S. 9 # 2, 3, 4
Due Oct 22 S. 10 # 1, 2, 4, 6, 9, 12 -- S. 11 # 4
Due Nov 12 S. 14 # 2, 5, 6

1. Prove that the category of finite-dimensional modules over the universal enveloping algebra U(g), for g a Lie algebra, is equivalent to the category of g-representations. More specifically, for finite-dimensional V with an action of U(g), extract an action of g on V, and vice versa. Observe that under this association, any map of U(g)-modules is identified with a map of the corresponding g-representations.
Due Dec 1 S. 13 # 4, 5 -- S. 20 # 5, 9 -- S. 21 # 3, 6

Fix g a semisimple Lie algebra with Cartan h
1. Consider weight graded g-modules V and V' (possibly infinite-dimensional). (a) Prove that any map f:V -> V' of g-modules is such that f(V_a)=V_a for all a\in h^*. (b) Prove that any submodule W in V is also weight graded. (c) Prove that if f: V -> W is a surjective module map, then W is also weight graded.

2. Construct a g-module which is not weight graded. [Hint: Induct such a module from the Borel.]