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

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.] |