Math 513 Fall 2024
Instructor: C. Negron
Time and location: 12:00-12:50pm, DMC 207
Office: KAP 444C
Office hours: Wed 3-4 pm
Book: Humphreys Introduction to Lie algebras and representation theory.
Syllabus: here
-- Lectures
week 1
-- Homework
All HW problems are out of Humphreys text (jk). To be turned in in class.
HW 1, do some of the problems (Due Sept 12):
1. [Important] Let U(sl2(C)) be the algebra with generators e, f, and h and relations given by the Lie bracket on sl2(C),
he - eh = 2e, hf - fh = -2f, ef - fe = h.
This is explicitly the tensor algebra T(sl2(C)), i.e. the free algebra generated by the vector space sl2(C), modulo the ideal generated by the relations (xy - yx) - [x,y], where x and y are arbitrary in sl2(C) and [x,y] denotes the bracket in sl2(C). We have the obvious linear map i: sl2(C) → U(sl2(C)) which sends each x in sl2(C) to its corresponding generator. [The algebra U(sl2(C)) is called the universal enveloping algebra for sl2(C).]
Prove that restricting along i provides an equivalence of categories
resi: U(sl2(C))-modfin.dim → rep(sl2(C)).
2. (a) Prove that U(sl2(C)) is spanned by monomials of the form ei hj fk, where the powers run over all nonnegative integers. [Hint: Define the degree of an element in U(sl2(C)) to be the minimal n so that a is in the space spanned by the products sl2(C)m with m ≤ n. Prove that for any monomial a = x1 ··· xn in which each of the xi is one of e, f, or h, we have a = ei hj fk + lower degree terms, where i + j + k = n.]
(b) Prove that the monomials ei hj fk are all linearly independent in U(sl2(C)), and hence form a basis of U(sl2(C)). [Hint: For any sum of monomials a which you want to show is nonzero, show that a is nonzero by finding a simple L(λ) on which a acts nontrivially.]
3. [Important] Define the character ch(V) of a sl2(C)-representation as the polynomial
ch(V) = Σn ≥ 0 dimC(Vn)·tn
in Z[t]. Prove that two sl2(C)-representations V and W are isomorphic if and only if ch(V) = ch(W).
4. [A good calculation] (a) For arbitrary λ in Z≥0 calculate the tensor product L(1) ⊗ L(λ), i.e. determine the multiplicities mτ in the simple expansion
L(1) ⊗ L(λ) = Στ ≥ 0 mτ·L(τ).
(b) For arbitrary λ and μ in Z≥0 similarly calculate the the product L(μ) ⊗ L(λ).
5. (a) For any lie algebra g, and g-representation V, prove that the linear dial V* is a g-representation under the action
x·ζ = ( v ↦ -ζ(x·v) ).
Note that there is a natural isomorphism of g-representations V ≅ (V*)*.
(b) Prove that there are natural isomorphisms
Homg(T ⊗ V, W) ≅ Homg(T, W ⊗ V*) and Homg(T ⊗ V, W) ≅ Homg(V, T* ⊗ W).
6. For simple g-representations L0 and L1, prove that there is a nonzero morphism from the trivial representation ξ: C → L0 ⊗ L1 if and only if L1 ≅ L0*.