Math 513 F24

Math 513 Fall 2024

Instructor: C. Negron
Time and location: 12:00-12:50pm, DMC 207
Office: KAP 444C
Office hours: Wed 2-3 pm or Wed 3-4 pm (but not both, on any given week)
Book: Humphreys Introduction to Lie algebras and representation theory.

Syllabus: here

-- Lectures

set 1, set 2 [Updated 10.10.24], set 3 [Nov 14], set 4 [Nov 14].

-- Final

For the final you have two options:
A) Write a 3--10 page paper about a Lie-theoretic topic of your choosing. Due ≤ Dec 13.
B) Give a 10 minute presentation about a Lie-theoretic topic of your choosing. (These presentations will occur on the last day of class. If you want to do this let me know.)

Possible topics include, real forms of simple Lie algebras, semisimple algebraic groups, classical super groups or super Lie algebras, appearances of Lie theory in physics or computer science, the Lie group to Lie algebra functor, compact Lie groups over the reals, the representation theory of GL_n(C), Tannakian reconstruction, the representation theory of Sp₄(C), simple groups of Lie type, Deligne-Lusztig theory, representations of SL₂(F) for F of finite characteristic and algebraically closed, diagramatic presentations of representation categories, etc.
You can write about quantum groups, but no presentations.

-- Homework

All HW problems are out of Humphreys text (jk). Report (presumed) errors via email or ask in class. To be turned in in class.

Vlad solutions: HW 1, HW 2, HW 3.

HW 5, do 4 or so of the following (Due Mon Dec 2):

1. Show that any base Δ for a root system (E,Φ) is of the form Δ(μ) for some regular element μ in E.

2. Let (E,Φ) be an arbitrary root system. Show that (E,Φ) admits a unique decomposition into a sum of irreducible subsystems
(E,Φ) = (E₁,Φ₁) ⊕ ... ⊕ (E_t, Φ_t).

3. Let (E,Φ) and (E',Φ') be irreducible root systems. Prove that (E,Φ) is isomorphic to (E',Φ') if and ony if their Dynkin diagrams are isomorphic. Prove the same result when (E,Φ) and (E',Φ') are not assumed to be irreducible.

4. Suppose a root system decomposes into a sum of subsystems
(E,Φ) = (E₁,Φ₁) ⊕ (E₂,Φ₂),
and let W, W₁, W₂ be the respective Weyl groups. Prove that W decomposes as a product W = W₁ × W₂.

5. Let h be a (finite-dimensional) abelian Lie algebra and V be a possibly infinite-dimensional h-representation. Suppose h acts semisimply on V, i.e. that V decomposes into eigenspaces for the action of h. Prove directly that for any subrepresentation V' ⊆ V, h acts semisimply on V' as well.

6. Let g be a semisimple Lie algebra and x be a nilpotent element in g. Prove that the formal exponential exp(x) = Σ_n≥0 xⁿ/n! acts by a well-defined automorphism on each g-representation V, with inverse exp(-x). Prove that for each y in g, and g-representation V, the conjugate exp(x)·y·exp(-x) acts on V as the element exp(ad_x)(y).

7. Let g be a semisimple Lie algebra with choice of Cartan h and positive Borel b.
(a) Prove that each simple b-representation is the 1-dimensional representation C(ζ) for some function ζ: h → C, i.e. the representation on which the Cartan acts via ζ and all the positive generators e_α all act trivially.
(b) Prove that h acts semisimply on the Verma module
M(ζ)=U(g)⊗_U(g)C(ζ).
(c) Use the PBW theorem to describe all weights μ for which the weight space M(ζ)_μ is nonzero, and observe that the weight space M(ζ)_ζ is 1-dimensional.
(d) Prove that M(ζ) is cyclically generated by its highest weight space M(ζ)_ζ.
(e) Prove that M(ζ) has a unique maximal submodule, and hence a unique simple quotient M(ζ) → L(ζ), and that L(ζ) is infinite-dimensional whenever ζ is not a dominant weight.

8. Consider semisimple g with a chosen Cartan and positive Borel. For a (finite-dimensional) g-representation V, let Π(V) ⊆ P denote the set of weights μ for which V_μ is nonzero.
(a) Prove that Π(V) is stable under the action of the Weyl group.
(b) Prove for any simple L(λ), we have L(λ)^*=L(w·λ) where w is the longest element in the Weyl group.

HW 4, do 3 or so of the following (Due Oct 28):

Humphreys Ch 8 # 1

Humphreys Ch 9 # 1, 2

1. Given a quotient π: g → g' of a semisimple Lie algebra g, prove that g' is either semisimple or zero.

2. Let g=g₁ × g₂ be a decomposition of a semisimple Lie algebra g into two semisimple factors g_i. For any g_i-representation V we consider V as a g-representation by restricting along the Lie algebra projection π_i: g → g_i.

(a) [corrected Nov-11-24] For V_i any g_i-representations, and T=Hom_g₂(C,V₂) the maximal trivial subrep in V₂, prove that any map of g-representations V₁ → V₂ factors through the trivial representation V₁ → T → V₂.
(b) Let L_i and L'_i be simple g_i-representations. Prove that
dim_C Hom_g( L₁ ⊗ L₂, L'₁ ⊗ L'₂ ) = 1 if both L_i ≅ L'_i, and dim_C Hom_g( L₁ ⊗ L₂, L'₁ ⊗ L'₂ ) = 0 otherwise.
(c) Prove that, for any g-representation V and g₂-representation L, the space Hom_g₂(L,V) is a g₁-representation under the expected action
x·f = ( m ↦ x·f(m) - f(x·m) ).
Prove furthermore that the evaluation map
ev: Hom_g₂(L,V) ⊗ L → V, f ⊗ m ↦ f(m),
is a map of g-representations.
(d) Prove that for every pair of simples L_i over the g_i, the product L₁ ⊗ L₂ is a simple g-representation.
(e) Prove that any simple g-representation M is of the form M ≅ L₁ ⊗ L₂ for uniquely determined L₁ and L₂.
(f) Observe that for a semisimple Lie algebra g, with simple decomposition g = g₁ ×· ·· × g_t, the simples in g are precisely the products
L₁ ⊗ ··· ⊗ L_t
of simples over the factors g_i.

HW 3, do 4 or 5 of the following (Due Monday Oct 14):

Humphreys Ch 3 # 2, 6, 7, 10

Ch 4 # 3, 6

Ch 5 # 1, 4, 5, 6, 8

You may also return to problems from HW 1 & 2.

HW 2, do some of the problems (Due Sept 23):

1. [Important] Prove that the Lie algebra sl_n(C) is simple. [Hint: Use the root decomposition and weight arguments to show that the adjoint representation is simple.]

2. [A good exercise] Show that the adjoint representation for sl_n(C) is self-dual. [Hint: Look at the highest weights.]

3. [A good exercise] (a) Observe that the trace form tr: sl_n(C) ⊗ sl_n(C) → C, tr(x,y) = Tr(xy), is an sl_n(C)-invariant bilinear form on sl_n(C). (Here sl_n(C)-invariance means that tr is a map of sl_n(C)-representations.)
(b) For x in sl_n(C)_μ and y in sl_n(C)_λ, prove that nonvanishing of tr(x,y) implies λ = - μ.
(c) Show that the trace form is non-degenerate, and restricts to a non-degenerate form on the Cartan. [Hint: The trace form is associated to a map of representations sl_n(C) → sl_n(C)^*, by HW 1 #5, say.]
(d) Show that any invariant form β on sl_n(C) is obtained as a scaling β = c·tr at some c in C. [Hint: How many maps between sl_n(C) and its dual are there?]
(e) Observe that the trace form induces a sl_n(C)-invariant non-degenerate form к on the dual sl_n(C)^* as well, and that any such form is unique up to scaling.

4. [Important] Let's call an element t in sl_n(C) semisimple if it acts semisimply on the standard representation 𝕍 = Cⁿ, i.e. if it acts via a diagonalizable matrix. Prove that t is semisimple if and only if the adjoint operator ad_t = [t, - ]: sl_n(C) → sl_n(C) is semisimple.

5. [Kind of important] Let t be a Lie subalgebra in sl_n(C) satisfying the following:
(a) t is abelian.
(b) Every element t in t is semisimple.
(c) t is a maximal Lie subalgebra in sl_n(C) which satisfies (a) and (b), with respect to inclusions of subalgebras.
Prove that there is an invertible matrix A in SL_n(C) so that AtA^-1 = h, where h is the standard Cartan.

Remark. Let us note here that, at all A in SL_n(C), the multiplicative adjoint operator
Ad_A: sl_n(C) → sl_n(C), Ad_A(x) := A·x·A^-1,
is a Lie algebra automorphism. Such a maximal semisimple subalgebra as in # 2 is called a Cartan subalgebra in sl_n(C), and the above problem establishes uniqueness of the Cartan subalgebra up to the natural adjoint action of the associated group SL_n(C).

6. Let L = L(ω_k) be a fundamental simple over sl_n(C). Prove that the m-th symmetric power S^m(L) is simple at all positive m. [Begin with the standard representation L = 𝕍, then try to generalize your argument.]

7. Consider the group algebra Z[P], which we consider as the free Z-module with basis { t^λ : λ ∈ P } and multiplication t^λ· t^μ = t^{(λ + μ)}. Define the character of any sl_n(C)-represetnation V as
ch(V) = Σ_{λ ∈ P} dim_C(V_λ)·t^λ.

(a) Prove that, for arbitrary sl_n(C)-representations V and W, we have V ≅ W if and only if ch(V) = ch(W).
(b) [A good calculation] Calculate the characters of the fundamental simples ch(L(ω_k)), for 1 ≤ k ≤ n-1, when n = 3 and 4.

8. Take α₁ and α₂ the standard simple roots for sl₃(C), and γ= α₂ + α₂ remaining positive root. Show that there is a Lie algebra automorphisms φ: sl₃(C) → sl₃(C) which stabilizes the Cartan, φ(h) = h, but which has
φ(e_α₁) = f_α₁, φ(e_α₂) = e_γ, φ(e_γ) = e_α₂, φ(f_α₁) = e_α₁, φ(f_α₂) = f_γ, φ(f_γ) = f_α₂ .

This is to say, there are Lie algebra automorphisms which permute the root spaces.

HW 1, do some of the problems (Due Sept 11):

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
res_i: U(sl₂(C))-mod_fin.dim → rep(sl₂(C)).

2. (a) Prove that U(sl₂(C)) is spanned by monomials of the form eⁱ h^j f^k, 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 ··· x_n in which each of the x_i is one of e, f, or h, we have a = eⁱ h^j f^k + lower degree terms, where i + j + k = n.]

(b) Prove that the monomials eⁱ h^j f^k 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) = Σ_{n ∈ Z} dim_C(V_n)·tⁿ
in Z[t, t^-1]. 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 m_τ in the simple expansion
L(1) ⊗ L(λ) = Σ_{τ ≥ 0} m_τ·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 Hom_g(T ⊗ V, W) ≅ Hom_g(T, W ⊗ V^*) and Hom_g(T ⊗ V, W) ≅ Hom_g(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^*.