Math 513 Fall 2025
Instructor: C. Negron
Time and location: 12:00-12:50 pm, KAP414
Office: KAP 444C
Office hours: Fridays
Book: Humphreys Introduction to Lie algebras and representation theory.
Syllabus: here
-- Lecture notes
set 1, set 2, set 3, set 4.
-- 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 GLn(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
Report (presumed) errors via email or ask in class. To be turned in in class. Don't turn in garbage.
HW 3, do some of the problems (Due Sept X):
1. [Important] Prove that the Lie algebra sln(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 sln(C) is self-dual. [Hint: Look at the highest weights.]
3. [A good exercise] (a) Observe that the trace form tr: sln(C) ⊗ sln(C) → C, tr(x,y) = Tr(xy), is an sln(C)-invariant bilinear form on sln(C). (Here sln(C)-invariance means that tr is a map of sln(C)-representations.)
(b) For x in sln(C)μ and y in sln(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 sln(C) → sln(C)*, by HW 1 #5, say.]
(d) Show that any invariant form β on sln(C) is obtained as a scaling β = c·tr at some c in C. [Hint: How many maps between sln(C) and its dual are there?]
(e) Observe that the trace form induces a sln(C)-invariant non-degenerate form к on the dual sln(C)* as well, and that any such form is unique up to scaling.
4. [Important] Let's call an element t in sln(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 adt = [t, - ]: sln(C) → sln(C) is semisimple.
5. [Kind of important] Let t be a Lie subalgebra in sln(C) satisfying the following:
(a) t is abelian.
(b) Every element t in t is semisimple.
(c) t is a maximal Lie subalgebra in sln(C) which satisfies (a) and (b), with respect to inclusions of subalgebras.
Prove that there is an invertible matrix A in SLn(C) so that AtA-1 = h, where h is the standard Cartan.
Remark. Let us note here that, at all A in SLn(C), the multiplicative adjoint operator
AdA: sln(C) → sln(C), AdA(x) := A·x·A-1,
is a Lie algebra automorphism. Such a maximal semisimple subalgebra as in # 2 is called a Cartan subalgebra in sln(C), and the above problem establishes uniqueness of the Cartan subalgebra up to the natural adjoint action of the associated group SLn(C).
6. Let L = L(ωk) be a fundamental simple over sln(C). Prove that the m-th symmetric power Sm(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 { tλ : λ ∈ P } and multiplication tλ· tμ = t(λ + μ). Define the character of any sln(C)-represetnation V as
ch(V) = Σλ ∈ P dimC(Vλ)·tλ.
(a) Prove that, for arbitrary sln(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(ωk)), 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(eα1) = fα1, φ(eα2) = eγ, φ(eγ) = eα2, φ(fα1) = eα1, φ(fα2) = fγ, φ(fγ) = fα2 .
This is to say, there are Lie algebra automorphisms which permute the root spaces.
HW 2, do some of the problems (Due Sept 17):
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] (a) Define the character ch(V) of a sl2(C)-representation as the polynomial
ch(V) = Σn ∈ Z dimC(Vn)·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).
(b) Prove that all sl2(C)-representations are self-dual.
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. Let 𝒪an=𝒪an(W) and 𝒪alg=𝒪alg(W) be the algebras of analytic and algebraic functions on W, for a finite-dimensional vector space W. The latter algebra has basis free expression 𝒪alg=Sym(W*) and, in a basis {w1,…,wn} for W with corresponding character functions zi(wj)=δij in W*, 𝒪an consists of analytic functions in the zi.
(a) Prove that every algebra derivation d: 𝒪alg → 𝒪alg (resp. d: 𝒪an → 𝒪an) is of the form d=∑j fj(z)·∂/∂zj in which all the fi are algebraic functions (resp. analytic functions) on W, and the index i runs from 1 to n. [Note: The case with analytic functions is not so trivial.]
(b) Prove that the apparent linear map Der(𝒪X) → Vect(W) to analytic vector fields on W, ∑i fj(z)·∂/∂zj ↦ ∑i fj(z)·∂/∂zj, is an injective map of Lie algebras when X=alg, and an isomorphism of Lie algebras when X=an. (Here derivations are given their standard commutator bracket.) For us, from this point on, we do not distinguish between "vector fields" and derivations on the algebra of functions.
(c) Let C× act on W via scaling, and consider the corresponding action of C× on Vect(W). (In terms of derivations, C× acts on 𝒪an by scaling the generators then on derivations by conjugation ζ·d = ζ·d(ζ-1·-).) Prove that the map to scale-invariant vector fields gl(W*) → Vect(W)C×, A ↦ ∑j(∑iaijzi)·∂/∂zj, is an isomorphism of Lie algebras.
6. Let 𝕍 be the standard representation for sl2(C).
(a) Let sl2(C) act on 𝒪alg(𝕍*)=Sym(V) by derivations, i.e. via the representation and the canonical embedding
sl2(C) → gl(𝕍) → Vect(𝕍*).
Prove that each homogeneous degree n component Sym(𝕍)_n is an sl2(C)-subrepresentation in Sym(𝕍), and write down the sl2(C)-action explicitly.
(b) Prove that homogeneous degree n functions Sym(𝕍)n are identified with the corresponding highest weight simple Sym(𝕍)n = L(n).
HW 1, do some of the problems (Due Sept 5):
1. (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).
2. 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*.
3. Consider a complex algebraic group G, so that we have both classes of analytic and algebraic functions G → C. An algebraic (resp. analytic) representation for G is a finite-dimensional vector space V equipped with a map of algebraic (resp. analytic) groups
φ: G → GL(V).
Equivalently, if we choose a basis on V and corresponding identification GL(V)=GLn(C), then φ = [φij] where the φij: G → C are all algebraic (resp. analytic) functions on G.
(a) Consider the multiplicative group Gm=C×, whose algebraic functions are simply polynomials in z and z-1, i.e. rational functions of the form p(z)/zn where p(z) is a polynomial and n is some non-negative integer. Prove that any algebraic representation
φ: Gm → GL(V)
is diagonalizable. [Hint: What kind of matrices must occur when we restrict to the roots up unity in C×? What about S1 ⊆ C×? Thinking about the off-diagonal entries as functions in z, how many zeros are they allowed to have?]
(b) Prove that any representations φ of Gm is necessarily, after an advantageous choice of coordinates, of the form φ(z) = diag{zm1,…,zmn} for integers mi.
(c) Classify all algebraic representations of the product group Gm×…×Gm.
(d) For a group G, a character is a 1-dimensional algebraic representation, i.e. a map of algebraic groups χ: G → Gm. Note that we can multiply characters in the apparent way, χ·χ'(g):= χ(g)·χ'(g), so that the collection of characters forms an abelian group Char(G). Prove that the character group of the n-fold product of the multiplicative group Char(Gm×…×Gm) is isomorphic to Zn.
(e) Given an arbitrary G-representation V and a character χ, define the subrepresentation
Vχ = { v in V: g·v = χ(g)·v for all g in G} ⊆ V.
Prove that any representation of the product G=Gm×…×Gm decomposes into a sum over the characters
V = Vm1⊕ … ⊕ Vmk.
(f) For a product of Gm's, do you think there is any analytic representations which are not algebraic?
4. For h the n-dimensional abelian Lie algebra, classify all simple h-representations. [Hint: Compare with modules for the polynomial algebra.] For dimension reasons and abelianosity, observe that Lie(Gm×…×Gm)=h. Compare your classifications from #4 and #5.
5. Consider SLn(C). If xij: SLn(C) → C is the function which picks out the ij-th matic entry, the collection of algebraic functions on SLn(C) are simply polynomial functions in the xij, along with the implicit relation det[xij]=1. We consider the case n=2.
For Ga = C the additive group, along with the expected polynomial functions, check that we have algebraic group maps
E, F: Ga → SL2(C), E(z) =[1 z // 0 1], F(z) = [1 0 // z 1]
and
H: Gm → SL2(C), H(z) = diag{z, z-1}.
Observe that any SL2(C)-representation decomposes into integral character spaces for the action of the diagonals,
V = Vm1⊕ … ⊕ Vmk.
(a) Calculate the commutator relations H(ζ)E(z)H(ζ)-1 = ? and H(ζ)F(z)H(ζ)-1 = ?.
(b) Convince yourself that SL2(C) is generated, as a discrete group, by the elements E(z), F(z), and H(z), for independently varying z's. Convince yourself further that SL2(C) is generated by just E(z) and F(z).
(c) Prove that there is a unique, algebraic, 1-dimension SL2(C)-representation (namely the trivial one), and that there is a unique nontrivial 2-dimensional representation (namely the standard representation). Prove that there are only two nontrivial 3-dimensional representations (namely triv ⊕ standard and sl2(C) with the conjugation action of SL2(C), i.e. the adjoint representation).