Pre-Talbot seminar, lecture 4
David Jordan - Quantum groups
Notes by Scott Carnahan
I hope you’re familiar with tensor categories. A tensor category isbraided if it is equipped with natural isomorphisms cV,W : V ⊗ W →W ⊗ V for all objects V, W , and they satisfy c12 c23 c12 = c23 c12 c23 forevery triple. This is the structure you need to have a representation of
the braid group. If cV,W ◦ cW,V = id, then we just have a symmetrictensor category.
Universal R-matrix
Let H be a Hopf algebra. We want H-mod to form a braided tensor
category. For motivation, suppose we already have a braided tensor
structure on H-mod, and consider the regular representation. ThencH,H : H ⊗ H → H ⊗ H takes 1 ⊗ 1 to some element cH,H (1 ⊗ 1), andwe define R = τ12 cH,H (1 ⊗ 1) ∈ H ⊗ H, where τ12 is the plain switchingmap.
Proposition We can express the braiding using R, via cV,W = τV,W ◦µR , where µR is just multiplication by the R-matrix.
The upshot is that H-mod admits a braided tensor category structure
if and only if there exists R ∈ H ⊗ H satisfying some conditions. Theconditions can be reconstructed by looking at the axioms for braiding.
Bi-cross products of Hopf algebras
This concerns Drinfeld’s construction of Uq using the quantum double.
A pair (X, A) of Hopf algebras is called matched if we have(1) a left action . : A ⊗ X → X(2) a right action / : A ⊗ X → A(3) a condition: . and / make X and A into module-coalgebras
over each other.
(4) some more technical conditions involving coproducts and composition (enough to put a Hopf algebra structure on X ⊗ A).
Proposition We can construct a Hopf algebra structure X ./ A onX ⊗ A by
• specifying that the inclusions iX : X → X ./ A given by x 7→x ⊗ 1, and iA : A → X ./ A given by a 7→ 1 ⊗ a are Hopf algebra
homomorphisms• specifying cross relations: a · x = (a1 . x1 )(a2 / x2 ) - note thatthe first factor is in X, and the second in A.
12Here, the subscripts denote Sweedler notation with tacit summation:∆(a) = a1 ⊗ a2 . The cross relations allow us to switch the order offactors.
Example: Group factorizations. Let G = H · K (so any g ∈ G canbe uniquely written as hk for some h ∈ H, k ∈ K), where H and K
are not necessarily normal subgroups of G (standard examples: Sylow
subgroups of A4 × S3 , and A4 · C5 = A5 ). Suppose someone gives youkh, and you want to put it into the other order. Let g 0 = kh, and use
the uniqueness of decomposition to get g 0 = h0 k 0 . We can define theaction of H and K on each other by k . h := h1 and k / h := k1 . Then
C[G] ∼= C[H] ./ C[K].
Quantum Double
Let H be a Hopf algebra. Then H has an adjoint action on itself, by(ad a) . h := a1 hS(a2 ) and h / (ad a) := S(a1 )ha2 . In group land, wehave adg . h = ghg −1 .
Proposition This is a module-algebra action: (ad a) . xy = (ad a1 ) . x ·(ad a2 ) . y. (For example, with groups we have gxyg −1 = gxg −1 gyg −1 .
This implies that H acts on H ∗ as a module coalgebra (if H is infinite
dimensional, we just ask for H ∗ to have a nondegenerate pairing). ThenH acts on (H op )∗ by module-coalgebra actions. By symmetry, since(((H op )∗ )op )∗ ∼= H, we get a module-coalgebra action of (H op )∗ on H.
This gives us a matched pair.
Definition: The quantum double of H is D(H) := (H op )∗ ./ H.
Why do we do this construction? We’ll see that the universal Rmatrix gives us an isomorphism R : H → H co−op , where H co−op is justH with the same multiplication but opposite comultiplication. Thequantum double construction makes this self-dual.
'To construct the R-matrix, take id ∈ End(H) → (H op )∗ ⊗ H,λH,Hand let ρ := λH,H (id). If H is infinite-dimensional, then λH,H is notnecessarily a well-defined map, but ρ can still be defined formally by
its action on finite dimensional representations as a power series ofnilpotents. We choose a basis such that ρP= ei ⊗ ei , and let R =(i(H op )∗ ⊗ iH )(ρ). Then we can write R = i (1 ⊗ ei ) ⊗ (ei ⊗ 1). If
you’ve ever seen an R-matrix for Uq (g) written explicitly, you might
wonder what happened to the huge mess. I’ll get to that soon.
We can recast this construction categorically. The existence of theR-matrix for the double follows from the universal property of theDrinfeld center of the category Rep H, i.e., Rep D(H) is the universal
braided tensor category with a monoidal functor to Rep H.
Claim: Uq (g) ∼= D(Uq (b± ))/some stuff.

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We can decompose Uq (g) as a vector space into Uq (n− ) ⊗ Uq (h) ⊗Uq (n+ ), with Uq (b− ) = Uq (n− ) ⊗ Uq (h) and Uq (b+ ) = Uq (n+ ) ⊗ Uq (h).
For g = sln , these are generated by the lower and upper triangularmatrices with trace zero, and they intersect on the torus of diagonals.
The explicit Hopf algebra structure can be found on line or in any
textbook on quantum groups.
The Killing form on the Lie algebra gives a nondegenerate pairing
hb+ , b− i → C, so we can write b− ∼= (b+,op )∗ , where the op comes fromthe transpose anti-involution.
Identifying the tori gives us a surjection D(Uq (b+ ))  Uq (g), andthe rather nice R-matrix we saw above gets taken to something rathernasty. You might see some red flags involving finite-dimensionality
with respect to this map, but it can be made rigorous.
Three contexts for quantum groups(1) Formal power series U~ (g) - defined by Drinfeld.
(2) Rational case Uq (g) - Lusztig and Jimbo noticed that setting q =e~ gives rational functions. The Hopf algebra is not quasitriangular, but if we restrict to finite dimensional modules,
get aP wekbraiding, because the formal R-matrix looks like k E ⊗ F k ,and the pieces act nilpotently.
(3) Roots of unity - this has a large center, so if we work in a
relative setting, we only need to worry about finite dimensional
behaviors.
Duality and covariance
In classical groups, we have a commutative algebra O(G) of functions
with a nondegenerate pairing hU (g), O(G)i → C, given by differential
operators acting on germs of functions at the identity. A natural question is, if we replace U (g) by Uq (g), what should replace O(G)?
There are two answers:(1) (Majid’s first braided reconstruction theorem) We can get O(G)from Tannaka-Krein. It is the coordinate ring of the automorphism group of the fiber functor U (g) → V ect. We cando something similar to get Oq (G), where our fiber functorUq (g) → V ect is adjusted so that the essential image has anontrivial braiding pushed forward. Since V ect has no nontrivial braidings, this braiding cannot be extended to all of V ect.
We get a noncommutative Hopf algebra.
(2) (Algebra of matrix coefficients) Let V be a Uq (g)-representation,and let v ∈ V and f ∈ V ∗ . Then we have functionals cf,v ∈Uq (g)∗ , defined by cf,v (u) = f (u · v). Here, the star just means
vector space dual. We write Oq (G) for the span of all cf,v inside4Uq (g) as V ranges over all finite dimensional Uq (g)-modules V .
There is a multiplication: cf,v · cg,w = cf ⊗g,v⊗w , which makes
this space a subalgebra. Actually, we get n or 2n copies of thealgebra we want, but there is a preferred one, called Oq+ (G).
Proposition: Uq (g)-mod has a projective element, i.e., every finite
dimensional (projective?) module lives in some V ⊗n .
A consequence of this is that Oq (G) is a quotient of the tensor algebra
T (V ∗ ⊗ V ) by some relations encoding the embeddings of representations W in the tensor powers of V . For G = SLn you get a relation,
detq = 1.
L
We can decompose: Oq (G) ∼= V simple V ∗ ⊗ V , and Uq (g) acts on
both sides, generalizing the actions of left and right vector fields. Oq (G)is a module-algebra for these actions, and when q = 1, this implies
O(G) is covariant with respect to the adjoint action. However, whenq 6= 1, Oq (G) is not commutative, and the left and right actions do notcombine to give an adjoint action.
Majid’s solution: BOq (G), the braided covariantized algebra.
Again, we have two definitions.
(1) (Fancy version) For any faithful exact braided tensor functor
Uq -mod → C, we get a Tannakian reconstruction of a Hopfalgebra in C. One obvious choice is the identity functor on Uq mod, and its automorphisms form a Hopf algebra BOq in Uq mod. Majid’s second reconstruction theorem asserts that thisis unique. Note that this is not a Hopf algebra in vector spaces- you use the braided tensor structure to get compatibility ofmultiplication and comultiplication.
(2) (Potatoes version) BOq ∼= {cf,v } as a coalgebra. We change themultiplication to be compatible with the braiding, since multiplying an element of V ∗ ⊗ V by an element of W ∗ ⊗ W involves
switching them. Writing R = r+ ⊗ r− , we have cf,v · cg,w =cf ⊗r− g,r+ v⊗w . This is still generated by the defining module.
What is this good for? Well, this is the punchline of the talk.
Proposition(1) Both Oq and BOq have the property that all Uq -modules arecomodules for Oq and BOq . For BOq -comodules, we get a wellbehaved braided diagram calculus.
(2) BOq is covariant for the adjoint action of Uq .
(3) BOq is braided commutative, unlike Oq .

5Theorem There is a unique isomorphism of algebras K : BOq → Uqlf ,called the “Fourier transform,” where the lf means the locally finitepart.
As q → 1, this degenerates, and is no longer an isomorphism. The
fact that it exists is a strange thing, since we constructed BOq as adual to Uq . One consequence of this theorem is that an analogue of
harmonic analysis works in the quantum context, and it is similar tothe case of abelian groups.
This theory was worked out by Majid (and independently by severalothers), and you can find it in this hot pink textbook that he wrote.