introduction to quantum groups and tensor categories · graduate voa seminar, feb/mar 2016 1got...

Introduction to Quantum Groups and Tensor Categories

Introduction toQuantum Groups and Tensor Categories

Johannes Flake1

Rutgers University

Graduate VOA Seminar, Feb/Mar 2016

1Got questions or comments? Just get in touch with him.


Outline

1 Hopf Algebras and Tensor Categories

2 Quasitriangular Hopf algebras and Ribbon Hopf Algebras

3 Quantum Groups at Roots of Unity


Hopf Algebras and Tensor Categories

Outline






“A mathematican is a machine for turning coffee into theorems.”Alfred Renyi (often attributed to Paul Erdos)

“A comathematican is a machine for turning cotheoremsinto ffee.” communicated to the author by Fei Qi



(Co-)Algebras

k : our favorite commutative r1ng/field, all maps are k-linear.

Algebra: k-space A with η : k → A, µ : A⊗ A→ A

Coalgebra: k-space C with ε : C → k , ∆ : C → C ⊗ C

k ⊗ A A⊗ A A⊗ k A⊗ A⊗ A A⊗ A

A A⊗ A A

∼=

η⊗I

µ

I⊗η

∼=

µ⊗I

I⊗µ µ

µ

(co-)unitarity (co-)associativity

k ⊗ C C ⊗ C C ⊗ k C ⊗ C ⊗ C C ⊗ C

C C ⊗ C C

∼=ε⊗I

I⊗ε

∼=

∆⊗I∆ I⊗∆

∆

∆



Convolution

Sweedler’s Notation: ∀x ∈ C

∆(x) =n∑

i=1

x1,i ⊗ x2,i =: x1 ⊗ x2 ∈ C ⊗ C

coassociativity ⇒ “x1 ⊗ x2 ⊗ x3” is well-definedcounitarity ⇔ ε(x1)x2 = x = x1ε(x2)

Convolution: ∀f , g : C → A, f ∗ g := µ ◦ (f ⊗ g) ◦∆,i.e. (f ∗ g)(x) := f (x1)g(x2) ∀x ∈ C

Note that η ◦ ε : C → A is an identity element for ∗:∀f : C → A, x ∈ C ,

(f ∗ (η ◦ ε))(x) = f (x1)η(ε(x2)) = f (x1ε(x2))1 = f (x)

((η ◦ ε) ∗ f )(x) = f (ε(x1)x2) = f (x)



Bialgebras, Hopf algebras

Bialgebra: algebra and coalgebra with compatible structuremaps (η, µ are coalgebra maps, ε,∆ are algebra maps.)

Hopf algebra: bialgebra H with an antipode, that is a∗-inverse S of I as maps H → H. For all x ∈ H, this means

x1S(x2) = (I ∗ S)(x) = ε(x)1 = (S ∗ I )(x) = S(x1)x2

⇒ S is an antialgebra map and an anticoalgebra map,every bialgebra has at most one antipode.



Examples

Group algebra k[G ] for a group Gbasis: {g} for g ∈ Gεg = 1, ∆g = g ⊗ g , Sg = g−1

(“group-like element”)

Universal enveloping algebra U(g) for a Lie group gbasis: {xp1

1 · · · xpnn |p1, . . . , pn ≥ 0} for a basis x1, . . . , xn of g

εxi = 0, ∆xi = 1⊗ xi + xi ⊗ 1, Sxi = −xi(“primitive element”)

⇒ In both cases, S2 = I .Any cocommutative Hopf algebra over C is generated bygroup-likes and primitives.2

2Any cocommutative Hopf algebra over C is the semidirect/smash productHopf algebra of the group algebra of the group formed by its group-likes andthe universal enveloping algebra of the Lie algebra formed by its primitives.



Categories and their bialgebras

“Tannaka(-Krein) duality”, “reconstruction theorems”

Rep(A): category of modules of an algebra A of finiterank/dimension over k

Consider categories “of k-modules of finite rank/dimension”.

category Rep(...)

vector spaces/modules kmonoidal bialgebrarigid monoidal Hopf algebrarigid braided monoidal quasitriangular Hopf algebraRibbon Ribbon Hopf algebra



Tannaka-Krein duality

“For A an algebra and AMod its category of modules, and forAMod→ Vect the fiber functor that sends a module to itsunderlying vector space, we have a natural isomorphismEnd(AMod→ Vect) ' A in Vect.” 3

“The assignments

(C,F ) 7→ H = End(F ),H 7→ (Rep(H),Forget)

are mutually inverse bijections between (1) equivalence classes offinite tensor categories C with a fiber functor F , up to tensorequivalence and isomorphism of tensor functors, and (2)isomorphism classes of finite dimensional Hopf algebras over k.” 4

3https://ncatlab.org/nlab/show/Tannaka+duality4thm. 5.3.12 in Etingof, Gelaki, Nikshych, Ostrik: Tensor Categories.

https://ncatlab.org/nlab/show/Tannaka+duality


Quasitriangular Hopf algebras and Ribbon Hopf Algebras

Outline






R-matrices

We fix a Hopf algebra A over k .

∀V ,W k-spaces, τV ,W : V ⊗W →W ⊗ V , v ⊗ w 7→ w ⊗ v .

∀R ∈ A⊗2 we define elements in A⊗3:R12 := R ⊗ 1, R23 := 1⊗ R, R13 := (I ⊗ τ)(R ⊗ 1).

R ∈ A⊗2 is called (universal) R-matrix, if

1 R is invertible and τ ◦∆(a) = R∆(a)R−1

2 (I ⊗∆)R = R13R12

3 (∆⊗ I )R = R13R23

⇒ (ε⊗ I )R = (I ⊗ ε)R = 1⊗ 1, (S ⊗ I )R = (I ⊗ S−1)R = R−1

⇒ R12R13R23 = R23R13R12 “Yang-Baxter Equation”



Scribble (some proofs)

R =: R1 ⊗ R2 =: r1 ⊗ r2 ∈ A⊗2, summation implied (but not acoproduct!).

(∆⊗ I )R = R13R23 ⇔ R11 ⊗ R1

2 ⊗ R2 = r1 ⊗ R1 ⊗ r2R2 . . .

. . .⇒ ε(R11 )⊗ R1

2 ⊗ R2 = ε(r1)⊗ R1 ⊗ r2R2

⇒ 1⊗ R1 ⊗ R2 = 1⊗ ε(r1)R1 ⊗ r2R2

⇒ 1⊗ 1 = ε(r1)⊗ r2

. . .⇒ S(R11 )R1

2 ⊗ R2 = S(r1)R1 ⊗ r2R2

⇒ ε(R1)⊗ R2 = (S(r1)⊗ r2)(R1 ⊗ R2)

⇒ 1⊗ 1 = (S(r1)⊗ r2)R



Representations of quasitriangular Hopf algebras

If A has an R-matrix R, it is called quasitriangular.In this case, we define maps for all pairs of objects V ,W ∈ Rep(A):

cV ,W : V ⊗W →W ⊗ V , x 7→ τ(Rx) .

⇒ Then Rep(A) is a braided monoidal category with braiding c ,i.e. for any n ≥ 1, the braid group Bn acts on n-fold tensorproducts of A-modules via c .

u := µ ◦ (S ⊗ I ) ◦ τ(R) ∈ A⇒ u is invertible and S2(a) = uau−1, ∀a ∈ A(compare this with our examples for Hopf algebras above)

⇒ u−1 = (I ⊗ S2)τ(R), ε(u) = 1,∆u = (τ(R)R)−1(u ⊗ u)



Ribbon elements

We fix a quasitriangular Hopf algebra A with R-matrix R.

A central invertible v ∈ A is called universal twist or ribbonelement if

1 v2 = uS(u)

2 ε(v) = 1

3 ∆v = (τ(R)R)−1(v ⊗ v)

4 S(v) = v

Note: If v = ug−1 for a group-like g , then (2), (3) follow directlyand (1), (4) are equivalent.



Representations of ribbon Hopf algebras

If A has a Ribbon element v , it is called ribbon Hopf algebra.In this case, we define maps for all objects V ∈ Rep(A):

θV : V → V , x 7→ vx .

⇒ Then Rep(A) is a Ribbon category with twist θ, i.e. ∀V ,W ,

θV⊗W = cW ,V cV ,W (θV ⊗ θW )

(θV ⊗ IV ∗)bV = (IV ⊗ θV ∗)bV , where bV : k → V ⊗ V ∗.


Quantum Groups at Roots of Unity

Outline






Definition

quantum grouphere:= quantized universal enveloping algebra

(aij)1≤i ,j≤m the Cartan matrix of a simple Lie algebra g oftype ADE (⇒ aii = 2, aij = aji ∈ {0,−1} for i 6= j)

q ∈ C \ {0,±1}

Uq(g) generated by {Ei ,Fi ,Ki ,K−1i }1≤i≤m with relations:

[Ki ,Kj ] = 0 KiK−1i = 1 = K−1

i Ki

KiEj = qaijEjKi KiFj = q−aijFjKi [Ei ,Fj ] = δijKi − K−1

i

q − q−1

[Ei ,Ej ] = [Fi ,Fj ] = 0 if aij = 0

E 2i Ej − (q + q−1)EiEjEi + EjE

2i = 0

F 2i Fj − (q + q−1)FiFjFi + FjF

2i = 0

}if aij = −1



Definition/Theorem

Uq(g) is a Hopf algebra with

∆(Ei ) = Ei ⊗ 1 + Ki ⊗ Ei S(Ei ) = −K−1i Ei ε(Ei ) = 0 ,

∆(Fi ) = Fi ⊗ K−1i + 1⊗ Fi S(Fi ) = −FiKi ε(Fi ) = 0 ,

∆(Ki ) = Ki ⊗ Ki S(Ki ) = K−1i ε(Ki ) = 1 .

Assume q is a p-th root of unity, p ≥ 3, p′ :=

{p p odd

p/2 p even.

J := 〈Ep′

i ,Fp′

i ,Kpi − 1〉i as ideal in Uq(g).

⇒ Uq(g) := Uq(g)/J is a fin.-dim. ribbon quotient Hopf algebra.



Scribble (proof ideas)

We may verify that Uq(g) is a Hopf algebra, and that J is aHopf ideal. Hence Uq(g) is a Hopf algebra.

It is quasitriangular, because it is the quotient of a Drinfel’ddouble (see following slides).

Let (bij)i ,j := (aij)−1i ,j , bi :=

∑j bij , g := K−2b1

1 · · ·K−2bmm .

⇒ g is an invertible group-like in Uq(g),S2(a) = gag−1 for all a ∈ Uq(g)

Let u be the distinguished element of the quasitriangular Hopfalgebra Uq(g).⇒ v := ug−1 is central invertible and we may also verify thatSv = v . Hence, v is a ribbon element.



Drinfel’d double

Consider

A a fin.-dim. Hopf algebra with dual A∗

A0 := A∗ as algebra, but with ∆0 := τ ◦∆, S0 := S−1

⇒ ∃ Hopf algebra D(A) ' A⊗ A0 as k-spaces such that theidentifications A→ A⊗ 1 ⊂ D(A) and A0 → 1⊗ A0 ⊂ D(A), areHopf algebra maps and such that their images generate D(A) asalgebra.

D(A) is quasitriangular, with R the identity element in A⊗ A0

(A has to be finite-dimensional!).

Note: D(A) can be defined even if A is not finite-dimensional, andeven for two Hopf algebras with a suitable pairing.

Note also: D(A) is the Hopf algebra corresponding to the “center”of the tensor category Mod(A) by Tannaka-Krein duality.



Yetter-Drinfel’d modules, Radford’s biproduct/bosonization

For a Hopf algebra H, HHYD is the category of (left left)

(H,H)-bimodules V with compatibility condition

δ(h.v) = h1v−1Sh3 ⊗ h2.v0 ∀h ∈ H, v ∈ V ,

where δ is the coaction and δ(v) =: v−1 ⊗ v0.⇒ H

HYD is a braided monoidal category

∃ functor Radford’s biproduct/bosonization{“braided” Hopf algebra in H

HYD} → {Hopf algebra},A 7→ A#H.5

A#H contains H as Hopf subalgebra and A as subalgebra.

5Not to be confused with the semidirect/smash product which is sometimesdenoted identically. The latter one is a product of a Hopf algebra and a modulealgebra, and no comodule structure is involved.



Quantum groups revisited

H := k[Zm] = k[K1, . . . ,Km], V± := kn = ⊕mi=1E

±i k the

Yetter-Drinfel’d modules defined by Ki .E±j = q±aij and

δ(E±i ) = Ki ⊗ E±i .

T (V±) are braided Hopf algebras

adding the Serre relations ad1−aijE±i

(E±j ) = 0 to T (V±)

→ braided Hopf algebras U(n±)(“Borel part”; ad is to be taken in H

HYD)

bosonizations U(n±)#H→ Hopf algebras which are dual in the sense of A 7→ A0

Uq(g): Drinfel’d double D(U(n+)#H) modulo identificationof the two copies of H. Ei = E+

i , Fi = E−i .



Quantum groups revisited / Outlook

Drinfel’d doubles and quotients of quasitriangular Hopfalgebras are quasitriangular, so Uq(g) is quasitriangular

Generalizations of the quantum groups discussed here whichare still Ribbon Hopf algebras have been defined6.The fact that quantum groups and their generalizations areribbon Hopf algebras can be proved through general Hopfalgebra theory, as well7.

There are results on how braided tensor categories obtainedfrom conformal field theories can be studied through quantumgroups8.

6Majid, Double-bosonization of braided groups and the construction ofUq(g), 1996 / Heckenberger, Nichols Algebras (Lecture Notes), 2008 / ...

7Burciu, A class of Drinfeld doubles that are ribbon algebras, 2008.8see http://arxiv.org/pdf/0705.4267v2.pdf, for instance

http://arxiv.org/pdf/0705.4267v2.pdf


Summary

Summary

category Rep(...)

vector spaces / modules kmonoidal bialgebrarigid monoidal Hopf algebrarigid braided monoidal quasitriangular Hopf algebraRibbon Ribbon Hopf algebra*

(*) e.g. quantum groups

Quantum groups are quotients of Drinfel’d doubles of bosonizationsof universal enveloping algebras of Borel subalgebras of Lie algebrasin a category of Yetter-Drinfel’d modules. Roughly speaking.


References

For further reading

Turaev, Quantum Invariants of Knots and 3-Manifolds, 1994:chapter XI 1-3, 6.

Chari, Pressley, A Guide to Quantum Groups, 1995.

Majid, Foundations of Quantum Group Theory, 2000.

Drinfel’d, Quantum Groups, 1986, [here].

Reshetikhin, Turaev, Invariants of 3-manifolds via linkpolynomials and quantum groups, 1991, [here].

Heckenberger, Nichols Algebras (Lecture Notes), 2008, [here]:section 7, see also Simon Lentner’s MO answer [here].

http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0798.0820.ocr.pdf

https://www.researchgate.net/profile/Vladimir_Turaev/publication/226921233_Invariants_of_3_-_Manifolds_Via_Link_Polynomials_and_Quantum_Groups_Invent/links/02e7e5236efa1cdfc0000000.pdf

http://www.mi.uni-koeln.de/~iheckenb/na.pdf

http://mathoverflow.net/q/101803

introduction to quantum groups and tensor categories · graduate voa seminar, feb/mar 2016 1got...

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