Introduction to Quantum Groups and Tensor Categories

Introduction to Quantum Groups and Tensor Categories

Johannes Flake1

Rutgers University

Graduate VOA Seminar, Feb/Mar 2016

1Got questions or comments? Just get in touch with him. Introduction to Quantum Groups and Tensor Categories

Outline

1 Hopf Algebras and Tensor Categories

2 Quasitriangular Hopf algebras and Ribbon Hopf Algebras

3 Quantum Groups at Roots of Unity Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories

Outline

1 Hopf Algebras and Tensor Categories

2 Quasitriangular Hopf algebras and Ribbon Hopf Algebras

3 Quantum Groups at Roots of Unity Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories

“A mathematican is a machine for turning coffee into theorems.” Alfr´edR´enyi(often attributed to Paul Erd˝os)

“A comathematican is a machine for turning cotheorems into ffee.” communicated to the author by Fei Qi Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories

(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

η⊗I µ⊗I k ⊗ A A ⊗ A A ⊗ k A ⊗ A ⊗ A A ⊗ A I ⊗η ∼ = µ I ⊗µ µ =∼ µ A A ⊗ A A (co-)unitarity (co-)associativity k ⊗ C C ⊗ C I ⊗ε C ⊗ k C ⊗ C ⊗ C C ⊗ C ε⊗I ∆⊗I ∼ = ∆ I ⊗∆ ∆ =∼ C C ⊗ C C ∆ Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories

Convolution

Sweedler’s Notation: ∀x ∈ C n X ∆(x) = x1,i ⊗ x2,i =: x1 ⊗ x2 ∈ C ⊗ C i=1

coassociativity ⇒ “x1 ⊗ x2 ⊗ x3” is well-defined counitarity ⇔ 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) Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories

Bialgebras, Hopf algebras

Bialgebra: algebra and coalgebra with compatible structure maps (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. Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories

Examples

Group algebra k[G] for a group G basis: {g} for g ∈ G εg = 1, ∆g = g ⊗ g, Sg = g −1 (“group-like element”) Universal enveloping algebra U(g) for a Lie group g p1 pn basis: {x1 ··· xn |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 by group-likes and primitives.2 2 Any cocommutative Hopf algebra over C is the semidirect/smash product Hopf algebra of the group algebra of the group formed by its group-likes and the universal enveloping algebra of the Lie algebra formed by its primitives. Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories

Categories and their bialgebras

“Tannaka(-Krein) duality”, “reconstruction theorems” Rep(A): category of modules of an algebra A of finite rank/dimension over k Consider categories “of k-modules of finite rank/dimension”.

category Rep(...) vector spaces/modules k monoidal bialgebra rigid monoidal Hopf algebra rigid braided monoidal quasitriangular Hopf algebra Ribbon Ribbon Hopf algebra Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories

Tannaka-Krein duality

“For A an algebra and AMod its category of modules, and for AMod → Vect the fiber functor that sends a module to its underlying vector space, we have a natural isomorphism End(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 of finite tensor categories C with a fiber functor F , up to tensor equivalence and isomorphism of tensor functors, and (2) isomorphism classes of finite dimensional Hopf algebras over k.” 4

3https://ncatlab.org/nlab/show/Tannaka+duality 4thm. 5.3.12 in Etingof, Gelaki, Nikshych, Ostrik: Tensor Categories. Introduction to Quantum Groups and Tensor Categories Quasitriangular Hopf algebras and Ribbon Hopf Algebras

Outline

1 Hopf Algebras and Tensor Categories

2 Quasitriangular Hopf algebras and Ribbon Hopf Algebras

3 Quantum Groups at Roots of Unity Introduction to Quantum Groups and Tensor Categories Quasitriangular Hopf algebras and Ribbon Hopf Algebras

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” Introduction to Quantum Groups and Tensor Categories Quasitriangular Hopf algebras and Ribbon Hopf Algebras

Scribble (some proofs)

R =: R1 ⊗ R2 =: r 1 ⊗ r 2 ∈ A⊗2, summation implied (but not a coproduct!).

1 1 2 1 1 2 2 (∆ ⊗ I )R = R13R23 ⇔ R1 ⊗ R2 ⊗ R = r ⊗ R ⊗ r R ...

1 1 2 1 1 2 2 ... ⇒ ε(R1 ) ⊗ R2 ⊗ R = ε(r ) ⊗ R ⊗ r R ⇒ 1 ⊗ R1 ⊗ R2 = 1 ⊗ ε(r 1)R1 ⊗ r 2R2 ⇒ 1 ⊗ 1 = ε(r 1) ⊗ r 2

1 1 2 1 1 2 2 ... ⇒ S(R1 )R2 ⊗ R = S(r )R ⊗ r R ⇒ ε(R1) ⊗ R2 = (S(r 1) ⊗ r 2)(R1 ⊗ R2) ⇒ 1 ⊗ 1 = (S(r 1) ⊗ r 2)R Introduction to Quantum Groups and Tensor Categories Quasitriangular Hopf algebras and Ribbon Hopf Algebras

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 tensor products 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) Introduction to Quantum Groups and Tensor Categories Quasitriangular Hopf algebras and Ribbon Hopf Algebras

Ribbon elements

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

A central invertible v ∈ A is called universal twist or ribbon element if 1 v 2 = 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 directly and (1), (4) are equivalent. Introduction to Quantum Groups and Tensor Categories Quasitriangular Hopf algebras and Ribbon Hopf Algebras

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 . Introduction to Quantum Groups and Tensor Categories Quantum Groups at Roots of Unity

Outline

1 Hopf Algebras and Tensor Categories

2 Quasitriangular Hopf algebras and Ribbon Hopf Algebras

3 Quantum Groups at Roots of Unity Introduction to Quantum Groups and Tensor Categories Quantum Groups at Roots of Unity

Definition

here quantum group := quantized universal enveloping algebra

(aij )1≤i,j≤m the Cartan matrix of a simple Lie algebra g of type ADE (⇒ aii = 2, aij = aji ∈ {0, −1} for i 6= j) q ∈ C \{0, ±1} −1 Uq(g) generated by {Ei , Fi , Ki , Ki }1≤i≤m with relations:

−1 −1 [Ki , Kj ] = 0 Ki Ki = 1 = Ki Ki −1 Ki − K K E = qaij E K K F = q−aij F K [E , F ] = δ i i j j i i j j i i j ij q − q−1

[Ei , Ej ] = [Fi , Fj ] = 0 if aij = 0 2 −1 2  Ei Ej − (q + q )Ei Ej Ei + Ej Ei = 0 2 −1 2 if aij = −1 Fi Fj − (q + q )Fi Fj Fi + Fj Fi = 0 Introduction to Quantum Groups and Tensor Categories Quantum Groups at Roots of Unity

Definition/Theorem

Uq(g) is a Hopf algebra with

−1 ∆(Ei ) = Ei ⊗ 1 + Ki ⊗ Ei S(Ei ) = −Ki Ei ε(Ei ) = 0 , −1 ∆(Fi ) = Fi ⊗ Ki + 1 ⊗ Fi S(Fi ) = −Fi Ki ε(Fi ) = 0 , −1 ∆(Ki ) = Ki ⊗ Ki S(Ki ) = Ki ε(Ki ) = 1 . ( p p odd Assume q is a p-th root of unity, p ≥ 3, p0 := . p/2 p even p0 p0 p J := hEi , Fi , Ki − 1ii as ideal in Uq(g).

⇒ U˜q(g) := Uq(g)/J is a fin.-dim. ribbon quotient Hopf algebra. Introduction to Quantum Groups and Tensor Categories Quantum Groups at Roots of Unity

Scribble (proof ideas)

We may verify that Uq(g) is a Hopf algebra, and that J is a Hopf ideal. Hence U˜q(g) is a Hopf algebra. It is quasitriangular, because it is the quotient of a Drinfel’d double (see following slides).

−1 P −2b1 −2bm Let (bij )i,j := (aij )i,j , bi := j bij , g := K1 ··· Km . ⇒ g is an invertible group-like in U˜q(g), 2 −1 S (a) = gag for all a ∈ U˜q(g) Let u be the distinguished element of the quasitriangular Hopf algebra U˜q(g). ⇒ v := ug −1 is central invertible and we may also verify that Sv = v. Hence, v is a ribbon element. Introduction to Quantum Groups and Tensor Categories Quantum Groups at Roots of Unity

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 the identifications A → A ⊗ 1 ⊂ D(A) and A0 → 1 ⊗ A0 ⊂ D(A), are Hopf algebra maps and such that their images generate D(A) as algebra.

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, and even 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. Introduction to Quantum Groups and Tensor Categories Quantum Groups at Roots of Unity

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

H For a Hopf algebra H, H YD 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 ⇒ H YD is a braided monoidal category

∃ functor Radford’s biproduct/bosonization H 5 {“braided” Hopf algebra in H YD} → {Hopf algebra},A 7→ A#H. A#H contains H as Hopf subalgebra and A as subalgebra.

5Not to be confused with the semidirect/smash product which is sometimes denoted identically. The latter one is a product of a Hopf algebra and a module algebra, and no comodule structure is involved. Introduction to Quantum Groups and Tensor Categories Quantum Groups at Roots of Unity

Quantum groups revisited

m ± n m ± H := k[Z ] = k[K1,..., Km], V := k = ⊕i=1Ei k the ± ±aij Yetter-Drinfel’d modules defined by Ki .Ej = q and ± ± δ(Ei ) = Ki ⊗ Ei . T (V ±) are braided Hopf algebras 1−aij ± ± adding the Serre relations ad ± (Ej ) = 0 to T (V ) Ei → braided Hopf algebras U(n±) H (“Borel part”; ad is to be taken in H YD) 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 identification + − of the two copies of H. Ei = Ei , Fi = Ei . Introduction to Quantum Groups and Tensor Categories Quantum Groups at Roots of Unity

Quantum groups revisited / Outlook

Drinfel’d doubles and quotients of quasitriangular Hopf algebras are quasitriangular, so U˜q(g) is quasitriangular Generalizations of the quantum groups discussed here which are still Ribbon Hopf algebras have been defined6. The fact that quantum groups and their generalizations are ribbon Hopf algebras can be proved through general Hopf algebra theory, as well7. There are results on how braided tensor categories obtained from conformal field theories can be studied through quantum groups8.

6Majid, Double-bosonization of braided groups and the construction of Uq(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 Introduction to Quantum Groups and Tensor Categories Summary

Summary

category Rep(...) vector spaces / modules k monoidal bialgebra rigid monoidal Hopf algebra rigid braided monoidal quasitriangular Hopf algebra Ribbon Ribbon Hopf algebra* (*) e.g. quantum groups

Quantum groups are quotients of Drinfel’d doubles of bosonizations of universal enveloping algebras of Borel subalgebras of Lie algebras in a category of Yetter-Drinfel’d modules. Roughly speaking. Introduction to Quantum Groups and Tensor Categories 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 link polynomials and quantum groups, 1991, [here]. Heckenberger, Nichols Algebras (Lecture Notes), 2008, [here]: section 7, see also Simon Lentner’s MO answer [here].