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 " : 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 ⊗" C ⊗ k C ⊗ C ⊗ C C ⊗ C "⊗I ∆⊗I ∼ = ∆ I ⊗∆ ∆ =∼ C C ⊗ C C ∆ Introduction to Quantum Groups and Tensor Categories Hopf Algebras and Tensor Categories Convolution Sweedler's Notation: 8x 2 C n X ∆(x) = x1;i ⊗ x2;i =: x1 ⊗ x2 2 C ⊗ C i=1 coassociativity ) \x1 ⊗ x2 ⊗ x3" is well-defined counitarity , "(x1)x2 = x = x1"(x2) Convolution: 8f ; g : C ! A, f ∗ g := µ ◦ (f ⊗ g) ◦ ∆, i.e.( f ∗ g)(x) := f (x1)g(x2) 8x 2 C Note that η ◦ " : C ! A is an identity element for ∗: 8f : C ! A; x 2 C, (f ∗ (η ◦ "))(x) = f (x1)η("(x2)) = f (x1"(x2))1 = f (x) ((η ◦ ") ∗ f )(x) = f ("(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, "; ∆ are algebra maps.) Hopf algebra: bialgebra H with an antipode, that is a ∗-inverse S of I as maps H ! H. For all x 2 H, this means x1S(x2) = (I ∗ S)(x) = "(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: fgg for g 2 G "g = 1, ∆g = g ⊗ g, Sg = g −1 (\group-like element") Universal enveloping algebra U(g) for a Lie group g p1 pn basis: fx1 ··· xn jp1;:::; pn ≥ 0g for a basis x1;:::; xn of g "xi = 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. 8V ; W k-spaces, τV ;W : V ⊗ W ! W ⊗ V ; v ⊗ w 7! w ⊗ v. 8R 2 A⊗2 we define elements in A⊗3: R12 := R ⊗ 1, R23 := 1 ⊗ R, R13 := (I ⊗ τ)(R ⊗ 1). R 2 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 ) (" ⊗ I )R = (I ⊗ ")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 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 ::: ) "(R1 ) ⊗ R2 ⊗ R = "(r ) ⊗ R ⊗ r R ) 1 ⊗ R1 ⊗ R2 = 1 ⊗ "(r 1)R1 ⊗ r 2R2 ) 1 ⊗ 1 = "(r 1) ⊗ r 2 1 1 2 1 1 2 2 ::: ) S(R1 )R2 ⊗ R = S(r )R ⊗ r R ) "(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 2 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) 2 A ) u is invertible and S2(a) = uau−1, 8a 2 A (compare this with our examples for Hopf algebras above) ) u−1 = (I ⊗ S2)τ(R);"(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 2 A is called universal twist or ribbon element if 1 v 2 = uS(u) 2 "(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 2 Rep(A): θV : V ! V ; x 7! vx : ) Then Rep(A) is a Ribbon category with twist θ, i.e. 8V ; 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 2 f0; −1g for i 6= j) q 2 C n f0; ±1g −1 Uq(g) generated by fEi ; Fi ; Ki ; Ki g1≤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 "(Ei ) = 0 ; −1 ∆(Fi ) = Fi ⊗ Ki + 1 ⊗ Fi S(Fi ) = −Fi Ki "(Fi ) = 0 ; −1 ∆(Ki ) = Ki ⊗ Ki S(Ki ) = Ki "(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.