DOCSLIB.ORG

Introduction to Quantum Groups and Tensor Categories

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Introduction to Quantum Groups and Tensor Categories 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.
Load more

Recommended publications
  • From Non-Semisimple Hopf Algebras to Correlation Functions For
    ZMP-HH/13-2 Hamburger Beitr¨age zur Mathematik Nr. 468 February 2013 FROM NON-SEMISIMPLE HOPF ALGEBRAS TO CORRELATION FUNCTIONS FOR LOGARITHMIC CFT J¨urgen Fuchs a, Christoph Schweigert b, Carl Stigner c a Teoretisk fysik, Karlstads Universitet Universitetsgatan 21, S–65188 Karlstad b Organisationseinheit Mathematik, Universit¨at Hamburg Bereich Algebra und Zahlentheorie Bundesstraße 55, D–20146 Hamburg c Camatec Industriteknik AB Box 5134, S–65005 Karlstad Abstract We use factorizable finite tensor categories, and specifically the representation categories of factorizable ribbon Hopf algebras H, as a laboratory for exploring bulk correlation functions in arXiv:1302.4683v2 [hep-th] 22 Aug 2013 local logarithmic conformal field theories. For any ribbon Hopf algebra automorphism ω of H we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric Frobenius algebra. We derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters; as a crucial ingredient this involves the Cartan matrix of the category. We also show how for any candidate bulk state space of the type we consider, correlation functions of bulk fields for closed oriented world sheets of any genus can be constructed that are invariant under the natural action of the relevant mapping class group. 1 1 Introduction Understanding a quantum field theory includes in particular having a full grasp of its correla- tors on various space-time manifolds, including the relation between correlation functions on different space-times. This ambitious goal has been reached for different types of theories to a variable extent.
    [Show full text]
  • Bialgebras in Rel
    CORE Metadata, citation and similar papers at core.ac.uk Provided by Elsevier - Publisher Connector Electronic Notes in Theoretical Computer Science 265 (2010) 337–350 www.elsevier.com/locate/entcs Bialgebras in Rel Masahito Hasegawa1,2 Research Institute for Mathematical Sciences Kyoto University Kyoto, Japan Abstract We study bialgebras in the compact closed category Rel of sets and binary relations. We show that various monoidal categories with extra structure arise as the categories of (co)modules of bialgebras in Rel.In particular, for any group G we derive a ribbon category of crossed G-sets as the category of modules of a Hopf algebra in Rel which is obtained by the quantum double construction. This category of crossed G-sets serves as a model of the braided variant of propositional linear logic. Keywords: monoidal categories, bialgebras and Hopf algebras, linear logic 1 Introduction For last two decades it has been shown that there are plenty of important exam- ples of traced monoidal categories [21] and ribbon categories (tortile monoidal cat- egories) [32,33] in mathematics and theoretical computer science. In mathematics, most interesting ribbon categories are those of representations of quantum groups (quasi-triangular Hopf algebras)[9,23] in the category of finite-dimensional vector spaces. In many of them, we have non-symmetric braidings [20]: in terms of the graphical presentation [19,31], the braid c = is distinguished from its inverse c−1 = , and this is the key property for providing non-trivial invariants (or de- notational semantics) of knots, tangles and so on [12,23,33,35] as well as solutions of the quantum Yang-Baxter equation [9,23].
    [Show full text]
  • Unitary Modular Tensor Categories
    Penneys Math 8800 Unitary modular tensor categories 5. Unitary modular tensor categories 5.1. Braided fusion categories. Let C be a tensor category. Definition 5.1.1. A braiding on C is a family β of isomorphisms 8 9 > a b > <> => = βa;b : b ⊗ a ! a ⊗ b > > :> b a ;> a;b2C which satisfy the following axioms: • (naturality) for all f 2 C(b ! c) and g 2 C(a ! d), a c a c f = (id ⊗f) ◦ β = β ◦ (f ⊗ id ) = a a;b a;c a f b a b a d b d b g = (g ⊗ id ) ◦ β = β ◦ (id ⊗g) = b a;b d;b b g b a b a • (monoidality) For all b; c 2 C, a b c a b c a b⊗c a⊗b c = and = b⊗c a c a⊗b b c a c a b where we have suppressed the associators. More formally, the following diagrams should commute: id ⊗β b ⊗ (c ⊗ a) b a;c b ⊗ (a ⊗ c) α (b ⊗ a) ⊗ c α βa;b⊗idc (5.1.2) β (b ⊗ c) ⊗ a a;b⊗c a ⊗ (b ⊗ c) α (a ⊗ b) ⊗ c β ⊗id −1 (c ⊗ a) ⊗ b a;c b (a ⊗ c) ⊗ b α a ⊗ (c ⊗ b) α−1 ida ⊗βb;c (5.1.3) β −1 c ⊗ (a ⊗ b) a⊗b;c (a ⊗ b) ⊗ c α a ⊗ (b ⊗ c) 1 When C is unitary, we typically require a braiding to be unitary as well. Remark 5.1.4. By [Gal14], every braiding on a unitary fusion category is automatically unitary.
    [Show full text]
  • Fusion Categories and Modular Tensor Categories Seminar
    Fusion categories and modular tensor categories seminar Aaron Mazel-Gee Contents 1 Introduction { Part I (Ehud \Udi" Meir) 1 2 Introduction { Part II (Orit Davidovich) 4 2.1 Recap . .4 2.2 Commutativity . .4 2.3 Graphical calculus . .5 2.4 Modularity . .6 3 Dimensions in fusion categories (Orit) 6 3.1 Dimensions . .6 3.2 Modularity . .8 3.3 Drinfeld centers . .8 4 Frobenius-Perron dimension and module categories (Udi) 8 4.1 Frobenius-Perron dimension . .8 4.2 Module categories and Morita equivalence . .9 5 On Morita equivalence for fusion categories (Udi) 10 6 Ocneanu rigidity (Orit) 12 6.1 Unitary fusion categories . 12 6.2 Algebro-linear data of fusion categories . 13 6.3 Davydov-Yetter cohomology . 14 7 Group theoretical fusion categories (Udi) 15 1 Introduction { Part I (Ehud \Udi" Meir) In order to explain what a fusion category is, we will begin with an example. Example 1. Suppose G is a finite group and k = k is a field with char k = 0. Consider the category C = Rep(G), the finite-dimensional k-vector spaces with a G-action. What do we know about this category? 1. By Maschke's theorem, we know that C is semisimple: that is, every object is a sum of simple (indecomposable) objects. 2. It is k-linear: HomC(X; Y ) is a k-vector space, and it has kernels and cokernels. 3. C is a monoidal category: if V; W 2 C, then we have V ⊗W 2 C (by the diagonal action: g(v ⊗w) = gv ⊗gw)). This determines a functor C ×C ! C, which will satisfy certain axioms which we'll explore later.
    [Show full text]
  • Internal Reshetikhin-Turaev Topological Quantum Field Theories Mickaël Lallouche
    Internal Reshetikhin-Turaev Topological Quantum Field Theories Mickaël Lallouche To cite this version: Mickaël Lallouche. Internal Reshetikhin-Turaev Topological Quantum Field Theories. General Topol- ogy [math.GN]. Université Montpellier, 2016. English. NNT : 2016MONTS015. tel-01681854v2 HAL Id: tel-01681854 https://tel.archives-ouvertes.fr/tel-01681854v2 Submitted on 12 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Délivré par l’ Université de Montpellier Préparée au sein de l’école doctorale Information Structures Systèmes et de l’unité de recherche Institut Montpelliérain Alexander Grothendieck Spécialité: Mathématiques et modélisation Présentée par Mickaël Lallouche Théories des champs quantiques topologiques internes de type Reshetikhin-Turaev Soutenue le 31 octobre 2016 devant le jury composé de M. Stéphane BASEILHAC Université de Montpellier Président du jury M. Christian BLANCHET UP7D / UMPC Rapporteur (absent) M. Alain BRUGUIERES Université de Montpellier Directeur de thèse M. Louis FUNAR Université Grenoble Alpes Examinateur M. Gwénaël MASSUYEAU Université de Strasbourg Examinateur M. Christoph SCHWEIGERT Universität Hamburg Rapporteur M. Alexis VIRELIZIER Université de Lille I Co-directeur de thèse 2 Remerciements À chaque fois que je me mets à écrire, c’est à 1 000 mains.
    [Show full text]
  • Computing F-Matrices for Modular Tensor Categories
    Computing F-Matrices for Modular Tensor Categories Galit Anikeeva Advisors: Daniel Bump, Guillermo Antonio Aboumrad August 29, 2020 Abstract In this paper, we find algorithms to compute the F-matrices for modular tensor categories, and in particular relevant anyon theories. Moreover, we implement this algorithm in SageMath. Contents 1 Introduction 2 2 Background 3 2.1 Modular Tensor Categories . .3 2.1.1 Fusion category . .3 2.1.2 Braided Monoidal Category . .5 2.1.3 Ribbon Category . .6 2.1.4 Monoidal Tensor Category . .8 2.1.5 Interpretation in TQC . .8 2.1.6 R-coefficients . .8 2.1.7 Ocneanu rigidity . .9 bc 2.2 Theories with Na 2 f0; 1g ................................9 2.3 Groebner basis . 10 3 Proposal 10 3.1 Eliminating leftover gauges . 11 3.2 Optimizations . 11 3.2.1 Creating the ideals . 11 3.2.2 Reducing and recomputing the ideal . 11 4 Future Work 12 5 Acknowledgements 12 A Some F matrices. 13 A.1 A1.............................................. 13 A.1.1 k =1 ........................................ 13 A.1.2 k =2 ........................................ 14 A.1.3 k =3 ........................................ 14 1 A.2 A2.............................................. 16 A.3 B2.............................................. 17 A.3.1 k =1 ........................................ 17 A.4 D3.............................................. 17 A.4.1 k =1 ........................................ 17 A.5 G2.............................................. 18 A.5.1 k =1 ........................................ 18 A.6 E8.............................................. 18 A.6.1 k =2 ........................................ 18 1 Introduction Topological quantum computing [1] is a proposal for a quantum computer that is theoretically distinct from (but computationally equivalent [2] to) the usual gate/circuit model of quantum computing. This proposal deals with topological states of matter.
    [Show full text]
  • Modular Categories∗
    Modular Categories∗ M. M¨uger IMAPP, Radboud University Nijmegen The Netherlands February 13, 2012 1 Introduction Modular categories, as well as the (possibly) more general non-degenerate braided fusion categories, are braided tensor categories that are linear over a field and satisfy some natural additional axioms, like existence of duals, semisimplicity, finiteness, and an important non-degeneracy condition. (Precise definitions will be given later.) There are several reasons to study modular categories: • As will hopefully become clear, they are rather interesting mathematical structures in themselves, well worth being studied for intrinsic reasons. For example, there are interesting number theoretic aspects. • Among the braided fusion categories, modular categories are the opposite extreme of the symmetric fusion categories, which are well known to be closely related to finite groups. Studying these two extreme cases is also helpful for understanding and classifying those braided fusion categories that are neither symmetric nor modular. • Modular categories serve as input datum for the Reshetikhin-Turaev construction of topological quantum field theories in 2+1 dimensions and therefore give rise to invariants of smooth 3-manifolds. This goes some way towards making Witten's interpretation of the Jones polynomial via Chern-Simons QFT rigorous. (But since there still is no complete rigorous non-perturbative construction of the Chern-Simons QFTs by conventional quantum field theory methods, there also is no proof of their equivalence to the RT-TQFTs constructed using the representation theory of quantum groups.) • Modular categories arise as representation categories of loop groups and, more generally, of rational chiral conformal quantum field theories. In chiral CQFT, the field theory itself, its representation category, and the conformal characters form a remarkably tightly connected structure.
    [Show full text]
  • The Trace on Projective Representations of Quantum Groups
    THE TRACE ON PROJECTIVE REPRESENTATIONS OF QUANTUM GROUPS NATHAN GEER AND BERTRAND PATUREAU-MIRAND Abstract. For certain roots of unity, we consider the categories of weight modules over three quantum groups: small, un-restricted and unrolled. The first main theorem of this paper is to show that there is a modified trace on the projective modules of the first two categories. The second main theorem is to show that category over the unrolled quantum group is ribbon. Partial results related to these theorems were known previously. Introduction L H L For an odd ordered root of unity ξ and lattice L, let Uξ , Uξ and U ξ be the unrestricted, unrolled H and small quantum groups, respectively (see Section 3). Let Codd (resp. Codd, resp. C odd) be L H L the category of Uξ (resp. U , resp. U ξ ) weight modules. The usual construction of quantum invariants do not directly apply to these categories because of the following obstructions: the categories are not semi-simple and have vanishing quantum dimensions. Partial results overcoming these obstructions have been obtained in [9, 17, 18, 19, 20, 22]. In this paper we generalize some of these results using a new concept called generically semi-simple (loosely meaning the category is graded and semi-simple on a dense portion of the graded pieces). In [3, 6], the results of this paper will be used in future work to construct topological invariants. The first part of this paper contains a general theory with two main theorems which extend properties of generic simple objects to general properties in the full category.
    [Show full text]
  • QUANTUM GROUPS and KNOT THEORY: WEEK 46 This Week We
    QUANTUM GROUPS AND KNOT THEORY: WEEK 46 This week we treat the first 2 sections of Chapter 6 of [1]. We give additional proofs and definitions (our paragraph numbering refers to the numbering in [1]). A good additional source for this material is [2, Chapter 1]. 1. From Ribbon categories to topological invariants of links 1. Ribbon categories. 1.1. Duality in monoidal categories. A left duality ∗ in a strict monoidal category C = ∗ ∗ (C; ⊗;I) is a function V ! V on Obj(C) together with morphisms bV : I ! V ⊗ V and ∗ dV : V ⊗ V ! I (called co-evaluation and evaluation respectively) such that for all V : (1.1) (idV ⊗ dV )(bV ⊗ idV ) = idV (1.2) (dV ⊗ idV ∗ )(idV ∗ ⊗ bV ) = idV ∗ Proposition 1.1. Given a morphism f : U ! V in C we define its transpose f ∗ : V ∗ ! U ∗ by ∗ (1.3) f := (dV ⊗ idU ∗ )(idV ∗ ⊗ f ⊗ idU ∗ )(idV ∗ ⊗ bU ) This gives rise to a functor ∗ : C!Cop. ∗ Proof. It is easy to check that idV = idV ∗ . Let f : V ! W and g : U ! V . Then (fg)∗ =(dW ⊗ idU ∗ )(idW ∗ ⊗ fg ⊗ idU ∗ )(idW ∗ ⊗ bU ) =(dW ⊗ idU ∗ )(idW ∗ ⊗ f ⊗ idU ∗ )(idW ∗ ⊗ idV ⊗ idU ∗ )(idW ∗ ⊗ g ⊗ idU ∗ )(idW ∗ ⊗ bU ) =(dW ⊗ idU ∗ )(idW ∗ ⊗ f ⊗ idU ∗ )(idW ∗ ⊗ idV ⊗ dV ⊗ idU ∗ ) (idW ∗ ⊗ bV ⊗ idV ⊗ idU ∗ )(idW ∗ ⊗ g ⊗ idU ∗ )(idW ∗ ⊗ bU ) =(dW ⊗ idU ∗ )(idW ∗ ⊗ idW ⊗ dV ⊗ idU ∗ )(idW ∗ ⊗ f ⊗ idV ∗ ⊗ idV ⊗ idU ∗ ) (idW ∗ ⊗ idV ⊗ idV ∗ ⊗ g ⊗ idU ∗ )(idW ∗ ⊗ bV ⊗ idU ⊗ idU ∗ )(idW ∗ ⊗ bU ) =(dW ⊗ idU ∗ )(idW ∗ ⊗ idW ⊗ dV ⊗ idU ∗ )(idW ∗ ⊗ idW ⊗ idV ∗ ⊗ g ⊗ idU ∗ ) (idW ∗ ⊗ f ⊗ idV ∗ ⊗ idU ⊗ idU ∗ )(idW ∗ ⊗ bV ⊗ idU ⊗ idU ∗ )(idW ∗ ⊗ bU ) =(dV ⊗ idU ∗ )(idV ∗ ⊗ g ⊗ idU ∗ )(idV ∗ ⊗ bU )(dW ⊗ idV ∗ )(idW ∗ ⊗ f ⊗ idV ∗ )(idW ∗ ⊗ bV ) =g∗f ∗ 1 2 QUANTUM GROUPS AND KNOT THEORY: WEEK 46 ∗ ∗ ∗ Proposition 1.2.
    [Show full text]
  • Skein Categories
    Skein Categories Juliet Cooke December 3, 2020 Universit´eCatholique de Louvain Introduction f f g g Internal Skein algebra ! Skein categories ! skein algebras # Factorisation homology or;t FV × Disc2 Catk SkV (Σ) or;t Mfld2 1 Skein Algebras The Kauffman bracket skein algebra SkAlgq(Σ) of the oriented smooth surface Σ is the Q(q) module of formal linear combinations of links up to isotopy modulo the Kauffman bracket skein relations 1 − 1 = q 2 + q 2 ; = −q − q−1: Multiplication is given by stacking. It is an invariant of framed links and it can be renormalised to give the Jones polynomial. 2 Coloured Ribbon Graphs Let V be a (strict) ribbon category: monoidal product ⊗ : V × V ! V braiding β : V × V ! V × V twist θ : V!V duals X ∗ with unit η : 1 ! V ∗ ⊗ V and counit : V ⊗ V ∗ ! 1 maps 3 Category of Coloured Ribbons A coloured ribbon diagram of the surface Σ is an embedding of a ribbon graph into Σ × [0; 1] such that unattached bases are sent to Σ × f0; 1g. f RibbonV (Σ) is the k-linear category: g Objects Finite collections of disjoint, framed, coloured, directed points in Σ Morphisms Finite k-linear combinations of coloured ribbon diagram which are compatible which the points attached to up to isotopy which preserves ribbon graph structure. 4 Evaluation Function Theorem (Turaev) There is a full surjective ribbon functor 3 eval : RibbonV ([0; 1] ) !V V V V Id V* V βV,W θV ηV εV 5 Skein Category (Walker, Johnson-Freyd) The skein category SkV (Σ) is the k-linear category of coloured ribbons RibbonV (Σ) modulo the following relation on morphisms X λi Fi ∼ 0 i if there exists an orientation preserving embedding E : [0; 1]3 ,! Σ × [0; 1] such that ! X eval λi Fi j[0;1]3 = 0; i the coloured ribbon diagrams Fi are identical outside the cube, and they only intersect the cube with strands transversely at the top and the bottom.
    [Show full text]
  • Tensor Categories: a Selective Guided Tour
    Tensor categories: A selective guided tour Michael M¨uger Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen The Netherlands June 21, 2010 Abstract These are the, somewhat polished and updated, lecture notes for a three hour course on tensor categories, given at the CIRM, Marseille, in April 2008. The coverage in these notes is relatively non-technical, focusing on the essential ideas. They are meant to be accessible for beginners, but it is hoped that also some of the experts will find something interesting in them. Once the basic definitions are given, the focus is mainly on categories that are linear over a field k and have finite dimensional hom-spaces. Connections with quantum groups and low dimensional topology are pointed out, but these notes have no pretension to cover the latter subjects to any depth. Essentially, these notes should be considered as annotations to the extensive bibliography. We also recommend the recent review [43], which covers less ground in a deeper way. 1 Tensor categories These informal notes are an outgrowth of the three hours of lectures that I gave at the Centre International de Rencontres Mathematiques, Marseille, in April 2008. The original version of text was projected to the screen and therefore kept maximally concise. For this publication, I have corrected the language where needed, but no serious attempt has been made to make these notes conform with the highest standards of exposition. I still believe that publishing them in this form has a purpose, even if only providing some pointers to the literature. 1.1 Strict tensor categories We begin with strict tensor categories, despite their limited immediate applicability.
    [Show full text]
  • Arxiv:Math/0301142V3 [Math.QA] 12 Feb 2020 X N H Quantity the and Wss(Ntelnug F[3)Frctgre Fti Type
    ON BRAIDED TENSOR CATEGORIES OF TYPE BCD IMRE TUBA AND HANS WENZL Abstract. We give a full classification of all braided semisimple tensor categories whose Grothendieck semiring is the one of Rep O(∞) (formally), Rep O(N), Rep Sp(N) or of one of its associ- ated fusion categories. If the braiding is not symmetric, they are completely determined by the eigenvalues of a certain braiding morphism, and we determine precisely which values can occur in the various cases. If the category allows a symmetric braiding, it is essentially determined by the dimension of the object corresponding to the vector representation. 1. Introduction Braided tensor categories have played a prominent role in various areas in recent years, such as conformal field theory, string theory, operator algebras and low-dimensional topology. Important examples have been constructed in a mathematically rigorous way using the representation theory of quantum groups, loop groups and Kac-Moody algebras. This naturally leads to the question of classifying such categories. We solve this question in this paper for braided categories associated to the representation categories of orthogonal and symplectic groups, and various generalizations of them. It has been shown in [23] that any rigid semisimple tensor category whose Grothendieck semiring is equivalent to the one of Rep SU(N) must necessarily be equivalent to the category Rep(Uq slN ), with q not a root of unity, up to N possible choices of a twist; here Uq slN is the Drinfeld-Jimbo q- deformation of the universal enveloping algebra U slN . The present paper proves a similar statement for a braided tensor category whose Grothendieck semiring is isomorphic to the one of a full orthogonal or a symplectic group.C It will be convenient to formulate the result in a slightly different way in this case: Let X be the object in corresponding to the vector representation of an orthogonal or symplectic group.
    [Show full text]