MIT 18 769 - Quantum traces (11 pages)

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Quantum traces



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Quantum traces

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11
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Massachusetts Institute of Technology
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18 769 - Topics in Lie Theory: Tensor Categories
Topics in Lie Theory: Tensor Categories Documents
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76 1 37 Quantum traces Let C be a rigid monoidal category V be an object in C and a Hom V V De ne the left quantum trace 1 37 1 TrLV a evV a IdV coevV End 1 Similarly if a Hom V trace 1 37 2 V then we can de ne the right quantum TrR V a ev V Id V a coev V End 1 In a tensor category over k TrL a and TrR a can be regarded as elements of k When no confusion is possible we will denote TrLV by TrV The following proposition shows that usual linear algebra formulas hold for the quantum trace Proposition 1 37 1 If a Hom V V b Hom W W then 1 T rVL a T rVR a 2 T rVL W a b TrLV a TrLW b in additive categories L 3 T rVL W a b TrLV a TrW b L R 4 If c Hom V V then TrV ac TrLV c a TrR V ac TrV ca Similar equalities to 2 3 also hold for right quantum traces Exercise 1 37 2 Prove Proposition 1 37 1 If C is a multitensor category it is useful to generalize Proposi tion 1 37 1 2 as follows Proposition 1 37 3 If a Hom V V and W V such that a W W then TrLV a TrLW a TrLV W a That is Tr is additive on exact sequences The same statement holds for right quantum traces Exercise 1 37 4 Prove Proposition 1 37 3 1 38 Pivotal categories and dimensions De nition 1 38 1 Let C be a rigid monoidal category A pivotal structure on C is an isomorphism of monoidal functors a Id That is a pivotal structure is a collection of morphisms aX X X natural in X and satisfying aX Y aX aY for all objects X Y in C De nition 1 38 2 A rigid monoidal category C equipped with a piv otal structure is said to be pivotal Exercise 1 38 3 1 If a is a pivotal structure then aV aV 1 Hence aV a V 77 2 Let C Rep H where H is a nite dimensional Hopf alge bra Show that pivotal structures on C bijectively correspond to group like elements of H such that gxg 1 S 2 x for all x H Let a be a pivotal structure on a rigid monoidal category C De nition 1 38 4 The dimension of an object X with respect to a is dima X Tr aX End 1 Thus in a tensor category over k dimensions are elements of k Also it follows from Exercise 1 38 3 that dima V dima V Proposition 1 38 5 If C is a tensor category then the function X dima X is a character of the Grothendieck ring Gr C Proof Proposition 1 37 3 implies that dima is additive on exact se quences which means that it gives rise to a well de ned linear map from Gr C to k The fact that this map is a character follows from the obvious fact that dima 1 1 and Proposition 1 37 1 3 Corollary 1 38 6 Dimensions of objects in a pivotal nite tensor cat egory are algebraic integers in k 12 Proof This follows from the fact that a character of any ring that is nitely generated as a Z module takes values in algebraic integers 1 39 Spherical categories De nition 1 39 1 A pivotal structure a on a tensor category C is spherical if dima V dima V for any object V in C A tensor category is spherical if it is equipped with a spherical structure Since dima is additive on exact sequences it su ces to require the property dima V dima V only for simple objects V Theorem 1 39 2 Let C be a spherical category and V be an object of 1 C Then for any x Hom V V one has T rVL aV x TrR V xaV 1 Proof We rst note that TrR X aX dima X for any object X by Proposition 1 37 1 1 and Exercise 1 38 3 1 Now let us prove the proposition in the special case when V is semisimple Thus V i Yi Vi where Vi are vector spaces and Yi are simple objects Then x 12If k has positive characteristic by an algebraic integer in k we mean an element of a nite sub eld of k 78 i xi IdVi with xi Endk Yi and a IdYi aVi by the functoriality of a Hence TrLV ax Tr xi dim Vi 1 TrR Tr xi dim Vi V xa This implies the result for a semisimple V Consider now the general case Then V has the coradical ltration 1 39 1 0 V0 V1 V2 Vn V such that Vi 1 Vi is a maximal semisimple subobject in V Vi This ltration is preserved by x and by a i e a Vi Vi Since traces are additive on exact sequences by Proposition 1 37 3 this implies that the general case of the required statement follows from the semisimple case Exercise 1 39 3 i Let Aut IdC be the group of isomorphism classes of monoidal automorphisms of a monoidal category C Show that the set of isomorphism classes of pivotal structures on C is a torsor over Aut IdC and the set of isomorphism classes of spherical structures on C is a torsor over the subgroup Aut IdC 2 in Aut IdC of elements which act by 1 on simple objects 1 40 Semisimple multitensor categories In this section we will more closely consider semisimple multitensor categories which have some important additional properties compared to the general case 1 41 Isomorphism between V and V Proposition 1 41 1 Let C be a semisimple multitensor category and let V be an object in C Then V V Hence V V Proof We may assume that V is simple We claim that the unique simple object X such that Hom 1 V X 0 is V Indeed Hom 1 V X Hom X V which is non zero if and only if X V i e X V Similarly the unique simple object X such that Hom V X 1 0 is V But since C is semisimple dimk Hom 1 V X dimk Hom V X 1 which implies the result Remark 1 41 2 As noted in Remark 1 27 2 the result of Proposi tion 1 41 1 is false for non semisimple categories Remark 1 41 3 Proposition 1 41 1 gives rise to the following ques tion 79 Question 1 41 4 Does any semisimple tensor category admit a piv otal structure A spherical structure This is the case for all known examples The general answer is un known to us at the moment of writing even for ground elds of char acteristic zero Proposition 1 41 5 If C is a semisimple tensor category and a V V for …


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