MIT 18 769 - Quantum traces (11 pages)

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Quantum traces



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Quantum traces

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Pages:
11
School:
Massachusetts Institute of Technology
Course:
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



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