Download e-book for kindle: Lectures on Tensor Categories and Modular Functors by Bojko Bakalov
By Bojko Bakalov
This ebook provides an exposition of the relatives one of the following 3 subject matters: monoidal tensor different types (such as a class of representations of a quantum group), third-dimensional topological quantum box idea, and 2-dimensional modular functors (which clearly come up in 2-dimensional conformal box theory). the subsequent examples are mentioned intimately: the class of representations of a quantum team at a root of harmony and the Wess-Zumino-Witten modular functor. the concept that those issues are comparable first seemed within the physics literature within the research of quantum box conception. Pioneering works of Witten and Moore-Seiberg brought on an avalanche of papers, either actual and mathematical, exploring numerous features of those kinfolk. Upon getting ready to lecture at the subject at MIT, notwithstanding, the authors found that the present literature used to be tough and that there have been gaps to fill. The textual content is fully expository and finely succinct. It gathers effects, fills current gaps, and simplifies a few proofs. The e-book makes a big addition to the present literature at the subject. it'd be compatible as a path textual content on the advanced-graduate point
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Additional resources for Lectures on Tensor Categories and Modular Functors
4). , the asymptotics of solutions of KZ equations. However, it is not too difficult. 17)) and confirms the equivalence of D(g, κ) and C(g, κ) as ribbon categories (cf. 6). , the maps δ are not uniquely defined by the associativity and commutativity isomorphisms. Here is an example. 8. 10). 3. This definition ∼ of δ coincides with the canonical isomorphism of vector spaces V −→ V ∗∗ . h Let Z = (−1) . It is obvious that for every sl2 -module V , the map Z : V → V commutes with the action of sl2 ; thus, Z is a functorial isomorphism (of the identity functor).
Let C be a ribbon category. Fix objects V1 , . . 16) X = ((V1 ⊗ V2 ) ⊗ V4 ) ⊗ ((V1∗∗∗ ⊗ 1) ⊗ ∗ V2 ) ⊗ · · · where we take the tensor product of V1 , . . , Vn in arbitrary order and allow repetitions, arbitrary number of left and right stars and 1’s. To each expression X as above we assign a sequence F (X) of arrows and labels by the following rule: to an object ∗···∗ V ∗···∗ we assign ↓V if the total number of 42 2. RIBBON CATEGORIES stars is odd and ↑V if it is even. All 1’s are skipped. 16) we assign the sequence V1 V2 V4 V1 V2 For two such expressions X1 and X2 consider all morphisms ϕ : X1 → X2 which can be obtained as a composition of the elementary morphisms α±1 , λ±1 , ρ±1 , σ ±1 , e, i, δ ±1 , as well as a number of other morphisms of C.
Let us first check the identity S 2 = C. We have: i (S 2 )ij = Sik Skj = k k di dk DD i = k j di D2 j 56 3. 21) and p+ p− = D2 , di = di∗ . 1. 10. This proves that (ST )3 = p+ /p− S 2 . 17), it is easy to see that (C 2 )ij = δij θV−1 ∗ = (θH )ij . i ⊗V i We cannot say that S, T give a projective representation of the modular group in H, since θH is not a constant. However, θH becomes a constant after restriction to an isotypic component of H. Equivalently, let us fix a simple object U in our category and consider the space Hom(U, H) = i∈I Hom(U, Vi ⊗ Vi∗ ).
Lectures on Tensor Categories and Modular Functors by Bojko Bakalov