Lectures on tensor categories and modular functor

introduction

Bojko Bakalov and Alexander Kirillov, Jr.

chapter 1 Braided Tensor Categories

def 1.1.3

An object \(U\) in an abelian category \(\mathcal{C}\) is called simple if any injection \(V\hookrightarrow U\) is either 0 or isomorphism

An abelian category \(\mathcal{C}\) is called semisimple if any object \(V\) is isomorphic to a direct sum of simple ones : \[ V\cong \oplus_{i\in I}N_iV_i \] where \(V_i\) are simple objects, \(I\) is the set of isomoprhism classes of non-zero simple objects in \(\mathcal{C}\), \(N_i\in \mathbb{Z}_{+}\) and only a finite number of \(N_i\) are non-zero.

quantum groups at roots of unity

\(\kappa=k+h^{\vee}\) \[h^{\vee}\] is the dual Coxeter number for \(\mathfrak{g}\)
\(U_q(\mathfrak{g})\) for \(q=e^{\pi i/m\kappa}\)
\(\mathcal{C}(\mathfrak{g},\kappa)\) category of finite dimensional representations over \(\mathbb{C}\) with weight decompositions
\(\mathcal{C}(\mathfrak{g},\kappa)\) is a very complicated category; in particular, it is not semisimple
\(C=\{\lambda\in P_{+}|(\lambda+\rho,\theta^{\vee})<\kappa \}\)
we want to extract a semisimple part with simple objects \(V_{\lambda},\lambda\in C\)
so let us introduce the category \(\mathcal{T}\) of tilting modules

examples of modular tensor category

the category \(\mathcal{C}^{\rm int}\equiv \mathcal{C}^{\rm int}(\mathfrak{g},\kappa)\) be the category with objects tilting modules and morphisms

\[ \operatorname{Hom}_{\mathcal{C}^{\rm int}}(V,W)=\operatorname{Hom}_{\mathcal{T}}(V,W)/{\text {negligible morphisms}} \] over \(U_q(\mathfrak{g})\) for \(q=e^{\pi i/m\kappa}\)

\(\mathcal{O}_{k}^{\rm int}\) the category of integrable modules of level k over the affine Lie algebra \(\hat{\mathfrak{g}}\)

main results

thm 3.3.20

\(\mathcal{C}^{\rm int}\) is a modular tensor category with simple objects \(V_{\lambda}\) (\(\lambda\in C\)).

thm 7.0.1

The category \(\mathcal{O}_{k}^{\rm int}\) has a structure of a modular tensor category. (the tensor product is called the fusion product)

thm 7.0.2

The category \(\mathcal{O}_{k}^{\rm int}\) is eqiuvalent to the category \(\mathcal{C}^{\rm int}(\mathfrak{g},\kappa)\) as a modular tensor category for \(\kappa=k+h^{\vee}\), where \(h^{\vee}\) is the dual Coxeter number for \(\mathfrak{g}\)

thm 7.9.10

The sheaves of coinvariants \(\tau(C, p, V_i), \, V_i \in \mathcal{O}_k^{\text{int}}\) form a modular functor with additive central charge \(c\).

memo

articles

Finkelberg, M. “An Equivalence of Fusion Categories.” Geometric and Functional Analysis 6, no. 2 (1996): 249–67. doi:10.1007/BF02247887.
Andersen, Henning Haahr, and Jan Paradowski. 1995. “Fusion Categories Arising from Semisimple Lie Algebras.” Communications in Mathematical Physics 169 (3) (May 1): 563–588. doi:10.1007/BF02099312.