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.