[Paper Review] From Subfactors to Categories and Topology II. The quantum double of tensor categories and subfactors

Michael Mueger|ArXiv.org|Nov 19, 2001

Algebraic structures and combinatorial models20 references4 citations

TL;DR

This paper generalizes the quantum double construction from finite-dimensional Hopf algebras to arbitrary semisimple spherical tensor categories over an algebraically closed field. It proves that the center ${\cal Z}({\cal C})$ is a modular category with dimension $(\dim{\cal C})^2$, establishes a Morita equivalence between ${\cal Z}({\cal C})$ and ${\cal C} \otimes {\cal C}^{\mathrm{op}}$, and verifies the Gelfand–Kazhdan conjecture on the number of simple objects in ${\cal Z}({\cal C})$ matching the torus TQFT state space dimension.

ABSTRACT

We are concerned with the center (=quantum double) of tensor categories and prove generalizations of several results proven previously for quantum doubles of Hopf algebras. We consider F-linear tensor categories C with simple unit and finitely many isomorphism classes of simple objects. We assume that C is either a *-category (i.e. there is a positive *-operation on the morphisms) or semisimple and spherical over an algebraically closed field F. In the latter case we assume that dim C=sum_i d(X_i)^2 is non-zero, where the summation runs over the isomorphism classes of simple objects. We prove: (i) Z(C) is a semisimple spherical (or *-) category. (ii) Z(C) is weakly monoidally Morita equivalent (in the sense of math.CT/0111204) to C X C^op. This implies dim Z(C)=(dim C)^2. (iii) We analyze the simple objects of Z(C) in terms of certain finite dimensional algebras, of which Ocneanu's tube algebra is the smallest. We prove the conjecture of Gelfand and Kazhdan according to which the number of simple objects of Z(C) coincides with the dimension of the state space H_{S^1 imes S^1} of the torus in the triangulation TQFT built from C. (iv) We prove that Z(C) is modular and we compute the Gauss sums Delta_+/-(Z(C))=sum_i theta(X_i)^{+/- 1}d(X_i)^2=dim C. (v) Finally, if C is already modular then Z(C)\simeq C X C~, where C~ is the tensor category C with the braiding c~_{X,Y}=c_{Y,X}^{-1}.

Motivation & Objective

• To extend the quantum double construction from Hopf algebras to general semisimple spherical tensor categories.
• To prove that the center ${\cal Z}({\cal C})$ of such a category is modular.
• To establish a weak Morita equivalence between ${\cal Z}({\cal C})$ and ${\cal C} \otimes {\cal C}^{\mathrm{op}}$, implying $\dim{\cal Z}({\cal C}) = (\dim{\cal C})^2$.
• To verify the Gelfand–Kazhdan conjecture: the number of simple objects in ${\cal Z}({\cal C})$ equals the dimension of the torus TQFT state space.
• To compute the Gauss sums $\Delta_{\pm}({\cal Z}({\cal C}))$ and show they equal $\dim{\cal C}$.

Proposed method

• Define the center ${\cal Z}({\cal C})$ of a tensor category ${\cal C}$ as the braided category of objects with natural isomorphisms satisfying the Drinfel'd center axioms.
• Use the vector space isomorphism ${\mathfrak{S}}$ on $\bigoplus_{i,j} \mathrm{Hom}_{\cal C}(X_i X_j, X_j X_i)$ to represent the $S$-matrix of ${\cal Z}({\cal C})$.
• Leverage sphericity and duality to show that ${\mathfrak{S}}$ is an order-four isomorphism and stabilizes the center of the endomorphism algebra.
• Construct a basis of $S$-matrix eigenvectors using $d(X)^{-1} z_{(X,e_X)}$ for simple objects $(X,e_X)$ in ${\cal Z}({\cal C})$.
• Apply the duality equation and trace properties to compute matrix elements of the $S$-matrix via the map ${\mathfrak{S}}$.
• Use the Gauss sum formula $\Delta_{\pm}({\cal Z}({\cal C})) = \sum_i \theta(X_i)^{\pm 1} d(X_i)^2$ and prove it equals $\dim{\cal C}$.

Experimental results

Research questions

• RQ1 Is the center ${\cal Z}({\cal C})$ of a semisimple spherical tensor category ${\cal C}$ modular?
• RQ2 Does the dimension of ${\cal Z}({\cal C})$ satisfy $\dim{\cal Z}({\cal C}) = (\dim{\cal C})^2$?
• RQ3 Is the number of simple objects in ${\cal Z}({\cal C})$ equal to the dimension of the state space of the torus in the TQFT constructed from ${\cal C}$?
• RQ4 Are the Gauss sums $\Delta_{\pm}({\cal Z}({\cal C}))$ equal to $\dim{\cal C}$?
• RQ5 Is ${\cal Z}({\cal C})$ weakly Morita equivalent to ${\cal C} \otimes {\cal C}^{\mathrm{op}}$?

Key findings

• The center ${\cal Z}({\cal C})$ is a modular category when ${\cal C}$ is a semisimple spherical tensor category with finite-dimensional, non-degenerate dimension over an algebraically closed field.
• The dimension of ${\cal Z}({\cal C})$ is exactly $(\dim{\cal C})^2$, as proven via weak Morita equivalence to ${\cal C} \otimes {\cal C}^{\mathrm{op}}$.
• The Gauss sums of ${\cal Z}({\cal C})$ satisfy $\Delta_{+}({\cal Z}({\cal C})) = \Delta_{-}({\cal Z}({\cal C})) = \dim{\cal C}$.
• The number of simple objects in ${\cal Z}({\cal C})$ equals the dimension of the state space of the torus in the triangulation TQFT built from ${\cal C}$, confirming the Gelfand–Kazhdan conjecture.
• If ${\cal C}$ is already modular, then ${\cal Z}({\cal C}) \simeq_{\mathrm{br}} {\cal C} \otimes \tilde{{\cal C}} \simeq_{\mathrm{br}} {\cal C} \otimes {\cal C}^{\mathrm{op}}$, where $\tilde{{\cal C}}$ is ${\cal C}$ with reversed braiding.