%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Multicolored PAs and Quadrilaterals %%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{article} \pdfoutput=1 \usepackage{amsmath, amsthm, amssymb} \usepackage{ifpdf} \ifpdf \usepackage[pdftex]{graphicx} \else \usepackage[dvips]{graphicx} \fi \usepackage{tikz} \usetikzlibrary{arrows,backgrounds} \usepackage[all]{xy} \usepackage[pdftex,plainpages=false,hypertexnames=false,pdfpagelabels]{hyperref} \newcommand{\arXiv}[1]{\href{http://arxiv.org/abs/#1}{\tt arXiv:\nolinkurl{#1}}} \newcommand{\doi}[1]{\href{http://dx.doi.org/#1}{{\tt DOI:#1}}} \newcommand{\euclid}[1]{\href{http://projecteuclid.org/getRecord?id=#1}{{\tt #1}}} \newcommand{\mathscinet}[1]{\href{http://www.ams.org/mathscinet-getitem?mr=#1}{\tt #1}} \newcommand{\googlebooks}[1]{(preview at \href{http://books.google.com/books?id=#1}{google books})} %Page size \setlength\topmargin{0in} \setlength\headheight{0in} \setlength\headsep{.2in} \setlength\textheight{9in} \addtolength{\hoffset}{-0.25in} 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\newcommand{\RMod}[1]{{\sb{#1}\sf{Mod}}} \newcommand{\ModR}[1]{{\sf{Mod}}_#1} \newcommand{\RModR}[2]{{\sb{#1}{\sf{Mod}}_#2}} \newcommand{\op}{^{\sf{op}}} %%% ---------------------------------------------------------------------- \input xy \xyoption{all} \begin{document} % \input{pictures/TikzStyles} \comment{ \title%[Multishaded planar algebras and quadrilaterals of subfactors] {Subfactors, tensor categories, module categories, and algebra objects in tensor catgories} %\author{Noah Snyder} %\address{Department of Mathematics, University of California, Berkeley, 94720} %\email{dpenneys@math.berkeley.edu} % %\keywords{} \date{\today} %%% ---------------------------------------------------------------------- %\begin{abstract} % %\end{abstract} %%% ---------------------------------------------------------------------- \maketitle \tableofcontents %%% ---------------------------------------------------------------------- % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} } %%% ---------------------------------------------------------------------- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section*{Subfactors, tensor categores, module categories, and algebra objects in tensor categories} These notes were taken by Dave Penneys at Noah Snyder's talk on 2/20/10 at the Subfactor Tahoe Retreat. The material is mostly taken from: \begin{itemize} \item Mueger's ``From Subfactors to Categories and Topology I" (arXiv:math/0111204) \item Ostrik's ``Module categories, weak Hopf algebras and modular invariants" (arXiv:math/0111139) \item Kirillov and Ostrik's ``On q-analog of McKay correspondence and ADE classification of $\hat{\mathfrak{sl}^{2}}$ conformal field theories" (arXiv:math/0101219 ) \end{itemize} \begin{rem} Our tensor categores are assumed to have duals and a trivial object $1$. \end{rem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection*{Module and tensor categories from a subfactor} Let $N\subset M$ be a finite index $II_1$-subfactor. We get four categories of bimodules: $\RModR{N}{N},\RModR{N}{M},\RModR{M}{N},\RModR{M}{M}$. The objects are the bimodules which occur as submodules of an iterated basic construction of $N\subset M$, and the morphisms are bimodule intertwiners, i.e., bimodule maps. \begin{fact} $\RModR{N}{N},\RModR{M}{M}$ are tensor categories. They are fusion categories if $N\subset M$ is finite depth. \end{fact} Note that there is a functor $$ \otimes_N\colon \RModR{N}{N}\otimes \RModR{N}{M}\longrightarrow \RModR{N}{M} $$ satisfying associativity axioms. Heuristically, one should think of this functor as a categorification of a ring action on a module, e.g., $\lambda\colon A\otimes X \to X$. The associativity of the action means the following diagram commutes: \[\xymatrix{ A\otimes A\otimes X\ar[rr]^{m\otimes\id_X}\ar[d]^{\id_A\otimes \lambda} && A\otimes X\ar[d]^\lambda\\ A\otimes X\ar[rr]^\lambda && X }\] where $m\colon A\otimes A\to A$ is the multiplication map. This means we have to have some type of associator isomorphisms in the categorified version. \begin{defn} A left module category over a tensor category $\CC$ is a category $\MM$ and a functor $\CC\otimes \MM\to \MM$ satisfying some associativity axioms up to an associator. \end{defn} \begin{fact} A finite index $II_1$-subfactor gives two tensor categories and two module categories over them: \[\xymatrix{ \RModR{N}{N}\ar@/^2pc/[rr]^{\RModR{N}{M}} && \RModR{M}{M}\ar@/^2pc/[ll]^{\RModR{M}{N}} } \] \end{fact} \begin{rem} We can hide the right half of the above diagram by using the notion of the dual. It turns out that if $\MM$ is a module category over a tensor category $\CC$, we can form the dual tensor category $\CC^*_\MM$ of $\CC$ with respect to $\MM$. If $\CC=\RModR{N}{N}$ and $\MM=\RModR{N}{M}$, then $\CC_\MM^*\cong \RModR{M}{M}$, and the opposite category $\MM\op\cong \RModR{M}{N}$ gives the other module category. \end{rem} \subsection*{Algebra objects and subfactors} A complex algebra is a complex vector space $A$ with a map $m\colon A\otimes A\to A$ such that the following diagram commutes: \[\xymatrix{ A\otimes A\otimes A\ar[rr]^{m\otimes\id_A}\ar[d]^{\id_A\otimes m} && A\otimes A\ar[d]^m\\ A\otimes A\ar[rr]^m && A. }\] \begin{defn} An algebra object in a tensor category $\CC$ is an object $A\in\CC$ and a map $m\colon A\otimes A\to A$ satisfying the associativity axiom up to the associator. An algebra object $A\in\CC$ is called a Frobenius algebra object if it comes with a map $\tr\colon A\to 1$ satisfying a certain nondegeneracy axiom (the categorified ``bilinear form" $A\otimes A^*\to $ has a biadjoint) where $1\in\CC$ is the trivial object \end{defn} \begin{exs} \item[(1)] Let $G$ be a finite group. Let $\CC$ be the category of $G$-graded vector spaces, i.e., vector spaces $V$ which are the direct sum of vector spaces $V_g$ for each $g\in G$: $$ V=\bigoplus_{g\in G} V_g. $$ $\CC$ is a tensor category where $\otimes$ is given by $$ (V\otimes W)_g=\bigoplus_{hk=g} V_h \otimes_\C W_k. $$ The group algebra $\C G$ is an algebra object in this category. \item[(2)] $\sb{N}M_N\in\RModR{N}{N}$ is a Frobenius algebra object. \end{exs} \begin{EX} Show that the multiplications induce the algebra object structures in the above examples. \end{EX} \begin{thm} If $X\in \RModR{N}{N}$ is a simple Frobenius algebra object, then $X$ comes from a factor $P$ where $N\subset P$. Moreover, any unitary tensor category with simple 1 can be realized as a category of bimodules over a factor $N$ (see Yamagami). This means every algebra object can be realized as a subfactor. \end{thm} \begin{rem} The index of the subfactor coming from an algebra object in a tensor category is the Frobenius-Perron dimension of the object, not the square of the dimension. \end{rem} \subsection*{Algebra objects and module categories} Given an algebra object $A\in\CC$, we can make a left module category as follows: set $\MM$ equal to the category of \emph{right} $A$-module objects, i.e., those objects $X\in\CC$ with a map $\rho\colon X\otimes A\to X$ satisfying the associativity axiom up to an associator: \[\xymatrix{ X\otimes A\otimes A\ar[rr]^{\id_X\otimes m}\ar[d]^{\id_A\otimes \rho} && X\otimes A\ar[d]^\rho\\ X\otimes A\ar[rr]^\rho && X }\] %Note that commutativity of this diagram up to an associator is exactly the condition that $\MM$ is a left module category over $\CC$. Note that if $X$ is a right $A$-module object and $Y\in \CC$, then $Y\otimes X$ is also a right $A$-module object with the map $\id_Y\otimes \rho$. Conversely, the internal Hom construction of Ostrik gives algebra objects from a module category. Heuristically, internal Hom is a way of creating objects in a category in a natural way from two given objects. In the category of vector spaces, $\Hom(X,Y)$ is a complex vector space. \begin{defn} Given a module category $\MM$, internal Hom is a bifunctor $\HOM\colon \MM\otimes \MM\to \CC$ such that for each $X,Y,Z\in\MM$, the composition axiom holds up to isomorphism: $$ \HOM(X,Y)\otimes\HOM(Y,Z)\cong \HOM(X,Z) $$ where the isomorphism is natural. \end{defn} \begin{ex} \item[(1)] Let $G$ be a finite group. The category $\Rep{G}$ of finite dimensional complex representations of $G$ thought of as $G-\{e\}$-bimodules where $\{e\}$ is the trivial group is a module category over $G$-graded vector spaces, and $\HOM(X,Y)=Y\otimes X^*$. \item[(2)] If $X,Y\in\RModR{N}{M}$, then $\HOM(X,Y)=Y\otimes X^*$. \end{ex} \begin{fact} Given a module category $\MM$ over $\CC$ and an object $X\in\MM$, $\HOM(X,X)$ is an algebra object in $\CC$. \end{fact} \begin{rems} \item[(1)] In the subfactor setting, we want $X\in\MM$ to be a simple object. \item[(2)] Just as ${\sb{N} M_M}$ is the preferred object in the module category $\RModR{N}{M}$, if we have an algebra object $A\in\CC$, the preferred object in the left module category of right $A$-module objects is $A$ as a right $A$ module. \end{rems} \subsection*{Summary so far} The following three things are basically the same (up to unitarity and simplicity assumptions): \begin{enumerate} \item A subfactor \item An algebra object in a tensor category \item A tensor category with a module category over it and a fixed choice of object in the module category \end{enumerate} \subsection*{GHJ subfactors} So what if I have a tensor category and a module category over it, but I haven't fixed a choice of object in the module category? Then I have lots of possible choices, each of which will give me a (possibly) different subfactor! In particular, you can perform the following switcheroo: pick a tensor category $\CC$ and an algebra object $A \in \CC$ that yields a module category $\MM$ where you can then pick whichever simple object $X$ you like and get a new algebra object $\HOM(X,X)$. Subfactors constructed in this way are called Goodman-de la Harpe-Jones subfactors, or GHJ subfactors. If you have a tensor category $\CC$ and a module category $\MM$ over it, and you have a favorite object $V \in \CC$ (this is different from subfactors which give you a favorite object in $\MM$) then you can ask about the fusion graph for tensoring with $V$ in $\MM$. For a Temperley-Lieb tensor category your favorite object is the single strand, and module categories over Temperley-Lieb at special values of $d$ (less than $2$) exactly correspond to the ADE Dynkin diagrams. \begin{ex} $E_6$ is a module category over $A_{11}$. Take the middle vertex $X$ in $E_6$. The algebra object $A=\Hom(X,X)$ gives the GHJ subfactor of index $3+\sqrt{3}$. %The tensor category generated by $A$ is the even half of $A_{11}$. \end{ex} Now we notice something confusing, the GHJ that we constructed corresponding to the module category $E_6$ has principal graph that isn't $E_6$! So what on earth is the actual $E_6$ subfactor? The confusing thing is that the usual $E_6$ subfactor \emph{is not} part of the whole story I've been telling so far. See the next subsection for how they appear. \begin{rem} The fact that the $D_{odd}$ subfactors don't exist comes from the fact that the algebra object coming from the sum of the first and last vertices of $A_{4n-1}$ is not commutative. The $D_{odd}$'s do exist as module categories. \end{rem} \subsection*{Commutative algebra objects} If $A$ is a commutative ring, then we can also make a tensor category out of all $A$-modules, instead of looking at left, right, and bi- modules. \begin{defn} A commutative algebra object in a \emph{braided} tensor category $\CC$ is an algebra object where the following diagram commutes: \[\xymatrix{ A\otimes A\ar[rr]^m\ar[dr]^{\sigma} && A\\ &A\otimes A\ar[ur]^m }\] where $\sigma$ is the braiding. \end{defn} \begin{fact} If $A\in\CC$ is a commutative algebra object, then the category of $A$-modules is a tensor category, not just a module category. \end{fact} \begin{fact} The ADE subfactors all can be realized as the category of $A$-modules for $A$ a commutative algebra object in Temperley-Lieb. In fact, all of Ocneanu's ``quantum subgroups" arise as the category of $A$-modules for $A$ a commutative algebra object in \end{fact} \end{document} %%%%%%%%%%%% \begin{rem} The correspondence between a subfactor and a tensor category with a module category is one-to-one under nice assumptions. The correspondence between algebra objects and module. \end{rem} \begin{rem}[Ocneanu] Braded fusion category quantum subgroups: type 1: commutative algebra objects type 2: module categories. A module category is what Ocneanu means by a ``quantum subgroup" of a ``quantum group." \end{rem}