Antony Wassermann
In this announcement we explain how to extend the von Neumann algebra definition of fusion to the case of twisted loop groups. This project completes research started by my student Robert Verrill in his 2001 Cambridge Ph.D. thesis.
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We describe the tube algebra and its representations in the cases of diagonal and Bisch-Haagerup subfactors possibly with a scalar 3-cocycle obstruction. We show that these categories are additively equivalent to the direct product over conjugacy classes of representation category of a centralizer subgroup (corresponding to the conjugacy class) twisted by a scalar 2-cocycle obtained from the 3-cocycle obstruction.
The quantum double of the Haagerup subfactor, the first irreducible finite depth subfactor with index above 4, is the most obvious candidate for exotic modular data. We show that its modular data DHg fits into a family $D^ωHg_{2n+1}$, where $n\ge 0$ and $ω\in \bbZ_{2n+1}$. We show $D^0 Hg_{2n+1}$ is related to the subfactors Izumi hypothetically associates to the cyclic groups $Z_{2n+1}$. Their modular data comes equipped with canonical and dual canonical modular invariants; we compute the corresponding alpha-inductions etc. In addition, we show there are (respectively) 1, 2, 0 subfactors of I…
We study L^2-Betti numbers for von Neumann algebras, as defined by D. Shlyakhtenko and A. Connes, in the presence of a bi-finite correspondence and prove a proportionality formula.
This paper is the third of a series giving a self-contained way from the Neveu-Schwarz algebra to a new series of irreducible subfactors. Here we introduce the local von Neumann algebra of the Neveu-Schwarz algebra, to obtain Jones-Wassermann subfactors for each representation of the discrete series. Then using primary fields we prove the irreducibility of these subfactors; to next compute the Connes fusion ring and obtain the explicit formula of the subfactors indices.
By using Ocneanu's result on the classification of all irreducible connections on the Dynkin diagrams, we show that the dual principal graphs as well as the fusion rules of bimodules arising from any Goodman-de la Harpe-Jones subfactors are obtained by a purely combinatorial method. In particular we obtain the dual principal graph and the fusion rule of bimodules arising from the Goodman-de la Harpe-Jones subfactor corresponding to the Dynkin diagram $E_8$. As an application, we also show some subequivalence among $A$-$D$-$E$ paragroups.
If H is a Hopf algebra whose square of the antipode is the identity, $v\inł(V)\otimes H$ is a corepresentation, and $π:H\toł(W)$ is a representation, then $u=(id\otimesπ)v$ satisfies the equation $(t\otimes id)u^{-1}=((t\otimes id)u)^{-1}$ of the vertex models for subfactors. A universal construction shows that any solution $u$ of this equatio n arises in this way. A more elaborate construction shows that there exists a ``minimal'' triple $(H,v,π)$ satisfying $(id\otimesπ)v=u$. This paper is devoted to the study of this latter construction of Hopf algebras. If $u$ is unitary we construct a $\c…
Surprisingly Richard Thompson's groups have recently appeared in Jones' subfactor theory. Vaughan Jones is famous for linking theories that are a priori completely disconnected; for instance, his celebrated polynomial for links emanating from subfactor theory. This note is about a new beautiful story in mathematics which results from a fortunate accident in the land of quantum field theory.
We give a subfactor construction for a $II_{1}$ factor M which is not anti-isomorphic to itself. The $II_{1}$ factor we consider is essentially the same as the example previously given by Connes. However, our construction uses the recently developed theory of free group factors. We show that there exists an inclusion of $II_{1}$ factors $A\subset B$ which by iteration of the Jones basic construction produces $M$ as the enveloping algebra. Here A is a free group factor and B is isomorphic to the crossed product of A by an action of a finite group. By using a Connes' argument involving the invar…
We show that the mathematical structures in a recent work of Bultinck-Mariena-Williamson-Sahinoglu-Haegemana-Verstraete are the same as those of flat symmetric bi-unitary connections and the tube algebra in subfactor theory. More specifically, a system of flat symmetric bi-unitary connections arising from a subfactor with finite index and finite depth satisfies all their requirements for tensors and the tube algebra for such a subfactor and the anyon algebra for such tensors are isomorphic up to the normalization constants. Furthermore, the matrix product operator algebras arising from tensors…
Growing out of the initial connections between subfactors and knot theory that gave rise to the Jones polynomial, Jones' axiomatization of the standard invariant of an extremal finite index $II_1$ subfactor as a spherical $C^*$-planar algebra, presented in arXiv:math.QA/9909027, is the most elegant and powerful description available. We make the natural extension of this axiomatization to the case of finite index subfactors of arbitrary type. We also provide the first steps toward a limited planar structure in the infinite index case. The central role of rotations, which provide the main non-t…
We define a canonical relative commutant planar algebra from a strongly Markov inclusion of finite von Neumann algebras. In the case of a connected unital inclusion of finite dimensional C*-algebras with the Markov trace, we show this planar algebra is isomorphic to the bipartite graph planar algebra of the Bratteli diagram of the inclusion. Finally, we show that a finite depth subfactor planar algebra is a planar subalgebra of the bipartite graph planar algebra of its principal graph.
We give a characterization of extremal irreducible discrete subfactors $(N\subseteq M, E)$ where $N$ is type ${\rm II}_1$ in terms of connected W*-algebra objects in rigid C*-tensor categories. We prove an equivalence of categories where the morphisms for discrete inclusions are normal $N-N$ bilinear ucp maps which preserve the state $τ\circ E$, and the morphisms for W*-algebra objects are categorical ucp morphisms. As an application, we get a well-behaved notion of the standard invariant of an extremal irreducible discrete subfactor, together with a subfactor reconstruction theorem. Thus our …
In this paper, we develop the theory of bimodules over von Neumann algebras, with an emphasis on categorical aspects. We clarify the relationship between dualizability and finite index. We also show that, for von Neumann algebras with finite-dimensional centers, the Haagerup L^2-space and Connes fusion are functorial with respect to homorphisms of finite index. Along the way, we describe a string diagram notation for maps between bimodules that are not necessarily bilinear.
Given a II$_1$-subfactor $A\subset B$ of arbitrary index, we show that the rectangular GICAR category, also called the rectangular planar rook category, faithfully embeds as $A-A$ bimodule maps among the bimodules $\bigotimes_A^n L^2(B)$. As a corollary, we get a lower bound on the dimension of the centralizer algebras $A_0'\cap A_{2n}$ for infinite index subfactors, and we also get that $A_0'\cap A_{2n}$ is nonabelian for $n\geq 2$, where $(A_n)_{n\geq 0}$ is the Jones tower for $A_0=A\subset B=A_1$. We also show that the annular GICAR/planar rook category acts as maps amongst the $A$-central…
We show that given a rigid C*-tensor category, there is an equivalence of categories between normalized irreducible Q-systems, also known as connected unitary Frobenius algebra objects, and compact connected W*-algebra objects. Although this result could be proved as a corollary of our previous article on realizations of algebra objects and discrete subfactors, we prove it here directly via categorical methods without passing through subfactor theory.
Vaughan Jones discovered unexpected connections between Richard Thompson's group and subfactor theory while attempting to construct conformal field theories (in short CFT). Among other this founded Jones' technology: a powerful new method for constructing actions of fraction groups which had numerous applications in mathematical physics, operator algebras, group theory and more surprisingly in knot theory and noncommutative probability theory. We propose and outline a program in the vein of Jones' work but where the Thompson group is replaced by a family of groups that we name forest-skein gro…
We consider certain categorical structures that are implicit in subfactor theory. Making the connection between subfactor theory (at finite index) and category theory explicit sheds light on both subjects. Furthermore, it allows various generalizations of these structures, e.g. to arbitrary ground fields, and the proof of new results about topological invariants in three dimensions. The central notion is that of a Frobenius algebra in a tensor category A, which reduces to the classical notion if A=F-Vect, where F is a field. An object X in A with two-sided dual X^ gives rise to a Frobenius alg…
We revisit the connection between index and relative entropy for an inclusion of finite von Neumann algebras. We observe that the Pimsner-Popa index connects to sandwiched Renyi $p$-relative entropy for all $1/2\le p\le \infty$, including Umegaki's relative entropy at $p=1$. Based on that, we introduce a new notation of relative entropy to a subalgebra which generalizes subfactors index. This relative entropy has application in estimating decoherence time of quantum Markov semigroups.
We give a proof of an analogue of Connes' Hochschild character theorem for twisted spectral triples obtained from twisting a spectral triple by scaling automorphisms, under some suitable conditions. We also survey some of the properties of twisted spectral triples that are known so far.
Twisted spectral triples are a twisting of the notion of spectral triple aiming at dealing with some type III geometric situations. In the first part of the paper, we give a geometric construction of the index map of a twisted spectral triple in terms of $σ$-connections on finitely generated projective modules. This makes it more transparent the analogy with the indices of Dirac operators with coefficients in vector bundles. In the second part, we give a direct construction of the Connes-Chern character of a twisted spectral, both in the invertible and non-invertible cases. Combining these two…
Twisted Roe algebras and their $K$-theory
本文提出扭曲Roe代数与带系数的扭曲粗Baum-Connes猜想，证明在粗纤维化结构下该猜想可由底空间与纤维传递，并应用于粗嵌入群扩张，得出粗Baum-Connes猜想成立。
Connes-embeddability and twisted group von Neumann algebras
本文定义余循环-超线性群，证明其包含特定扩张与中心扩张，并在多种群运算下封闭，从而推广 Connes-可嵌入性。
The twisted coarse Baum--Connes conjecture and relative hyperbolic groups
本文引入稳定粗代数并提出扭曲粗Baum--Connes猜想，证明其在粗等价等操作下保持，并应用于相对双曲群，得出群满足猜想当且仅当其每个双曲子群满足。
We study the twisted K-theory and K-homology of some infinite dimensional spaces, like SU(\infty), in the bivariant setting. Using a general procedure due to Cuntz we construct a bivariant K-theory on the category of separable σ-C^*-algebras that generalizes both twisted K-theory and K-homology of (locally) compact spaces. We construct a bivariant Chern--Connes type character taking values in bivariant local cyclic homology. We analyse the structure of the dual Chern--Connes character from (analytic) K-homology to local cyclic cohomology under some reasonable hypotheses. We also investigate th…
We construct the geometric Baum-Connes assembly map for twisted Lie groupoids, that means for Lie groupoids together with a given groupoid equivariant $PU(H)-$principle bundle. The construction is based on the use of geometric deformation groupoids, these objects allow in particular to give a geometric construction of the associated pushforward maps and to establish the functoriality. The main results in this paper are to define the geometric twisted K-homology groups and to construct the assembly map. Even in the untwisted case the fact that the geometric twisted K-homology groups and the geo…
The goal of this paper is to construct a calculus whose higher indices are naturally elements in the twisted K-theory groups for Lie groupoids. Given a Lie groupoid $G$ and a $PU(H)$-valued groupoid cocycle, we construct an algebra of projective pseudodifferential operators. The subalgebra of regularizing operators identifies with the naturally associated smooth convolution algebra of the associated twisted gerbe. We develop the associated symbolic calculus, symbol short exact sequences and existence of parametrices. In particular the algebra of projective operators appears as a quantization o…
We show that the real Baum-Connes conjecture for abelian groups, possibly twisted by a cocycle, explains the isomorphisms of (twisted) KR-groups that underlie all T-dualities of torus orientifold string theories.
For an orbifold X and $α\in H^3(X, Z)$, we introduce the twisted cohomology $H^*_c(X, α)$ and prove that the Connes-Chern character establishes an isomorphism between the twisted K-groups $K_α^* (X) \otimes C$ and twisted cohomology $H^*_c(X, α)$. This theorem, on the one hand, generalizes a classical result of Baum-Connes, Brylinski-Nistor, and others, that if X is an orbifold then the Chern character establishes an isomorphism between the K-groups of X tensored with C, and the compactly-supported cohomology of the inertia orbifold. On the other hand, it also generalizes a recent result of Ad…
Flatness of $α$-induced bi-unitary connections and commutativity of Frobenius algebras
该研究针对辫子酉融合范畴中的双酉联络开展$\alpha$-归纳分析，证明诱导后联络的平坦性等价于原Frobenius代数的交换性，补全了此前结论的逆命题，还建立了平坦部分与交换Frobenius子代数的精细对应。
This paper is the continuation of Part I, expanding previous results of math.DG/9803051. This paper uses techniques in noncommutative geometry as developed by Alain Connes in order to study the twisted higher index theory of elliptic operators on orbifold covering spaces of compact good orbifolds, which are invariant under a projective action of the orbifold fundamental group. We also compute the range of the higher cyclic traces on $K$-theory for cocompact Fuchsian groups, which is then applied to determine the range of values of the Connes-Kubo Hall conductance in the discrete model of the q…
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