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Claudia Pinzari

Publications and source records attributed to Claudia Pinzari.

At least 19 recordsLinked to original sources
arXiv2026

The braided Doplicher-Roberts program and the Finkelberg-Kazhdan-Lusztig equivalence: A historical perspective, recent progress, and future directions

Our recent approach to the Finkelberg-Kazhdan-Lusztig equivalence theorem centers on the construction of a fiber functor associated with the categories in the equivalence theorem, which in turn explains the underlying algebraic and analytic structure of the corresponding weak Hopf algebra in a new sense. We provide a non-technical and historical overview of the core arguments behind our proof, discuss these structural properties, and its applications to rigidity and unitarizability of braided fusion categories arising from conformal field theory. We conclude proposing some natural directions for future research.

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arXiv2025

Constructing equivalences between quantum group fusion categories and Huang-Lepowsky modular categories via quantum gauge groups

This paper provides a unified framework resolving two long-standing problems: the intrinsic construction of global quantum gauge groups for braided tensor $C^*$-categories (the Doplicher-Roberts problem) and the direct proof of the Finkelberg equivalence theorem at positive integer levels (the Huang problem). In our previous work, we solved both problems for the WZW model across all Lie types by constructing a unitary modular tensor category structure on the module category of an affine vertex operator algebra at positive integer level, together with a quantum gauge group for our analytic structure. Specifically, we utilized the global quantum gauge group A_{W}(\mathfrak{g},q) to equip the Zhu algebra of the affine vertex operator algebra V_{\mathfrak{g}_{k}} with a unitary coboundary weak quasi-Hopf algebra structure with a 3-coboundary associator. This relies on an isometric analytic Drinfeld twist and Wenzl's continuous de-quantization curve. In the present paper, we address the Huang problem specifically for the Huang-Lepowsky tensor structure. We provide a complete identification of our modular tensor category structure with the Huang-Lepowsky structure for all the classical Lie types and $G_2$ via generalized quantum Schur-Weyl duality. This unified approach establishes rigidity directly from the quantum group fusion category, completely bypassing reliance on the Verlinde formula, the monodromy of the Knizhnik-Zamolodchikov equations, negative-level shifting typically required in the VOA setting, and the Jones index used in the conformal net setting. Consequently, our framework restores the natural categorical hierarchy where local rigidity precedes global modular properties entirely within the vertex operator algebra setting.

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arXiv2021

Weak quasi-Hopf algebras, C*-tensor categories and conformal field theory, and the Kazhdan-Lusztig-Finkelberg theorem

We develop Doplicher-Roberts quantum group duality program for the WZW model within the framework of vertex operator algebras. We establish that a weak quasi-fibre structure on a functor preserving a Drinfeld coboundary symmetry naturally extends a symmetric functor under permutation symmetry. Utilizing Wenzl's functor associated with the unitary quantum group fusion category, we construct a weak tensor structure, yielding a new class of unitary coboundary weak Hopf $C^*$-algebras for all Lie types and levels. Via a specialized Drinfeld twist and the Wenzl de-quantization curve, this structure is transported onto the Zhu algebra--which consequently becomes a unitary coboundary weak quasi-Hopf $C^*$-algebra with a 3-coboundary associator--providing a uniform, self-contained construction of unitary rigid braided tensor categories for categories of affine VOA modules at positive integer levels. Furthermore, we analyze the type A case via classification methods based on Kazhdan--Wenzl theory and our weak Hopf algebra framework, providing key insight into the determination of associativity from the braiding in the general case. We develop a cohomology theory for braided tensor categories with a generating object enabling a complete identification of our ribbon braided tensor structure with the constructions of Huang and Lepowsky for the classical Lie types and G_2, while bypassing their original reliance on the KZ equations and the Verlinde formula entirely. Our methods solve several long-standing problems: Galindo's question on the uniqueness of unitary tensor structures, Kirillov's conjecture on the positivity of a certain Hermitian form on the module category of an affine Lie algebra by Beilinson-Feigin-Mazur, the quantum group structure on the Zhu algebra sought by Frenkel and Zhu, and provide a direct proof of the Kazhdan-Lusztig-Finkelberg equivalence settling an open problem of Huang.

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arXiv2017

Quasi-coassociative C*-quantum groupoids of type A and modular C*-categories

We construct a new class of finite-dimensional C^*-quantum groupoids at roots of unity q=e^{iπ/\ell}, with limit the discrete dual of the classical SU(N) for large orders. The representation category of our groupoid turns out to be tensor equivalent to the well known quotient C^*-category of the category of tilting modules of the non-semisimple quantum group U_q({\mathfrak sl}_N) of Drinfeld, Jimbo and Lusztig. As an algebra, the C^*-groupoid is a quotient of U_q({\mathfrak sl}_N). As a coalgebra, it naturally reflects the categorical quotient construction. In particular, it is not coassociative, but satisfies axioms of the weak quasi-Hopf C^*-algebras: quasi-coassociativity and non-unitality of the coproduct. There are also a multiplicative counit, an antipode, and an R-matrix. For this, we give a general construction of quantum groupoids for complex simple Lie algebras {\mathfrak g}\neq E_8 and certain roots of unity. Our main tools here are Drinfeld's coboundary associated to the R-matrix, related to the algebra involution, and certain canonical projections introduced by Wenzl, which yield the coproduct and Drinfeld's associator in an explicit way. Tensorial properties of the negligible modules reflect in a rather special nature of the associator. We next reduce the proof of the categorical equivalence to the problems of establishing semisimplicity and computing dimension of the groupoid. In the case {\mathfrak g}={\mathfrak sl}_N we construct a (non-positive) Haar-type functional on an associative version of the dual groupoid satisfying key non-degeneracy properties. This enables us to complete the proof.

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arXiv2017

Polynomial growth of discrete quantum groups, topological dimension of the dual and *-regularity of the Fourier algebra

Banica and Vergnioux have shown that the dual discrete quantum group of a compact simply connected Lie group has polynomial growth of order the real manifold dimension. We extend this result to a general compact group and its topological dimension, by connecting it with the Gelfand-Kirillov dimension of an algebra. Furthermore, we show that polynomial growth for a compact quantum group G of Kac type implies *-regularity of the Fourier algebra A(G), that is every closed ideal of C(G) has a dense intersection with A(G). In particular, A(G) has a unique C*-norm.

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arXiv2015

Ergodic actions of compact quantum groups from solutions of the conjugate equations

We use a tensor C*-category with conjugates and two quasitensor functors into the category of Hilbert spaces to define a *-algebra depending functorially on this data. If one of them is tensorial, we can complete in the maximal C*-norm. A particular case of this construction allows us to begin with solutions of the conjugate equations and associate ergodic actions of quantum groups on the C*-algebra in question. The quantum groups involved are A_u(Q) and B_u(Q).

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arXiv2013

Embedding ergodic actions of compact quantum groups on C*-algebras into quotient spaces

The notion of compact quantum subgroup is revisited and an alternative definition is given. Induced representations are considered and a Frobenius reciprocity theorem is obtained. A relationship between ergodic actions of compact quantum groups on C*-algebras and topological transitivity is investigated. A sufficient condition for embedding such actions in quantum quotient spaces is obtained.

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arXiv2012

Connected components of compact matrix quantum groups and finiteness conditions

We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1. We give necessary and sufficient conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A_u(F) is not of Lie type. We discuss an example arising from the compact real form of U_q(sl_2) for q<0.

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arXiv2011

Growth rates of dimensional invariants of compact quantum groups and a theorem of Hoegh-Krohn, Landstad and Stormer

We give local upper and lower bounds for the eigenvalues of the modular operator associated to an ergodic action of a compact quantum group on a unital C*-algebra. They involve the modular theory of the quantum group and the growth rate of quantum dimensions of its representations and they become sharp if other integral invariants grow subexponentially. For compact groups, this reduces to the finiteness theorem of Hoegh-Krohn, Landstad and Stormer. Consequently, compact quantum groups of Kac type admitting an ergodic action with a non-tracial invariant state must have representations whose dimensions grow exponentially. In particular, S_{-1}U(d) acts ergodically only on tracial C*-algebras. For quantum groups with non-involutive coinverse, we derive a lower bound for the parameters 0<λ<1 of factors of type III_λthat can possibly arise from the GNS representation of the invariant state of an ergodic action with a factorial centralizer.

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arXiv2010

A rigidity result for extensions of braided tensor C*-categories derived from compact matrix quantum groups

Let G be a classical compact Lie group and G_μthe associated compact matrix quantum group deformed by a positive parameter μ(or a nonzero and real μin the type A case). It is well known that the category Rep(G_μ) of unitary f.d. representations of G_μis a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_μ) to another braided tensor C*-category with irreducible tensor unit is full if |μ|\neq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows that the Temperley--Lieb category generated by an object of dimension >2 can not be embedded properly into a larger category with the same objects as a braided tensor C*-subcategory.

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arXiv2010

A theory of induction and classification of tensor C*-categories

This paper addresses the problem of describing the structure of tensor C*-categories M with conjugates and irreducible tensor unit. No assumption on the existence of a braided symmetry or on amenability is made. Our assumptions are motivated by the remark that these categories often contain non-full tensor C*-subcategories with conjugates and the same objects admitting an embedding into the Hilbert spaces. Such an embedding defines a compact quantum group by Woronowicz duality. An important example is the Temperley--Lieb category canonically contained in a tensor C*-category generated by a single real or pseudoreal object of dimension bigger than 2. The associated quantum groups are the universal orthogonal quantum groups of Wang and Van Daele. Our main result asserts that there is a full and faithful tensor functor from M to a category of Hilbert bimodule representations of the compact quantum group. In the classical case, these bimodule representations reduce to the G-equivariant Hermitian bundles over compact homogeneous G-spaces, with G a compact group. Our structural results shed light on the problem of whether there is an embedding functor of M into the Hilbert spaces. We show that this is related to the problem of whether a classical compact Lie group can act ergodically on a non-type I von Neumann algebra. In particular, combining this with a result of Wassermann shows that an embedding exists if M is generated by a pseudoreal object of dimension 2.

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arXiv2009

Ergodic actions of S_μU(2) on C*-algebras from II_1 subfactors

To a proper inclusion N\subset M of II_1 factors of finite Jones index [M:N], we associate an ergodic C*-action of the quantum group S_μU(2). The deformation parameter is determined by -1<μ<0 and [M:N]=|μ+μ^{-1}|. The higher relative commutants can be identified with the spectral spaces of the tensor powers of the defining representation of the quantum group. This ergodic action may be thought of as a virtual subgroup of S_μU(2) in the sense of Mackey arising from the tensor category generated by M regarded as a bimodule over N. μis negative as M is a real bimodule.

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arXiv2006

A duality theorem for ergodic actions of compact quantum groups on C*-algebras

The spectral functor of an ergodic action of a compact quantum group G on a unital C*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor *-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C*-algebra. As an application, we associate an ergodic G-action on a unital C*-algebra to an inclusion of Rep(G) into an abstract tensor C*-category. If the inclusion arises from a quantum subgroup of G, the associated G-system is just the quantum quotient space. If G is a group and the category has permutation symmetry, the associated system is commutative, and therefore isomorphic to the classical quotient space by a closed subgroup of $G$. If a tensor C*-category has a Hecke symmetry making an object of dimension d and q-quantum determinant one then there is an ergodic action of S_qU(d) on a unital C*-algebra, having the spaces of intertwiners from the tensor unit to powers of the object as its spectral subspaces. The special case od S_qU(2) is discussed.

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arXiv2002

Regular Objects, Multiplicative Unitaries and Conjugation

The notion of left (resp. right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (resp. right) regular object, it can be interpreted as a category of corepresentations (resp. representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry. Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.

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arXiv2000

Noncommutative pressure and the variational principle in Cuntz-Krieger-type C*-algebras

We define a notion of dynamical pressure at a self-adjoint element for a contractive completely positive self-map of an exact C*-algebra which adopts Voiculescu's approximation approach to noncommutative entropy and extends the Voiculescu-Brown topological entropy and Neshveyev-Stormer unital-nuclear pressure. A variational inequality bounding the pressure below by the free energies with respect to the Sauvageot-Thouvenot entropy is established in two stages via the introduction of a local state approximation entropy, whose associated free energies function as an intermediate term. Pimsner C*-algebras furnish a framework for investigating the variational principle, which asserts the equality of the pressure and the supremum of the free energies over all dynamically invariant states. In one direction we extend Brown's result on the constancy of the Voiculescu-Brown entropy upon passing to the crossed product, and in another we show that the pressure of a self-adjoint element over the Markov subshift underlying the canonical map on a Cuntz-Kreiger algebra is equal to its classical pressure. The latter result is extended to a more general setting comprising an expanded class of Cuntz-Krieger-type Pimsner algebras, leading to the variational principle for self-adjoint elements in a diagonal subalgebra. Equilibrium states are constructed from KMS states under certain conditions in the case of Cuntz-Krieger algebras.

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arXiv1998

Ideal structure and simplicity of the C*-algebras generated by Hilbert bimodules

Pimsner introduced the C*-algebra O_X generated by a Hilbert bimodule X over a C*-algebra A. We look for additional conditions that X should satisfy in order to study simplicity and, more generally, the ideal structure of O_X when X is finite projective. We introduce two conditions: `(I)-freeness' and `(II)-freeness', stronger than the former, in analogy with [J. Cuntz, W. Krieger, Invent. Math. 56, 251-268] and [J. Cuntz, Invent. Math. 63, 25-40] respectively. (I)-freeness comprehend the case of the bimodules associated with an inclusion of simple C*-algebras with finite index, real or pseudoreal bimodules with finite dimension and the case of `Cuntz-Krieger bimodules'. If X satisfies this condition the C*-algebra O_X does not depend on the choice of the generators when A is faithfully represented. As a consequence, if X is (I)-free and A is X-simple, then O_X is simple. In the case of Cuntz-Krieger algebras, X-simplicity corresponds to irreducibility of the defining matrix. If A is simple and p.i. then O_X is p.i., if A is nonnuclear then O_X is nonnuclear. We therefore provide examples of (purely) infinite nonnuclear simple C*-algebras. Furthermore if X is (II)-free, we determine the ideal structure of O_X.

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