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Sergio Ciamprone

Publications and source records attributed to Sergio Ciamprone.

2 recordsLinked to original sources
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.

math.QAā†—
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^{iĻ€/\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.

math.OAā†—