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80
Tensor products of modules for a vertex operator algebras and vertex tensor categories
 in: Lie Theory and Geometry, in honor of Bertram Kostant
, 1994
"... In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announ ..."
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Cited by 70 (11 self)
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In this paper, we present a theory of tensor products of classes of modules for a vertex operator algebra. We focus on motivating and explaining new structures and results in this theory, rather than on proofs, which are being presented in a series of papers beginning with [HL4] and [HL5]. An announcement has also appeared [HL1].
Higherdimensional algebra II: 2Hilbert spaces
"... A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also ..."
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Cited by 60 (13 self)
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A 2Hilbert space is a category with structures and properties analogous to those of a Hilbert space. More precisely, we define a 2Hilbert space to be an abelian category enriched over Hilb with a ∗structure, conjugatelinear on the homsets, satisfying 〈fg,h 〉 = 〈g,f ∗ h 〉 = 〈f,hg ∗ 〉. We also define monoidal, braided monoidal, and symmetric monoidal versions of 2Hilbert spaces, which we call 2H*algebras, braided 2H*algebras, and symmetric 2H*algebras, and we describe the relation between these and tangles in 2, 3, and 4 dimensions, respectively. We prove a generalized DoplicherRoberts theorem stating that every symmetric 2H*algebra is equivalent to the category Rep(G) of continuous unitary finitedimensional representations of some compact supergroupoid G. The equivalence is given by a categorified version of the Gelfand transform; we also construct a categorified version of the Fourier transform when G is a compact abelian group. Finally, we characterize Rep(G) by its universal properties when G is a compact classical group. For example, Rep(U(n)) is the free connected symmetric 2H*algebra on one even object of dimension n. 1
Constructing the extended Haagerup planar algebra
, 2009
"... We construct a subfactor planar algebra, and as a corollary a subfactor, with the ‘extended Haagerup´ principal graph pair. This is the last open case from Haagerup’s 1993 list of potential principal graphs of subfactors with index in the range (4, 3 + √ 3). We prove that the subfactor planar algeb ..."
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Cited by 43 (17 self)
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We construct a subfactor planar algebra, and as a corollary a subfactor, with the ‘extended Haagerup´ principal graph pair. This is the last open case from Haagerup’s 1993 list of potential principal graphs of subfactors with index in the range (4, 3 + √ 3). We prove that the subfactor planar algebra with these principal graphs is unique. We give a skein theoretic description, and a description as a subalgebra generated by a certain element in the graph planar algebra of its principal graph. We give an explicit algorithm for evaluating closed diagrams using the skein theoretic description. This evaluation algorithm is unusual because intermediate steps may increase the number of generators in a diagram.
Quantum categories, star autonomy, and quantum groupoids
 in &quot;Galois Theory, Hopf Algebras, and Semiabelian Categories&quot;, Fields Institute Communications 43 (American Math. Soc
, 2004
"... Abstract A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. Generalizing this concept, we define the term "quantum category"in a braided monoidal category with equalizers dist ..."
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Cited by 39 (13 self)
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Abstract A useful general concept of bialgebroid seems to be resolving itself in recent publications; we give a treatment in terms of modules and enriched categories. Generalizing this concept, we define the term &quot;quantum category&quot;in a braided monoidal category with equalizers distributed over by tensoring with an object. The definition of antipode for a bialgebroid is less resolved in the literature. Our suggestion is that the kind of dualization occurring in Barr's starautonomous categories is more suitable than autonomy ( = compactness = rigidity). This leads to our definition of quantum groupoid intended as a &quot;Hopf algebra with several objects&quot;. 1.
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Cited by 38 (4 self)
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to &quot;strongly separable &quot; Frobenius algebras and &quot;weak monoidal Morita equivalence&quot;. Wreath products of Frobenius algebras are discussed.
The representation category of the quantum group of a nondegenerate bilinear form
 Comm. Algebra
"... We show that the representation category of the quantum group of a nondegenerate bilinear form is monoidally equivalent to the representation category of the quantum group SLq(2) for a wellchosen nonzero parameter q. The key ingredient for the proof of this result is the direct and explicit const ..."
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Cited by 33 (9 self)
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We show that the representation category of the quantum group of a nondegenerate bilinear form is monoidally equivalent to the representation category of the quantum group SLq(2) for a wellchosen nonzero parameter q. The key ingredient for the proof of this result is the direct and explicit construction of an appropriate Hopf bigalois extension. Then we get, when the base field is of characteristic zero, a full description of cosemisimple Hopf algebras whose representation semiring is isomorphic to the one of SL(2).
Categorical and combinatorial aspects of descent theory, [arXiv:math/0303175
"... There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict ncategories and outlined for weak ncategories. The categorical aspects concern ..."
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Cited by 27 (2 self)
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There is a construction which lies at the heart of descent theory. The combinatorial aspects of this paper concern the description of the construction in all dimensions. The description is achieved precisely for strict ncategories and outlined for weak ncategories. The categorical aspects concern the development of descent theory in low dimensions in order to provide a template for a theory in all dimensions. The theory involves nonabelian cohomology, stacks, torsors, homotopy, and higherdimensional categories. Many of the ideas are scattered through the literature or are folklore; a few are new. Section Headings
The monoidal EilenbergMoore construction and bialgebroids
"... Abstract. Monoidal functors U: C → M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on ”quantum groupoids ” we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi’s ×Rbialgebras, app ..."
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Cited by 22 (3 self)
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Abstract. Monoidal functors U: C → M with left adjoints determine, in a universal way, monoids T in the category of oplax monoidal endofunctors on ”quantum groupoids ” we derive Tannaka duality between left adjointable monoidal functors and bimonads. Bialgebroids, i.e., Takeuchi’s ×Rbialgebras, appear as the special case when T has also a right adjoint. Street’s 2category of monads then leads to a natural definition of the 2category of bialgebroids. Contents
Adjointable monoidal functors and quantum groupoids, Hopf algebras in noncommutative geometry and physics
 Lecture Notes in Pure and
"... Abstract. Every monoidal functor G: C → M has a canonical factorization through the category RMR of bimodules in M over some monoid R in M in which the factor U: C → RMR is strongly unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given b ..."
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Cited by 22 (2 self)
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Abstract. Every monoidal functor G: C → M has a canonical factorization through the category RMR of bimodules in M over some monoid R in M in which the factor U: C → RMR is strongly unital. Using this result and the characterization of the forgetful functors MA → RMR of bialgebroids A over R given by Schauenburg [15] together with their bimonad description given by the author in [18] here we characterize the ”long ” forgetful functors MA → RMR → M of both bialgebroids and weak bialgebras. 1.
Integral noncommutative spaces
 J. Algebra
"... Abstract. This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a noncommutative space. 1. ..."
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Cited by 18 (3 self)
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Abstract. This paper concerns the closed points, closed subspaces, open subspaces, weakly closed and weakly open subspaces, and effective divisors, on a noncommutative space. 1.