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Algebras and Hopf algebras IN BRAIDED CATEGORIES
, 1995
"... This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise superalgebras and superHopf algebras, as well as colourLie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras i ..."
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Cited by 87 (13 self)
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This is an introduction for algebraists to the theory of algebras and Hopf algebras in braided categories. Such objects generalise superalgebras and superHopf algebras, as well as colourLie algebras. Basic facts about braided categories C are recalled, the modules and comodules of Hopf algebras in such categories are studied, the notion of ‘braidedcommutative ’ or ‘braidedcocommutative ’ Hopf algebras (braided groups) is reviewed and a fully diagrammatic proof of the reconstruction theorem for a braided group Aut (C) is given. The theory has important implications for the theory of quasitriangular Hopf algebras (quantum groups). It also includes important examples such as the degenerate Sklyanin algebra and the quantum plane.
HigherDimensional Algebra I: Braided Monoidal 2Categories
 Adv. Math
, 1996
"... We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give lon ..."
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Cited by 53 (9 self)
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We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2categories and their relevance to 4d TQFTs and 2tangles. Then we give concise definitions of semistrict monoidal 2categories and braided monoidal 2categories, and show how these may be unpacked to give long explicit definitions similar to, but not quite the same as, those given by Kapranov and Voevodsky. Finally, we describe how to construct a semistrict braided monoidal 2category Z(C) as the `center' of a semistrict monoidal category C, in a manner analogous to the construction of a braided monoidal category as the center of a monoidal category. As a corollary this yields a strictification theorem for braided monoidal 2categories. 1 Introduction This is the first of a series of articles developing the program introduced in the paper `HigherDimensional Algebra and Topological Quantum Field Theory' [1], henceforth referred to as `HDA'. This program consists of generalizing algebraic concep...
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 43 (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
Galois theory for braided tensor categories and the modular closure
 Adv. Math
, 2000
"... Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC ..."
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Cited by 29 (6 self)
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Given a braided tensor ∗category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C ⋊ S. This construction yields a tensor ∗category with conjugates and an irreducible unit. (A ∗category is a category enriched over VectC with positive ∗operation.) A Galois correspondence is established between intermediate categories sitting between C and C ⋊S and closed subgroups of the Galois group Gal(C⋊S/C) = AutC(C⋊S) of C, the latter being isomorphic to the compact group associated to S by the duality theorem of Doplicher and Roberts. Denoting by D ⊂ C the full subcategory of degenerate objects, i.e. objects which have trivial monodromy with all objects of C, the braiding of C extends to a braiding of C⋊S iff S ⊂ D. Under this condition C⋊S has no nontrivial degenerate objects iff S = D. If the original category C is rational (i.e. has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category C ≡ C ⋊ D is called the modular closure of C since in the rational case it is modular, i.e. gives rise to a unitary representation of the modular group SL(2, Z). (In passing we prove that every braided tensor ∗category with conjugates automatically is a ribbon category, i.e. has a twist.) If all simple objects of S have dimension one the structure of the category C ⋊ S can be clarified quite explicitly in terms of group cohomology. 1
Uniqueness theorems for certain triangulated categories possessing an Adams spectral sequence
, 139
"... 1.2. The axioms ..."
Triangular Hopf algebras with the Chevalley property
 Michigan Journal of Mathematics
"... Triangular Hopf algebras were introduced by Drinfeld [Dr]. They are the Hopf algebras whose representations form a symmetric tensor category. In that sense, they are the class of Hopf algebras closest to group algebras. The structure of triangular Hopf algebras is far from ..."
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Cited by 26 (11 self)
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Triangular Hopf algebras were introduced by Drinfeld [Dr]. They are the Hopf algebras whose representations form a symmetric tensor category. In that sense, they are the class of Hopf algebras closest to group algebras. The structure of triangular Hopf algebras is far from
Weak Hopf Algebras II: Representation theory, dimensions, and the Markov trace
 J. Algebra
"... If A is a weak C∗Hopf algebra then the category of finite dimensional unitary representations of A is a monoidal C∗category with monoidal unit being the GNS representation Dε associated to the counit ε. This category has isomorphic left dual and right dual objects which leads, as usual, to the not ..."
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Cited by 20 (3 self)
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If A is a weak C∗Hopf algebra then the category of finite dimensional unitary representations of A is a monoidal C∗category with monoidal unit being the GNS representation Dε associated to the counit ε. This category has isomorphic left dual and right dual objects which leads, as usual, to the notion of dimension function. However, if ε is not pure the dimension function is matrix valued with rows and columns labelled by the irreducibles contained in Dε. This happens precisely when the inclusions AL ⊂ A and AR ⊂ A are not connected. Still there exists a trace on A which is the Markov trace for both inclusions. We derive two numerical invariants for each C∗WHA of trivial hypercenter. These are the common indices I and δ, of the Haar, respectively Markov conditional expectations of either one of the inclusions AL/R ⊂ A and ÂL/R ⊂ Â. In generic cases I> δ. In the special case of weak Kac algebras we show that I = δ is an integer. Submitted to J. Algebra
A polytope calculus for semisimple groups
 Duke Math. J
"... We define a collection of polytopes associated to a semisimple group G. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain region. The polytopes are defined as moment map images of algebraic cycles discovered by I. Mirković an ..."
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Cited by 20 (3 self)
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We define a collection of polytopes associated to a semisimple group G. Weight multiplicities and tensor product multiplicities may be computed as the number of such polytopes fitting in a certain region. The polytopes are defined as moment map images of algebraic cycles discovered by I. Mirković and K. Vilonen. These cycles are a canonical basis for the intersection homology of (the closures of the strata of) the loop Grassmannian. 1.