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110
Faulttolerant quantum computation by anyons
, 2003
"... A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation ..."
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Cited by 230 (3 self)
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A twodimensional quantum system with anyonic excitations can be considered as a quantum computer. Unitary transformations can be performed by moving the excitations around each other. Measurements can be performed by joining excitations in pairs and observing the result of fusion. Such computation is faulttolerant by its physical nature.
On the classification of finitedimensional pointed Hopf algebras
, 2006
"... We classify finitedimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are onedimensional, and whose group of grouplike elements G(A) is abelian such that all prime divisors of the order of G(A) are> 7. Since these Hopf algebras turn out to be def ..."
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Cited by 119 (14 self)
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We classify finitedimensional complex Hopf algebras A which are pointed, that is, all of whose irreducible comodules are onedimensional, and whose group of grouplike elements G(A) is abelian such that all prime divisors of the order of G(A) are> 7. Since these Hopf algebras turn out to be deformations of a natural class of generalized small quantum groups, our result can be read as an axiomatic description of generalized small quantum groups.
Pointed Hopf algebras
 In “New directions in Hopf algebras”, MSRI series Cambridge Univ
, 2002
"... Abstract. This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra gr A associated to the coradical filtration of A. The A0coinvariants elements form a braided Hopf alg ..."
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Cited by 99 (7 self)
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Abstract. This is a survey on pointed Hopf algebras over algebraically closed fields of characteristic 0. We propose to classify pointed Hopf algebras A by first determining the graded Hopf algebra gr A associated to the coradical filtration of A. The A0coinvariants elements form a braided Hopf algebra R in the category of Yetter–Drinfeld modules over the coradical A0 = Γ, Γ the group of grouplike elements of A, and gr A ≃ R#A0. We call the braiding of the primitive elements of R the infinitesimal braiding of A. If this braiding is of Cartan type [AS2], then it is often possible to determine R, to show that R is generated as an algebra by its primitive elements and finally to compute all deformations or liftings, that is pointed Hopf algebras such that gr A ≃ R#Γ. In the last chapter, as a concrete illustration of the method, we describe explicitly all finitedimensional pointed Hopf algebras A with abelian group of grouplikes G(A) and infinitesimal braiding of type An (up to some exceptional cases). In other words, we compute all the liftings of type An; this result is our main new
Anyons in an exactly solved model and beyond
, 2005
"... A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge f ..."
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Cited by 87 (2 self)
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A spin 1/2 system on a honeycomb lattice is studied. The interactions between nearest neighbors are of XX, YY or ZZ type, depending on the direction of the link; different types of interactions may differ in strength. The model is solved exactly by a reduction to free fermions in a static Z2 gauge field. A phase diagram in the parameter space is obtained. One of the phases has an energy gap and carries excitations that are Abelian anyons. The other phase is gapless, but acquires a gap in the presence of magnetic field. In the latter case excitations are nonAbelian anyons whose braiding rules coincide with those of conformal blocks for the Ising model. We also consider a general theory of free fermions with a gapped spectrum, which is characterized by a spectral Chern number ν. The Abelian and nonAbelian phases of the original model correspond to ν = 0 and ν = ±1, respectively. The anyonic properties of excitation depend on ν mod 16, whereas ν itself governs edge thermal transport. The paper also provides mathematical background on anyons as well as an elementary theory of Chern number for quasidiagonal matrices.
Diagonal Crossed Products by Duals of QuasiQuantum Groups
 Rev. Math. Phys
, 1999
"... A twosided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right Gcoactions on M, respectively. Denoting the associated commuting right and left actions ..."
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Cited by 36 (1 self)
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A twosided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right Gcoactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra ˆ G on M by ⊳ and ⊲, respectively, we define the diagonal crossed product M ⊲ ⊳ ˆ G to be the algebra generated by M and ˆ G with relations given by ϕm = (ϕ (1) ⊲m ⊳ ˆ S −1 (ϕ (3)))ϕ (2), m ∈ M, ϕ ∈ ˆ G. We give a natural generalization of this construction to the case where G is a quasi–Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct ∆ is nonunital). In these cases our diagonal crossed product will still be an associative algebra structure on M ⊗ ˆ G extending M ≡ M ⊗ ˆ1, even though the analogue of an ordinary crossed product M ⋊ ˆ G in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasiquantum
Correspondences of ribbon categories
, 2003
"... Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories ..."
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Cited by 34 (9 self)
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Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.
On pointed Hopf algebras associated to some conjugacy classes
 in Sn, Proc. Amer. Math. Soc
"... Abstract. We show that any pointed Hopf algebra with infinitesimal braiding associated to the conjugacy class of π ∈ Sn is infinitedimensional, if either the order of π is odd, or π is a product of disjoint cycles of odd order except for exactly two transpositions. ..."
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Cited by 33 (17 self)
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Abstract. We show that any pointed Hopf algebra with infinitesimal braiding associated to the conjugacy class of π ∈ Sn is infinitedimensional, if either the order of π is odd, or π is a product of disjoint cycles of odd order except for exactly two transpositions.
ON NICHOLS ALGEBRAS OF LOW DIMENSION
, 2000
"... This is a contribution to the classification program of pointed Hopf algebras. We give a generalization of the quantum Serre relations and propose a generalization of the Frobenius–Lusztig kernels in order to compute Nichols algebras coming from the abelian case. With this, we classify Nichols alg ..."
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Cited by 32 (3 self)
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This is a contribution to the classification program of pointed Hopf algebras. We give a generalization of the quantum Serre relations and propose a generalization of the Frobenius–Lusztig kernels in order to compute Nichols algebras coming from the abelian case. With this, we classify Nichols algebras B(V) with dimension < 32 or with dimension p³, p a prime number, when V lies in a Yetter–Drinfeld category over a finite group. With the so called Lifting Procedure, this allows to classify pointed Hopf algebras of index < 32 or p³.
FrobeniusSchur indicators and exponents of spherical categories
 Adv. Math
"... Abstract. We obtain two formulae for the higher FrobeniusSchur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, ..."
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Cited by 30 (10 self)
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Abstract. We obtain two formulae for the higher FrobeniusSchur indicators: one for a spherical fusion category in terms of the twist of its center and the other one for a modular tensor category in terms of its twist. The first one is a categorical generalization of an analogous result by Kashina, Sommerhäuser, and Zhu for Hopf algebras, and the second one extends Bantay’s 2nd indicator formula for a conformal field theory to higher degree. These formulae imply the sequence of higher indicators of an object in these categories is periodic. We define the notion of FrobeniusSchur (FS)exponent of a pivotal category to be the global period of all these sequences of higher indicators, and we prove that the FSexponent of a spherical fusion category is equal to the order of the twist of its center. Consequently, the FSexponent of a spherical fusion category is a multiple of its exponent by a factor not greater than 2. As applications of these results, we prove that the FSexponent of a semisimple quasiHopf algebra H has the same set of prime divisors as of dim(H) and it divides dim(H) 4. In addition, if H is a grouptheoretic quasiHopf algebra, the FSexponent of H divides dim(H) 2, and this upper bound is shown to be tight. 1.