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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 95 (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.
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 84 (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.
BEYOND SUPERSYMMETRY AND QUANTUM SYMMETRY (AN INTRODUCTION TO BRAIDEDGROUPS AND BRAIDEDMATRICES)
, 1993
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QUANTUM AND BRAIDED LIE ALGEBRAS
, 1993
"... We introduce the notion of a braided Lie algebra consisting of a finitedimensional vector space L equipped with a bracket [ , ] : L ⊗ L → L and a YangBaxter operator Ψ: L ⊗ L → L ⊗ L obeying some axioms. We show that such an object has an enveloping braidedbialgebra U(L). We show that every gener ..."
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Cited by 50 (28 self)
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We introduce the notion of a braided Lie algebra consisting of a finitedimensional vector space L equipped with a bracket [ , ] : L ⊗ L → L and a YangBaxter operator Ψ: L ⊗ L → L ⊗ L obeying some axioms. We show that such an object has an enveloping braidedbialgebra U(L). We show that every generic Rmatrix leads to such a braided Lie algebra with [ , ] given by structure constants c IJ K determined from R. In this case U(L) = B(R) the braided matrices introduced previously. We also introduce the basic theory of these braided Lie algebras, including the natural rightregular action of a braidedLie algebra L by braided vector fields, the braidedKilling form and the quadratic Casimir associated to L. These constructions recover the relevant notions for usual, colour and superLie algebras as special cases. In addition, the standard quantum deformations Uq(g) are understood as the enveloping algebras of such underlying braided Lie algebras with [ ,]
Construction of Field Algebras with Quantum Symmetry from Local Observables
, 1996
"... It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reco ..."
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Cited by 46 (5 self)
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It has been discussed earlier that ( weak quasi) quantum groups allow for conventional interpretation as internal symmetries in local quantum theory. From general arguments and explicit examples their consistency with (braid) statistics and locality was established. This work addresses to the reconstruction of quantum symmetries and algebras of field operators. For every algebra A of observables satisfying certain standard assumptions, an appropriate quantum symmetry is found. Field operators are obtained which act on a positive definite Hilbert space of states and transform covariantly under the quantum symmetry. As a substitute for Bose/Fermi (anti) commutation relations, these fields are demonstrated to obey local braid relation. Contents 1 Introduction 1 2 The Notion of Quantum Symmetry 5 3 Algebraic Methods for Field Construction 9 3.1 Observables and superselection sectors in local quantum field theory . . . . 10 3.2 Localized endomorphisms and fusion structure . . . . . ....
The Quantum Double as Quantum Mechanics
"... We introduce ∗structures on braided groups and braided matrices. Using this, we show that the quantum double D(Uq(su2)) can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in qMinkowski space (a threesphere in the Lorentz metric), and with the role of ..."
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Cited by 28 (21 self)
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We introduce ∗structures on braided groups and braided matrices. Using this, we show that the quantum double D(Uq(su2)) can be viewed as the quantum algebra of observables of a quantum particle moving on a hyperboloid in qMinkowski space (a threesphere in the Lorentz metric), and with the role of angular momentum played by Uq(su2). This provides a new example of a quantum system whose algebra of observables is a Hopf algebra. Furthermore, its dual Hopf algebra can also be viewed as a quantum algebra of observables, of another quantum system. This time the position space is a qdeformation of SL(2, R) and the momentum group is Uq(su ∗ 2) where su ∗ 2 is the Drinfeld dual Lie algebra of su2. Similar results hold for the quantum double and its dual of a general quantum group.
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 25 (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
On Matrix Quantum Groups Of Type A_n
, 1997
"... To a Hecke symmetry R there associate a matrix bialgebra ER and a matrix Hopf algebra HR , which are called function rings on the matrix quantum semigroup and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum gr ..."
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Cited by 19 (4 self)
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To a Hecke symmetry R there associate a matrix bialgebra ER and a matrix Hopf algebra HR , which are called function rings on the matrix quantum semigroup and matrix quantum groups associated to R. We show that for an even Hecke symmetry, the rational representations of the corresponding quantum group are absolutely reducible and compute the integral on the function ring of the quantum group, i.e, on HR . Further, we show that the fusion coefficients of simple representations depend only on the rank of the symmetry, and give the explicit formula for the rank or 8dim of HRcomodules. In the general case, we show that the quantum semigroup is "Zariski" dense in the quantum group. This enables us to study the semisimplicity of the associated quantum group in some case. 1.