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56
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.
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 [ ,]
Braided Quantum Field Theory
"... We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for npoint functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have n ..."
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Cited by 47 (6 self)
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We develop a general framework for quantum field theory on noncommutative spaces, i.e., spaces with quantum group symmetry. We use the path integral approach to obtain expressions for npoint functions. Perturbation theory leads us to generalised Feynman diagrams which are braided, i.e., they have nontrivial over and undercrossings. We demonstrate the power of our approach by applying it to φ 4theory on the quantum 2sphere. We find that the basic divergent diagram of the theory is regularised.
NicholsWoronowicz algebra model for Schubert calculus on Coxeter groups
 J. Algebra
"... Abstract. We realise the cohomology ring of a flag manifold, more generally the coinvariant algebra of an arbitrary finite Coxeter group W, as a commutative subalgebra of a certain Nichols algebra in the YetterDrinfeld category over W. This gives a braided Hopf algebra version of the corresponding ..."
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Cited by 26 (4 self)
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Abstract. We realise the cohomology ring of a flag manifold, more generally the coinvariant algebra of an arbitrary finite Coxeter group W, as a commutative subalgebra of a certain Nichols algebra in the YetterDrinfeld category over W. This gives a braided Hopf algebra version of the corresponding Schubert calculus. Our proof is based on methods of braided differential calculus rather than on working directly with the relations in the Nichols algebra, which are not known explicitly. We also discuss the relationship between FominKirillov quadratic algebras, KirillovMaeno bracket algebras and our construction. Contents
qEpsilon Tensor For Quantum And Braided Spaces
 J. Math. Phys
, 1994
"... We use the machinery of braided geometry to construct the ε `totally antisymmetric tensor' on a general braided vector space determined br Rmatrices, as introduced previously. This includes natural qEuclidean and qMinkowski spaces. Our formalism is completely covariant under the correspon ..."
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Cited by 20 (13 self)
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We use the machinery of braided geometry to construct the ε `totally antisymmetric tensor' on a general braided vector space determined br Rmatrices, as introduced previously. This includes natural qEuclidean and qMinkowski spaces. Our formalism is completely covariant under the corresponding quantum group such as SO_q(4) or SO_q(1, 3).
Solutions of Klein–Gordon and Dirac Equations on Quantum Minkowski Spaces
, 1995
"... Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein–Gordo ..."
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Cited by 19 (1 self)
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Covariant differential calculi and exterior algebras on quantum homogeneous spaces endowed with the action of inhomogeneous quantum groups are classified. In the case of quantum Minkowski spaces they have the same dimensions as in the classical case. Formal solutions of the corresponding Klein–Gordon and Dirac equations are found. The Fock space construction is sketched.
Integrals for braided Hopf algebras
 J. Pure Appl. Algebra
, 2000
"... Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided ve ..."
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Cited by 17 (3 self)
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Let H be a Hopf algebra in a rigid braided monoidal category with split idempotents. We prove the existence of integrals on (in) H characterized by the universal property, employing results about Hopf modules, and show that their common target (source) object IntH is invertible. The fully braided version of Radford’s formula for the fourth power of the antipode is obtained. Connections of integration with crossproduct and transmutation are studied. 1991 Mathematics Subject Classification. Primary 16W30, 18D15, 17B37; Secondary 18D35.
SOLUTIONS OF THE YANGBAXTER EQUATIONS FROM BRAIDEDLIE ALGEBRAS AND BRAIDED GROUPS
, 1993
"... We obtain an Rmatrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)Lie algebra or braidedLie algebra. The same result applies for every (super)Hopf algebra or braidedHopf algebra. We recover some known representations such as th ..."
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Cited by 16 (10 self)
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We obtain an Rmatrix or matrix representation of the Artin braid group acting in a canonical way on the vector space of every (super)Lie algebra or braidedLie algebra. The same result applies for every (super)Hopf algebra or braidedHopf algebra. We recover some known representations such as those associated to racks. We also obtain new representations such as a nontrivial one on the ring k[x] of polynomials in one variable, regarded as a braidedline. Representations of the extended Artin braid group for braids in the complement of S 1 are also obtained by the same method.
THE NICHOLS ALGEBRA OF A SEMISIMPLE YETTERDRINFELD MODULE
, 2008
"... We study the Nichols algebra of a semisimple YetterDrinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a “reflection” defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig’s ..."
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Cited by 16 (8 self)
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We study the Nichols algebra of a semisimple YetterDrinfeld module and introduce new invariants including the notions of real roots and the Weyl groupoid. The crucial ingredient is a “reflection” defined on arbitrary such Nichols algebras. Our construction generalizes the restriction of Lusztig’s automorphisms of quantized KacMoody algebras to the nilpotent part. As a direct application we complete the classifications of finitedimensional pointed Hopf algebras over S3, and of finitedimensional Nichols algebras over S4. This theory has led to surprising new results in the classification of finitedimensional pointed Hopf algebras with a nonabelian group of grouplike elements.
*Products on Quantum Spaces
, 2001
"... In this paper we present explicit formulas for the ∗product on quantum spaces which are of particular importance in physics, i.e., the qdeformed Minkowski space and the qdeformed Euclidean space in 3 and 4 dimensions, respectively. Our formulas are complete and formulated using the deformation pa ..."
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Cited by 16 (13 self)
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In this paper we present explicit formulas for the ∗product on quantum spaces which are of particular importance in physics, i.e., the qdeformed Minkowski space and the qdeformed Euclidean space in 3 and 4 dimensions, respectively. Our formulas are complete and formulated using the deformation parameter q. In addition, we worked out an expansion in powers of h = ln q up to second order, for all considered cases.