Papers by Bernhard Drabant
Cornell University - arXiv, Mar 11, 2021
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The subject of this article are cross product bialgebras without co-cycles. We establish a theory... more The subject of this article are cross product bialgebras without co-cycles. We establish a theory characterizing cross product bialgebras universally in terms of projections and injections. Especially all known types of biproduct, double cross product and bicross product bialgebras can be described by this theory. Furthermore the theory provides new families of (co-cycle free) cross product bialgebras. Besides the universal characterization we find an equivalent (co-)modular description of certain types of cross product bialgebras in terms of so-called Hopf data. With the help of Hopf data construction we recover again all known cross product bialgebras as well as new and more general types of cross product bialgebras. We are working in the general setting of braided monoidal categories which allows us to apply our results in particular to the braided category of Hopf bimodules over a Hopf algebra. Majid's double biproduct is seen to be a twisting of a certain tensor product bia...
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Letters in Mathematical Physics, 1992
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Zeitschrift f�r Physik C Particles and Fields, 1992
Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsUq(N), SOq(N) and the... more Inhomogeneous quantum groups corresponding to the homogeneous quantum groupsUq(N), SOq(N) and theq-deformed Lorentz group acting on affine quantum spaces are constructed.
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Communications in Mathematical Physics, 1992
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Communications in Mathematical Physics, 1992
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Classical and Quantum Gravity, 1989
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Acta Applicandae Mathematicae, 1996
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Journal of Physics A: Mathematical and General, 1996
We consider quantized derivation representations 0305-4470/29/11/012/img5 of Hopf algebras 0305-4... more We consider quantized derivation representations 0305-4470/29/11/012/img5 of Hopf algebras 0305-4470/29/11/012/img6 in some associative algebras 0305-4470/29/11/012/img7. The algebra of quantized differential operators 0305-4470/29/11/012/img8 of bialgebra representations, which are special representations by derivations, are constructed. For the quantum enveloping algebras 0305-4470/29/11/012/img9 of Lie algebras associated with the root systems 0305-4470/29/11/012/img10, 0305-4470/29/11/012/img11, 0305-4470/29/11/012/img12 and 0305-4470/29/11/012/img13 we define a deformation 0305-4470/29/11/012/img14 of the exterior algebra of forms, and by applying the above results it is shown that the quantized adjoint representation of 0305-4470/29/11/012/img9 induces a bialgebra representation 0305-4470/29/11/012/img16 of 0305-4470/29/11/012/img9 in 0305-4470/29/11/012/img14. The resulting algebra of differential operators 0305-4470/29/11/012/img19 is a deformation of the standard Koszul complex of Lie algebras. It admits an exterior derivative which is, in particular, a 0305-4470/29/11/012/img9-module morphism. Hence cohomology groups of 0305-4470/29/11/012/img9 relative to some 0305-4470/29/11/012/img9-module 0305-4470/29/11/012/img23 can be constructed.
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This paper proposes a tool for dimension reduction where the dimension of the original space is r... more This paper proposes a tool for dimension reduction where the dimension of the original space is reduced: a Principal Loading Analysis (PLA). PLA is a tool to reduce dimensions by discarding variables. The intuition is that variables are dropped which distort the covariance matrix only by a little. Our method is introduced and an algorithm for conducting PLA is provided. Further, we give bounds for the noise arising in the sample case.
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Abstract. Generalizing the notion of continuous Hilbert space representations of compact topologi... more Abstract. Generalizing the notion of continuous Hilbert space representations of compact topological groups we define unitary continuous correpresentations of C ∗-completions of compact quantum group Hopf algebras on arbitrary Hilbert spaces. It is proved that the unitary continuous correpresentations decompose in finite dimensional irreducible correpresentations. 1
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For an action α of a group G on an algebra R (over C), the crossed product R ×α G is the vector s... more For an action α of a group G on an algebra R (over C), the crossed product R ×α G is the vector space of R-valued functions with finite support in G, together with the twisted convolution product given by (ξη)(p) = ∑ ξ(q)αq(η(q −1 p)) q∈G where p ∈ G. This construction has been extended to the theory of Hopf algebras. Given an action of a Hopf algebra A on an algebra R, it is possible to make the tensor product R ⊗ A into an algebra by using a twisted product, involving the action. In this case, the algebra is called the smash product and denoted by R#A. In the group case, the action α of G on R yields an action of the group algebra CG as a Hopf algebra on R and the crossed R ×α G coincides with the smash product R#CG. In this paper we extend the theory of actions of Hopf algebras to actions of multiplier Hopf algebras. We also construct the smash product and we obtain results very similar as in the original situation for Hopf algebras. The main result in the paper is a duality theo...
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This is the central part of a series of three articles on cross product bialgebras. We present a ... more This is the central part of a series of three articles on cross product bialgebras. We present a universal theory of cross product bialgebras with cocycles and dual cocycles. The construction provides an equivalent (co-)modular cocyclic formulation. All known constructions as for instance bi- or smash, doublecross and bicross product bialgebras as well as double biproduct bialgebras and bicrossed or cocycle bicross product bialgebras are now united within a single theory. Furthermore our construction bears various novel types of cross product bialgebras.
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In the article we investigate balanced categories and balanced Hopf algebras and discuss applicat... more In the article we investigate balanced categories and balanced Hopf algebras and discuss applications in knot theory. In the first part we consider balanced categories and balanced Hopf algebras as well as ribbon and sovereign categories and Hopf algebras. Sovereign categories have been introduced in [9], sovereign Hopf algebras have been studied in [3]. From the reconstruction theoretical point of view they are the natrual objects in relation with sovereign categories [3]. We will generalize known results and find new relations between sovereign and ribbon structures. The notion of strong sovereignity will be introduced in order to formulate the one-to-one correspondence of ribbon and strong sovereign Hopf algebras. The result will be used to find equivalent conditions for the twist of a balanced Hopf algebra to be a ribbon twist. For every quasitriangular bialgebra a corresponding balanced bialgebra will be built which will be used to construct an example of a balanced category re...
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arXiv: Quantum Algebra, 1997
AbstractBraided non-commutative differential geometry is studied. In particular we investigate the... more AbstractBraided non-commutative differential geometry is studied. In particular we investigate the theory of (bi-covariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopfbimodules in braided categories are used to construct higher order bicovariant differential calculi over braidedHopf algebras out of first order ones. These graded objects are shown to be braided differential Hopf alge-bras with universal bialgebra properties. The article especially extends Woronowicz’s results on (bicovariant)differential calculi to the braided non-commutative case. Keywords: Braided category, Hopf algebra, Hopf bimodule, Differential calculus 1991 Mathematics Subject Classification: 16W30, 18D10, 18E10, 17B37 Introduction Differential geometry and group theory interact very fruitfully within the theory of Lie groups. Tangent Liealgebras, invariant differential forms, infinitesimal representations, principal bundles, gauge theory, etc. emergedout of this inter...
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Braided non-commutative differential geometry is studied. In particular we investigate the theory... more Braided non-commutative differential geometry is studied. In particular we investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided categories are used to construct higher order bicovariant differential calculi over braided Hopf algebras out of first order ones. These graded objects are shown to be braided differential Hopf algebras with universal bialgebra properties. The article especially extends Woronowicz’s results on (bicovariant) differential calculi to the braided non-commutative case.
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Braided non-commutative differential geometry is studied. We investigate the theory of (bicovaria... more Braided non-commutative differential geometry is studied. We investigate the theory of (bicovariant) differential calculi in braided abelian categories. Previous results on crossed modules and Hopf bimodules in braided categories are used to construct higher order bicovariant differential calculi over braided Hopf algebras out of first order ones. These graded objects are shown to be braided differential Hopf algebras with universal bialgebra properties. The article extends Woronowicz’s results on (bicovariant) differential calculi to the braided non-commutative case.
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Papers by Bernhard Drabant