κ-Poincaré–Hopf algebra and Hopf algebroid structure of phase space from twist

We unify κ-Poincaré algebra and κ-Minkowski spacetime by embedding them into quantum phase space. The quantum phase space has Hopf algebroid structure to which we apply the twist in order to get κ-deformed Hopf algebroid structure and κ-deformed Heisenberg algebra. We explicitly construct κ-Poincaré–Hopf algebra and κ-Minkowski spacetime from twist. It is outlined how this construction can be extended to κ-deformed super-algebra including exterior derivative and forms. Our results are relevant for constructing physical theories on noncommutative spacetime by twisting Hopf algebroid phase space structure.     

Generalization of Drinfeld Quantum Affine Algebras

Drinfeld gave a current realization of the quantum affine algebras as a Hopf algebra with a simple comultiplication for the quantum current operators. In this Letter, we will present a generalization of such a realization of quantum Hopf algebras. As a special case, we will choose the structure functions for this algebra to be elliptic functions to derive certain elliptic quantum groups as a Hopf algebra, which degenerates into quantum affine algebras if we take certain degeneration of the structure functions.     

On the quantum super Virasoro algebra

The quantum super-algebra structure on the deformed super Virasoro algebra is investigated. More specifically we established the possibility of defining a nontrivial Hopf super-algebra on both one and two-parameters deformed super Virasoro algebras.     

Quasi-integrable and superintegrable systems from a ‘deformed-mass’ two-photon algebra

A quantum deformation of the two-photon (or Schrödinger) Lie algebra is introduced in order to construct newn-dimensional classical Hamiltonian systems which have (n?2) functionally independent integrals of motion in involution; we say that such Hamiltonians define quasi-integrable systems. Furthermore, Hopf subalgebras of this quantum two-photon algebra (quantum extended Galilei and harmonic oscillator algebras) provide another set of (n?1) integrals of motion for Hamiltonians defined on these Hopf subalgebras, so that they lead to superintegrable systems.     

Duality on the quantum space(3) with two parameters

In this study, we introduce a dual Hopf algebra in the sense of Sudbery for the quantum space(3) whose coordinates satisfy the commutation relations with two parameters and we show that the dual algebra is isomorphic to the quantum Lie algebra corresponding to the Cartan-Maurer right invariant differential forms on the quantum space(3). We also observe that the quantum Lie algebra generators are commutative as those of the undeformed Lie algebra and the deformation becomes apparent when one studies the Leibniz rules for the generators.     

Twisted Heisenberg Doubles

We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted pairing. We state a Stone–von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction.     

Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra

We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisations of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra . This is a quantum group which has an limit as the functions on a classical but non-Abelian group and a limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality-like self-duality. The noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone. Received: 28 October 1998 / Accepted: 7 March 1999     

Quantum mechanics,knot theory,and quantum doubles

In previous papers we have described quantum mechanics as a matrix symplectic geometry and showed the existence of a braiding and Hopf algebra structure behind our lattice quantum phase space. The first aim of this work is to give the defining commutation relations of the quantum Weyl-Schwinger-Heisenberg group associated with our ℜ-matrix solution. The second aim is to describe the knot formalism at work behind the matrix quantum mechanics. In this context, the quantum mechanics of a particle-antiparticle system (pˉp) moving in the quantum phase space is viewed as a quantum double.     

Sheaves on Fibered Threefolds and Quiver Sheaves

We analyse the Dirichlet convolution ring of arithmetic number theoretic functions. It turns out to fail to be a Hopf algebra on the diagonal, due to the lack of complete multiplicativity of the product and coproduct. A related Hopf algebra can be established, which however overcounts the diagonal. We argue that the mechanism of renormalization in quantum field theory is modelled after the same principle. Singularities hence arise as a (now continuously indexed) overcounting on the diagonals. Renormalization is given by the map from the auxiliary Hopf algebra to the weaker multiplicative structure, called Hopf gebra, rescaling the diagonals.     

Braided Identities, Quantum Groups, and Clifford Algebras

A Lie algebra in a braided category is constructed within the algebra structure of the positive part of the Drinfeld—Jimbo quantum group of type An such that its universal enveloping algebra is a braided Hopf algebra. Similarities with Clifford algebras are discussed.     

Differential geometry of the Lie algebra of the quantum plane

We present a differential calculus on the extension of the quantum plane obtained by considering that the (bosonic) generator x is invertible and by working with polynomials in ln x instead of polynomials in x. We construct the quantum Lie algebra associated with this extension and obtain its Hopf algebra structure and its dual Hopf algebra.     

The Hopf Algebra Structure of GL(1, Hq) and the Isomorphism between SPq(1)and SUq(2)

In this Letter, we introduce the Hopf algebra structure of the quantum quaternionic group GL(1, H₁) and discuss the isomorphism between the quantum symplectic group SP_q(1) and the quantum unitary group SU_q(2).     

Associated quantum vector bundles and symplectic structure on a quantum space

We define a quantum generalization of the algebra of functions over an associated vector bundle of a principal bundle. Here the role of a quantum principal bundle is played by a Hopf-Galois extension. Smash products of an algebra times a Hopf algebra H are particular instances of these extensions, and in these cases we are able to define a differential calculus over their associated vector bundles without requiring the use of a (bicovariant) differential structure over H. Moreover, if H is coquasitriangular, it coacts naturally on the associated bundle, and the differential structure is covariant.We apply this construction to the case of the finite quotient of the SL _q(2) function Hopf algebra at a root of unity (q ³ = 1) as the structure group, and a reduced 2-dimensional quantum plane as both the base manifold and fibre, getting an algebra which generalizes the notion of classical phase space for this quantum space. We also build explicitly a differential complex for this phase space algebra, and find that levels 0 and 2 support a (co)representation of the quantum symplectic group. On this phase space we define vector fields, and with the help of the Sp _q structure we introduce a symplectic form relating 1-forms to vector fields. This leads naturally to the introduction of Poisson brackets, a necessary step to do classical mechanics on a quantum space, the quantum plane.     

On Kreimer's Hopf algebra structure of Feynman graphs

We reinvestigate Kreimer's Hopf algebra structure of perturbative quantum field theories with a special emphasis on overlapping divergences. Kreimer first disentangles overlapping divergences into a linear combination of disjoint and nested ones and then tackles that linear combination by the Hopf algebra operations. We present a formulation where the Hopf algebra operations are directly defined on any type of divergence. We explain the precise relation to Kreimer's Hopf algebra and obtain thereby a characterization of their primitive elements. Received: 9 July 1998 / Revised version: 21 September 1998 / Published online: 19 November 1998     

Quantum group gauge theory on quantum spaces

We construct quantum group-valued canonical connections on quantum homogeneous spaces, including aq-deformed Dirac monopole on the quantum sphere of Podles with quantum differential structure coming from the 3D calculus of Woronowicz onSU _q (2). The construction is presented within the setting of a general theory of quantum principal bundles with quantum group (Hopf algebra) fibre, associated quantum vector bundles and connection one-forms. Both the base space (spacetime) and the total space are non-commutative algebras (quantum spaces).Supported by St. John's College, Cambridge and KBN grant 202189101     

Hopf-type deformed oscillators,their quantum double and a q-deformed Calogero-Vasiliev algebra

A q-deformed oscillator Hopf algebra is presented and the quantum double construction is carried out to obtain an R-matrix. Investigation of the algebra's structure and Fock-type representation leads to a new q-deformed Calogero-Vasiliev algebra.     

Relating the approaches to quantised algebras and quantum groups

This paper constructs two representations of the quantum groupU _q g' by exploiting its quotient structure and the quantum double construction. Here the quantum group is taken as the dual to the quantised algebraU _q g, a one parameter deformation of the universal enveloping algebra of the Lie algebra g, as in Drinfel'd [6] and Jimbo [10]. From the two representations, the Hopf structure of the quantised algebraU _q g is reexpressed in a matrix format. This is the very structure given by Faddeev et al. [7], in their approach to defining quantum groups and quantised algebras via the quantisation of the function space of the associated Lie group to g.Supported by a SERC studentship     

Construction of a new quantum double and new solutions to the Yang-Baxter equation

A new quantum double is established from a new Hopf algebra and a new kind of quantum R-matrix is obtained.     

Direct Sum and Hyperbolic Complexification of Hopf Algebras

The problem of how to obtain a hyperboliccomplexification of a Hopf algebra from two known Hopfalgebras is discussed. We prove that this Hopf algebrais isomorphic to a direct sum of the two known Hopf algebras, and that therefore this direct sum isalso a Hopf algebra. In particular, the result can beapplied to quantum enveloping algebras.