Shahn Majid

Bicrossproduct structure of κ-Poincare group and non-commutative geometry

Abstract We show that the κ -deformed Poincare quantum algebra proposed for particle physics has the structure of a Hopf algebra bicrossproduct U(so (1, 3)) T . The algebra is a semidirect product of the classical Lorentz group so (1,3) acting in a formed way on the momentum sector T . The novel feature is that the coalgebra is also semidirect, with a backreaction of the momentum sector on the Lorentz rotations. Using this, we show that the κ -Poincare acts covariantly on a κ -Minkowski space, which we introduce. It turns out necessarily to be deformed and non-commutative. We also connect this algebra with a previous approach to Planck scale physics.

Quasitriangular Hopf Algebras and {Yang-Baxter} Equations

This is an informal introduction to the theory of quasitriangular Hopf algebras and its connections with physics. Basic properties and applications of Hopf algebras and Yang-Baxter equations are reviewed, with the quantum group Uq(sl2) as a frequent example. The development builds up to the representation theory of quasitriangular Hopf algebras. Much of the abstract representation theory is new, including a formula for the rank of a representation.

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 onSUq(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).

Examples of braided groups and braided matrices

Matrix braided groups are developed as an analog of the ‘‘coordinate functions’’ on a group or supergroup. The ±1 in the super case is replaced by braid statistics. There are braided group analogs of all the classical simple Lie groups as well as braided matrix groups and braided matrices B(R) for every regular solution R of the quantum Yang–Baxter equations. A direct verification of B(R) is provided and some of the simplest examples are computed in detail.

Hopf algebras for physics at the Planck scale

Applies ideas of non-commutative geometry to reformulate the classical and quantum mechanics of a particle moving on a homogeneous spacetime. The reformulation maintains an interesting symmetry between observables and states in the form of a Hopf algebra structure. In the simplest example both the dynamics and quantum mechanics are completely determined by the Hopf algebra consideration. The simplest example is a two-parameter algebra-co-algebra KAB which is the unique Hopf algebra extension, such as is possible, of the self-dual Hopf algebra of functions on flat phase space C(R*R). In the limit (A=0,B) the author recovers functions on a curved phase space with curvature proportional to B2 and in another limit (A= infinity , B= infinity ), A/B=h(cross), the author recovers quantum mechanics on R>or=0 with an absorbing wall at the origin. The algebra in this way corresponds to a toy model of quantum mechanics of a particle in one space dimension combined with gravity-like forces. It has an interesting Z2 symmetry interchanging A to or from B, and thereby, in some sense, the quantum element with the geometric element. The compatibility conditions that are solved are a generalisation of the classical Yang-Baxter equations.

BRAIDED MOMENTUM IN THE Q-POINCARE GROUP

The q‐Poincare group of M. Schlieker et al. [Z. Phys. C 53, 79 (1992)] is shown to have the structure of a semidirect product and coproduct B× SOq(1,3) where B is a braided‐quantum group structure on the q‐Minkowski space of four‐momentum with braided‐coproduct Δ_p=p⊗1+1⊗p. Here the necessary B is not a usual kind of quantum group, but one with braid statistics. Similar braided vectors and covectors V(R’), V*(R’) exist for a general R‐matrix. The abstract structure of the q‐Lorentz group is also studied.

QUANTUM AND BRAIDED LIE ALGEBRAS

Abstract We introduce the notion of a braided-Lie algebra consisting of a finite-dimensional vector space L equipped with a bracket [ , ]: L ⊗ L → L and Yang-Baxter operator Ψ: L ⊗ L → L ⊗ L obeying some axioms. We show that such an object has an enveloping braided-bialgebra U (L). We show that every generic R -matrix 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 right-regular action of a braided-Lie algebra L by braided vector fields, the braided-Killing form and the quadratic Casimir associated to L. These constructions recover the relevant notions for usual, colour and super-Lie algebras as special cases. In addition, the standard quantum deformations U q(g) are understood as the enveloping algebras of such underlying braided-Lie algebras with [ , ] on L ⊂ U q(g) given by the quantum adjoint action.

Projective Module Description of the q-Monopole

The Dirac q-monopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The Chern-Connes pairing of cyclic cohomology and K-theory is computed for the winding number -1. The non-triviality of this pairing is used to conclude that the quantum principal Hopf fibration is non-cleft. Among general results, we provide a left-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (Hopf-Galois extensions) their associated covariant derivatives on projective modules.

FREE BRAIDED DIFFERENTIAL CALCULUS, BRAIDED BINOMIAL THEOREM AND THE BRAIDED EXPONENTIAL MAP

Braided differential operators ∂i are obtained by differentiating the addition law on the braided covector spaces introduced previously (such as the braided addition law on the quantum plane). These are affiliated to a Yang–Baxter matrix R. The quantum eigenfunctions expR(x‖v) of the ∂i (braided‐plane waves) are introduced in the free case where the position components xi are totally noncommuting. A braided R‐binomial theorem and a braided Taylor theorem expR(a‖∂)f(x)=f(a+x) are proven. These various results precisely generalize to a generic R‐matrix (and hence to n dimensions) the well‐known properties of the usual one‐dimensional q‐differential and q‐exponential. As a related application, it is shown that the q‐Heisenberg algebra px−qxp=1 is a braided semidirect product C[x]×C[ p] of the braided line acting on itself (a braided Weyl algebra) and similarly for its generalization to an arbitrary R‐ matrix.

Algebraic q‐integration and Fourier theory on quantum and braided spaces

An algebraic theory of integration on quantum planes and other braided spaces is introduced. In the one‐dimensional case a novel picture of the Jackson q‐ integral as indefinite integration on the braided group of functions in one variable x is obtained. Here x is treated with braid statistics q rather than the usual bosonic or Grassmann ones. It is shown that the definite integral ∫x∞−x∞ can also be evaluated algebraically as multiples of the integral of a q‐Gaussian, with x remaining as a bosonic scaling variable associated with the q‐deformation. Further composing the algebraic integration with a representation then leads to ordinary numbers for the integral. Integration is also used to develop a full theory of q‐Fourier transformation F The braided addition Δx=x⊗1+1⊗x and braided‐antipode S is used to define a convolution product, and prove a convolution theorem. It is also proven that F2=S. The analogous results are proven on any braided group, including integration and Fourier transformation on quan...