arXiv subjects

Shahn Majid

Publications and source records attributed to Shahn Majid.

At least 91 records · Page 5Linked to original sources

arXiv2007

Matched pairs approach to set-theoretic solutions of the Yang-Baxter equation

We study set-theoretic solutions $(X,r)$ of the Yang-Baxter equations on a set $X$ in terms of the induced left and right actions of $X$ on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of abstract matched pair properties of the associated monoid $S(X,r)$ and we show that $r$ extends as a solution $(S(X,r),r_S)$. Finally, we study extensions of solutions both directly and in terms of matched pairs of their associated monoids. We also prove several general results about matched pairs of monoids $S$ of the required type, including iterated products $S\bowtie S\bowtie S$ equivalent to $r_S$ a solution, and extensions $(S\bowtie T,r_{S\bowtie T})$. Examples include a general `double' construction $(S\bowtie S,r_{S\bowtie S})$ and some concrete extensions, their actions and graphs based on small sets.

math.QA↗

arXiv2006

Bicrossproduct approach to the Connes-Moscovici Hopf algebra

We give a rigorous proof that the (codimension one) Connes-Moscovici Hopf algebra H_CM is isomorphic to a bicrossproduct Hopf algebra linked to a group factorisation of the group of positively-oriented diffeomorphisms of the real line. We construct a second bicrossproduct U_CM equipped with a nondegenerate dual pairing with H_CM. We give a natural quotient Hopf algebra of H_CM and Hopf subalgebra of U_CM which again are in duality. All these Hopf algebras arise as deformations of commutative or cocommutative Hopf algebras that we describe in each case. Finally we develop the noncommutative differential geometry of the quotient of H_CM by studying covariant first order differential calculi of small dimension over this algebra.

math.QA↗

arXiv2005

Noncommutative Model with Spontaneous Time Generation and Planckian bound

We illustrate the thesis that if time did not exist, we would have to create it if space is noncommutative, and extend functions by something like Schroedinger's equation. We propose that the phenomenon is a somewhat general mechanism within noncommutative geometry for `spontaneous time generation'. We show in detail how this works for the $su_2$ algebra $[x_i,x_j]=2\imathλε_{ij}{}^kx_k$ as noncommutative space, by explicitly adjoining the forced time variable. We find the natural induced noncommutative Schroedingers equation and show that it has the correct classical limit for a particle of some mass $m\ne 0$, which is generated as a second free parameter by the theory. We show that plane waves exist provided $|\vec p|< π/2λ$, i.e. we find a Planckian bound on spatial momentum. We also propose dispersion relations $|{\del p^0\over\del \vec p}|=|\tan(λ|\vec p|)|/mλ$ for the model and explore some elements of the noncommutative geometry. The model is complementary to our previous bicrossproduct one.

hep-th↗

arXiv2003

Noncommutative Differentials and Yang-Mills on Permutation Groups S_N

We study noncommutative differential structures on the group of permutations $S_N$, defined by conjugacy classes. The 2-cycles class defines an exterior algebra $Λ_N$ which is a super analogue of the Fomin-Kirillov algebra $\CE_N$ for Schubert calculus on the cohomology of the $GL_N$ flag variety. Noncommutative de Rahm cohomology and moduli of flat connections are computed for $N<6$. We find that flat connections of submaximal cardinality form a natural representation associated to each conjugacy class, often irreducible, and are analogues of the Dunkl elements in $\CE_N$. We also construct $Λ_N$ and $\CE_N$ as braided groups in the category of $S_N$-crossed modules, giving a new approach to the latter that makes sense for all flag varieties.

math.QA↗

arXiv2002

Noncommutative Ricci curvature and Dirac operator on $C_q[SL_2]$ at roots of unity

We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group $C_q[SL_2]$, using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for $q$ an odd $r$'th root of unity that its eigenvalues are given by $q$-integers $[m]_q$ for $m=0,1,...,r-1$ offset by the constant background curvature. We fully solve the Dirac equation for $r=3$.

math.QA↗

arXiv2001

Electromagnetism and Gauge Theory on the Permutation Group $S_3$

Using noncommutative geometry we do U(1) gauge theory on the permutation group $S_3$. Unlike usual lattice gauge theories the use of a nonAbelian group here as spacetime corresponds to a background Riemannian curvature. In this background we solve spin 0, 1/2 and spin 1 equations of motion, including the spin 1 or `photon' case in the presence of sources, i.e. a theory of classical electromagnetism. Moreover, we solve the U(1) Yang-Mills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli spaces of flat connections. We show that the Yang-Mills action has a simple form in terms of Wilson loops in the permutation group, and we discuss aspects of the quantum theory.

hep-th↗

arXiv2000

Quantum geometry of algebra factorisations and coalgebra bundles

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices $M_2(\C)=\C\Z_2\cdot\C\Z_2$. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct $q$-monopoles on all the Podleś quantum spheres $S^2_{q,s}$.

math.QA↗

arXiv1999

Waves on Noncommutative Spacetime and Gamma-Ray Bursts

Quantum group Fourier transform methods are applied to the study of processes on noncommutative Minkowski spacetime $[x^i,t]=\imathλx^i$. A natural wave equation is derived and the associated phenomena of {\it in vacuo} dispersion are discussed. Assuming the deformation scale $λ$ is of the order of the Planck length one finds that the dispersion effects are large enough to be tested in experimental investigations of astrophysical phenomena such as gamma-ray bursts. We also outline a new approach to the construction of field theories on the noncommutative spacetime, with the noncommutativity equivalent under Fourier transform to non-Abelianness of the `addition law' for momentum in Feynman diagrams. We argue that CPT violation effects of the type testable using the sensitive neutral-kaon system are to be expected in such a theory.

hep-th↗

arXiv1998

Twisting of quantum differentials and the Planck scale Hopf algebra

We show that the crossed modules and bicovariant different 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-quantisation of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra $\C[x]\bicross_{\hbar,\grav}\C[p]$. This is a quantum group which has an $\hbar\to 0$ limit as the functions on a classical but non-Abelian group and a $\grav\to 0$ 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 $\grav\to 0$ and is therefore not visible in flat space quantum mechanics alone.

math.QA↗

arXiv1998

Line bundles on quantum spheres

The (left coalgebra) line bundle associated to the quantum Hopf fibration of any quantum two-sphere is shown to be a finitely generated projective module. The corresponding projector is constructed and its monopole charge is computed. It is shown that the Dirac $q$-monopole connection on any quantum two-sphere induces the Grassmannian connection built with this projector.

math.QA↗

arXiv1997

Quantum differentials and the q-monopole revisited

The q-monopole bundle introduced previously is extended to a general construction for quantum group bundles with non-universal differential calculi. We show that the theory applies to several other classes of bundles as well, including bicrossproduct quantum groups, the quantum double and combinatorial bundles associated to covers of compact manifolds.

q-alg↗

arXiv1994

Introduction to Braided Geometry and $q$-Minkowski Space

We present a systematic introduction to the geometry of linear braided spaces. These are versions of $\R^n$ in which the coordinates $x_i$ have braid-statistics described by an R-matrix. From this starting point we survey the author's braided-approach to $q$-deformation: braided differentiation, exponentials, Gaussians, integration and forms, i.e. the basic ingredients for $q$-deformed physics are covered. The braided approach includes natural $q$-Euclidean and $q$-Minkowski spaces in R-matrix form.

hep-th↗

arXiv1994

*-Structures on Braided Spaces

$*$-structures on quantum and braided spaces of the type defined via an R-matrix are studied. These include $q$-Minkowski and $q$-Euclidean spaces as additive braided groups. The duality between the $*$-braided groups of vectors and covectors is proved and some first applications to braided geometry are made.

hep-th↗

arXiv1994

Duality Principle and Braided Geometry

We give an overview of a new kind symmetry in physics which exists between observables and states and which is made possible by the language of Hopf algebras and quantum geometry. It has been proposed by the author as a feature of Planck scale physics. More recent work includes corresponding results at the semiclassical level of Poisson-Lie groups and at the level of braided groups and braided geometry.

hep-th↗

arXiv1994

Some Remarks on the Quantum Double

We recall the abstract theory of Hopf algebra bicrossproducts and double cross products due to the author. We use it to develop some less-well known results about the quantum double as a twisting, as an extension and as $q$-Lorentz group.

hep-th↗

arXiv1994

Anyonic FRT construction

The Faddeev-Reshetikhin-Takhtajan method to construct matrix bialgebras from non-singular solutions of the quantum Yang-Baxter equation is extended to the anyonic or $\Z_n$-graded case. The resulting anyonic quantum matrices are braided groups in which the braiding is given by a phase factor.

hep-th↗

arXiv1994

q-Epsilon tensor for quantum and braided spaces

The machinery of braided geometry introduced previously is used now to construct the $ε$ `totally antisymmetric tensor' on a general braided vector space determined by R-matrices. This includes natural $q$-Euclidean and $q$-Minkowski spaces. The formalism is completely covariant under the corresponding quantum group such as $\widetilde{SO_q(4)}$ or $\widetilde{SO_q(1,3)}$. The Hodge $*$ operator and differentials are also constructed in this approach.

hep-th↗

arXiv1994

Algebraic {$q$}-Integration and Fourier Theory on Quantum and Braided Spaces

We introduce an algebraic theory of integration on quantum planes and other braided spaces. In the one dimensional case we obtain a novel picture of the Jackson $q$-integral as indefinite integration on the braided group of functions in one variable $x$. Here $x$ is treated with braid statistics $q$ rather than the usual bosonic or Grassmann ones. We show that the definite integral $\int x$ 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 our algebraic integration with a representation then leads to ordinary numbers for the integral. We also use our integration to develop a full theory of $q$-Fourier transformation $F$. We use the braided addition $Δx=x\otimes 1+1\otimes x$ and braided-antipode $S$ to define a convolution product, and prove a convolution theorem. We prove also that $F^2=S$. We prove the analogous results on any braided group, including integration and Fourier transformation on quantum planes associated to general R-matrices, including $q$-Euclidean and $q$-Minkowski spaces.

hep-th↗