arXiv subjects

S. Majid

Publications and source records attributed to S. Majid.

At least 19 recordsLinked to original sources

arXiv2022

Geometric Dirac operator on noncommutative torus and $M_2(\Bbb C)$

We solve for quantum-geometrically realised spectral triples or `Dirac operators' on the noncommutative torus $\Bbb C_\theta[T^2]$ and on the algebra $M_2(\Bbb C)$ of $2\times 2$ matrices with their standard quantum metrics and associated quantum Levi-Civita connections. For $\Bbb C_\theta[T^2]$, we obtain an even standard spectral triple but now uniquely determined by full geometric realisability. For $M_2(\Bbb C)$, we are forced to the flat quantum Levi-Civita connection and again obtain a natural fully geometrically realised even spectral triple. In both case there is also an odd spectral triple for a different choice of a sign parameter. We also consider an alternate quantum metric on $M_2(\Bbb C)$ with curved quantum Levi-Civita connection and find a natural 2-parameter of almost spectral triple in that $D$ fails to be antihermitian. In all cases, we split the construction into a local tensorial level related to the quantum geometry, where we classify the results more broadly, and the further requirements relating to the Hilbert space structure. We also illustrate the Lichnerowicz formula for $D^2$ which applies in the case of a full geometric realisation.

math.QA↗

arXiv2022

Quantum gravity on finite spacetimes and dynamical mass

We review quantum gravity model building using the new formalism of `quantum Riemannian geometry' to construct this on finite discrete spaces and on fuzzy ones such as matrix algebras. The formalism starts with a `differential structure' as a bimodule $Ω^1$ of differential 1-forms over the coordinate algebra $A$, which could be noncommutative. A quantum metric is a noncommutative rank (0,2) tensor in $Ω^1\otimes_AΩ^1$, for which we then search for a quantum Levi-Civita connection (this is no longer unique or guaranteed). We outline the three models which have so far been constructed in this formalism, commonalities among them, and issues going forward. One commonality is a uniform nonzero variance of metric expectation values in the strong gravity limit. We also outline and discuss the construction of quantum FLRW cosmology and black-hole backgrounds using quantum Riemannian geometry and other recent results. Among new results, we perform a Kaluza-Klein type analysis where we tensor classical spacetime coordinates with a finite quantum Riemannian geometry and we give an example where a scalar field on the total space appears as a multiplet of scalar fields on spacetime with a spread of dynamically generated masses.

gr-qc↗

arXiv2022

Quantum Riemannian geometry of the discrete interval and q-deformation

We solve for quantum Riemannian geometries on the finite lattice interval $\bullet-\bullet-\cdots-\bullet$ with $n$ nodes (the Dynkin graph of type $A_n$) and find that they are necessarily $q$-deformed with $q=e^{\imath\pi\over n+1}$. This comes out of the intrinsic geometry and not by assuming any quantum group in the picture. Specifically, we discover a novel `boundary effect' whereby, in order to admit a quantum-Levi Civita connection, the `metric weight' at any edge is forced to be greater pointing towards the bulk compared to towards the boundary, with ratio given by $(i+1)_q/(i)_q$ at node $i$, where $(i)_q$ is a $q$-integer. The Christoffel symbols are also q-deformed. The limit $q\to 1$ likewise forces the quantum Riemannian geometry of the natural numbers $\Bbb N$ to have rational metric multiples $(i+1)/i$ in the direction of increasing $i$. In both cases, there is a unique Ricci-scalar flat metric up to normalisation. Elements of quantum field theory and quantum gravity are exhibited for $n=3$ and for the continuum limit of the geometry of $\Bbb N$. The Laplacian for the scalar-flat metric becomes the Airy equation operator ${1\over x}{d^2\over d x^2}$ in so far as a limit exists. Scaling this metric by a conformal factor $e^{\psi(i)}$ gives a limiting Ricci scalar curvature proportional to ${e^{-\psi}\over x}{d^2 \psi\over d x^2}$.

math.QA↗

arXiv2020

Fuzzy and discrete black hole models

Using quantum Riemannian geometry, we solve for a Ricci=0 static spherically-symmetric solution in 4D, with the $S^2$ at each $t,r$ a noncommutative fuzzy sphere, finding a dimension jump with solutions having the time and radial form of a classical 5D Tangherlini black hole. Thus, even a small amount of angular noncommutativity leads to radically different radial behaviour, modifying the Laplacian and the weak gravity limit. We likewise provide a version of a 3D black hole with the $S^1$ at each $t,r$ now a discrete circle $\Bbb Z_n$, with the time and radial form of the inside of a classical 4D Schwarzschild black hole far from the horizon. We study the Laplacian and the classical limit $\Bbb Z_n\to S^1$. We also study the 3D FLRW model on $\Bbb R\times S^2$ with $S^2$ an expanding fuzzy sphere and find that the Friedmann equation for the expansion is the classical 4D one for a closed $\Bbb R\times S^3$ universe.

gr-qc↗

arXiv2020

Quantum gravity on polygons and $\Bbb R\times \Bbb Z_n$ FLRW model

We fully solve the quantum geometry of $\Bbb Z_n$ as a polygon graph with arbitrary metric lengths on the edges, finding a $*$-preserving quantum Levi-Civita connection which is unique for $n\ne 4$. As a first application, we numerically compute correlation functions for Euclideanised quantum gravity on $\Bbb Z_n$ for small $n$. We then study an FLRW model on $\Bbb R\times\Bbb Z_n$, finding the same expansion rate as for the classical flat FLRW model in 1+2 dimensions. We also look at particle creation on $\Bbb R\times \Bbb Z_n$ and find an additional $m=0$ adiabatic no particle creation expansion as well as the particle creation spectrum for a smoothed step expansion.

gr-qc↗

arXiv2020

Digital quantum groups

We find and classify all bialgebras and Hopf algebras or `quantum groups' of dimension $\le 4$ over the field $\Bbb F_2=\{0,1\}$. We summarise our results as a quiver, where the vertices are the inequivalent algebras and there is an arrow for each inequivalent bialgebra or Hopf algebra built from the algebra at the source of the arrow and the dual of the algebra at the target of the arrow. There are 314 distinct bialgebras, and among them 25 Hopf algebras with at most one of these from one vertex to another. We find a unique smallest noncommutative and noncocommutative one, which is moreover self-dual and resembles a digital version of $u_q(sl_2)$. We also find a unique self-dual Hopf algebra in one anyonic variable $x^4=0$. For all our Hopf algebras we determine the integral and associated Fourier transform operator, viewed as a representation of the quiver. We also find all quasitriangular or `universal R-matrix' structures on our Hopf algebras. These induce solutions of the Yang-Baxter or braid relations in any representation.

math.QA↗

arXiv2019

Finite noncommutative geometries related to $F_p[x]$

It is known that irreducible noncommutative differential structures over $\Bbb F_p[x]$ are classified by irreducible monics $m$. We show that the cohomology $H_{\rm dR}^0(\Bbb F_p[x]; m)=\Bbb F_p[g_d]$ if and only if ${\rm Tr}(m)\ne 0$, where $g_d=x^{p^d}-x$ and $d$ is the degree of $m$. This implies that there are ${p-1\over pd}\sum_{k|d, p\nmid k}μ_M(k)p^{d\over k}$ such noncommutative differential structures ($μ_M$ the Möbius function). Motivated by killing this zero'th cohomology, we consider the directed system of finite-dimensional Hopf algebras $A_d=\Bbb F_p[x]/(g_d)$ as well as their inherited bicovariant differential calculi $Ω(A_d;m)$. We show that $A_d=C_d\otimes_χA_1$ a cocycle extension where $C_d=A_d^ψ$ is the subalgebra of elements fixed under $ψ(x)=x+1$. We also have a Frobenius-fixed subalgebra $B_d$ of dimension $\frac{1}{d} \sum_{k | d} ϕ(k) p^\frac{d}{k}$ ($ϕ$ the Euler totient function), generalising Boolean algebras when $p=2$. As special cases, $A_1\cong \Bbb F_p(\Bbb Z/p\Bbb Z)$, the algebra of functions on the finite group $\Bbb Z/p\Bbb Z$, and we show dually that $\Bbb F_p\Bbb Z/p\Bbb Z\cong\Bbb F_p[L]/(L^p)$ for a `Lie algebra' generator $L$ with $e^L$ group-like, using a truncated exponential. By contrast, $A_2$ over $\Bbb F_2$ is a cocycle modification of $\Bbb F_2((\Bbb Z/2\Bbb Z)^2)$ and is a 1-dimensional extension of the Boolean algebra on 3 elements. In both cases we compute the Fourier theory, the invariant metrics and the Levi-Civita connections within bimodule noncommutative geometry.

math.QA↗

arXiv2017

Quasitriangular structure and twisting of the 2+1 bicrossproduct model

We show that the bicrossproduct model $C[SU_2^*]{\blacktriangleright\!\!\triangleleft} U(su_2)$ quantum Poincare group in 2+1 dimensions acting on the quantum spacetime $[x_i,t]=\imathλx_i$ is related by a Drinfeld and module-algebra twist to the quantum double $U(su_2)\ltimes C[SU_2]$ acting on the quantum spacetime $[x_μ,x_ν]=\imathλε_{μνρ}x_ρ$. We obtain this twist by taking a scaling limit as $q\to 1$ of the $q$-deformed version of the above where it corresponds to a previous theory of $q$-deformed Wick rotation from $q$-Euclidean to $q$-Minkowski space. We also recover the twist result at the Lie bialgebra level.

math.QA↗

arXiv2009

Hopf quasigroups and the algebraic 7-sphere

We introduce the notions of Hopf quasigroup and Hopf coquasigroup $H$ generalising the classical notion of an inverse property quasigroup $G$ expressed respectively as a quasigroup algebra $k G$ and an algebraic quasigroup $k[G]$. We prove basic results as for Hopf algebras, such as anti(co)multiplicativity of the antipode $S:H\to H$, that $S^2=\id$ if $H$ is commutative or cocommutative, and a theory of crossed (co)products. We also introduce the notion of a Moufang Hopf (co)quasigroup and show that the coordinate algebras $k[S^{2^n-1}]$ of the parallelizable spheres are algebraic quasigroups (commutative Hopf coquasigroups in our formulation) and Moufang. We make use of the description of composition algebras such as the octonions via a cochain $F$ introduced in \cite{Ma99}. We construct an example $k[S^7]\rtimes\Z_2^3$ of a Hopf coquasigroup which is noncommutative and non-trivially Moufang. We use Hopf coquasigroup methods to study differential geometry on $k[S^7]$ including a short algebraic proof that $S^7$ is parallelizable. Looking at combinations of left and right invariant vector fields on $k[S^7]$ we provide a new description of the structure constants of the Lie algebra $g_2$ in terms of the structure constants $F$ of the octonions. In the concluding section we give a new description of the $q$-deformation quantum group $\C_q[S^3]$ regarded trivially as a Moufang Hopf coquasigroup (trivially since it is in fact a Hopf algebra) but now in terms of $F$ built up via the Cayley-Dickson process.

math.QA↗

arXiv2009

Nonassociative Riemannian Geometry by Twisting

Many quantum groups and quantum spaces of interest can be obtained by cochain (but not cocycle) twist from their corresponding classical object. This failure of the cocycle condition implies a hidden nonassociativity in the noncommutative geometry already known to be visible at the level of differential forms. We extend the cochain twist framework to connections and Riemannian structures and provide examples including twist of the $S^7$ coordinate algebra to a nonassociative hyperbolic geometry in the same category as that of the octonions.

math.QA↗

arXiv2009

Bicrossproduct Hopf quasigroups

We recall the notion of Hopf quasigroups introduced previously. We construct a bicrossproduct Hopf quasigroup $kM\bicross k(G)$ from every group $X$ with a finite subgroup $G\subset X$ and IP quasigroup transversal $M\subset X$ subject to certain conditions. We identify the octonions quasigroup $G_O$ as transversal in an order 128 group $X$ with subgroup $Z_2^3$ and hence obtain a Hopf quasigroup $kG_O\lcocross k(Z_2^3)$ as a particular case of our construction.

math.QA↗

arXiv2009

*-Compatible Connections in Noncommutative Riemannian Geometry

We develop the formalism for noncommutative differential geometry and Riemmannian geometry to take full account of the *-algebra structure on the (possibly noncommutative) coordinate ring and the bimodule structure on the differential forms. We show that *-compatible bimodule connections lead to braid operators $σ$ in some generality (going beyond the quantum group case) and we develop their role in the exterior algebra. We study metrics in the form of Hermitian structures on Hilbert *-modules and metric compatibility in both the usual and a cotorsion form. We show that the theory works well for the quantum group $C_q[SU_2]$ with its 3D calculus, finding for each point of a 3-parameter space of covariant metrics a unique `Levi-Civita' connection deforming the classical one and characterised by zero torsion, metric-preservation and *-compatibility. Allowing torsion, we find a unique connection with classical limit that is metric-preserving and *-compatible and for which $σ$ obeys the braid relations. It projects to a unique `Levi-Civita' connection on the quantum sphere. The theory also works for finite groups and in particular for the permutation group $S_3$ where we find somewhat similar results.

math.QA↗

arXiv2007

Bar categories and star operations

We introduce the notion of `bar category' by which we mean a monoidal category equipped with additional structure formalising the notion of complex conjugation. Examples of our theory include bimodules over a $*$-algebra, modules over a conventional $*$-Hopf algebra and modules over a more general object which call a `quasi-$*$-Hopf algebra' and for which examples include the standard quantum groups $u_q(g)$ at $q$ a root of unity (these are well-known not to be a usual $*$-Hopf algebra). We also provide examples of strictly quasiassociative bar categories, including modules over `$*$-quasiHopf algebras' and a construction based on finite subgroups $H\subset G$ of a finite group. Inside a bar category one has natural notions of `$\star$-algebra' and `unitary object' therefore extending these concepts to a variety of new situations. We study braidings and duals in bar categories and $\star$-braided groups (Hopf algebras) {\em in} braided-bar categories. Examples include the transmutation $B(H)$ of a quasitriangular $*$-Hopf algebra and the quantum plane $C_q^2$ at certain roots of unity $q$ in the bar category of $\widetilde{u_q(su_2)}$-modules. We use our methods to provide a natural quasi-associative $C^*$-algebra structure on the octonions ${\mathbb O}$ and on a coset example. In the appendix we extend the Tannaka-Krein reconstruction theory to bar categories in relation to $*$-Hopf algebras.

math.QA↗

arXiv2007

Noncommutative Harmonic Analysis, Sampling Theory and the Duflo Map in 2+1 Quantum Gravity

We show that the $\star$-product for $U(su_2)$, group Fourier transform and effective action arising in [1] in an effective theory for the integer spin Ponzano-Regge quantum gravity model are compatible with the noncommutative bicovariant differential calculus, quantum group Fourier transform and noncommutative scalar field theory previously proposed for 2+1 Euclidean quantum gravity using quantum group methods in [2]. The two are related by a classicalisation map which we introduce. We show, however, that noncommutative spacetime has a richer structure which already sees the half-integer spin information. We argue that the anomalous extra `time' dimension seen in the noncommutative geometry should be viewed as the renormalisation group flow visible in the coarse-graining in going from $SU_2$ to $SO_3$. Combining our methods we develop practical tools for noncommutative harmonic analysis for the model including radial quantum delta-functions and Gaussians, the Duflo map and elements of `noncommutative sampling theory'. This allows us to understand the bandwidth limitation in 2+1 quantum gravity arising from the bounded $SU_2$ momentum and to interpret the Duflo map as noncommutative compression. Our methods also provide a generalised twist operator for the $\star$-product.

hep-th↗

arXiv2007

Quantisation of twistor theory by cocycle twist

We present the main ingredients of twistor theory leading up to and including the Penrose-Ward transform in a coordinate algebra form which we can then `quantise' by means of a functorial cocycle twist. The quantum algebras for the conformal group, twistor space CP^3, compactified Minkowski space CMh and the twistor correspondence space are obtained along with their canonical quantum differential calculi, both in a local form and in a global *-algebra formulation which even in the classical commutative case provides a useful alternative to the formulation in terms of projective varieties. We outline how the Penrose-Ward transform then quantises. As an example, we show that the pull-back of the tautological bundle on CMh pulls back to the basic instanton on S^4\subset CMh and that this observation quantises to obtain the Connes-Landi instanton on θ-deformed S^4 as the pull-back of the tautological bundle on our θ-deformed CMh. We likewise quantise the fibration CP^3--> S^4 and use it to construct the bundle on θ-deformed CP^3 that maps over under the transform to the θ-deformed instanton.

math.QA↗

arXiv2006

Algebraic approach to quantum gravity II: noncommutative spacetime

We provide a self-contained introduction to the quantum group approach to noncommutative geometry as the next-to-classical effective geometry that might be expected from any successful quantum gravity theory. We focus particularly on a thorough account of the bicrossproduct model noncommutative spacetimes of the form [t,x_i]=i λx_i and the correct formulation of predictions for it including a variable speed of light. We also study global issues in the Poincaré group in the model with the 2D case as illustration. We show that any off-shell momentum can be boosted to infinite negative energy by a finite Lorentz transformaton.

hep-th↗

arXiv2006

Algebraic approach to quantum gravity III: noncommmutative Riemannian geometry

This is a self-contained introduction to quantum Riemannian geometry based on quantum groups as frame groups, and its proposed role in quantum gravity. Much of the article is about the generalisation of classical Riemannian geometry that arises naturally as the classical limit; a theory with nonsymmetric metric and a skew version of metric compatibilty. Meanwhile, in quantum gravity a key ingredient of our approach is the proposal that the differential structure of spacetime is something that itself must be summed over or `quantised' as a physical degree of freedom. We illustrate such a scheme for quantum gravity on small finite sets.

hep-th↗

arXiv2005

Semi-classical differential structures

We semiclassicalise the standard notion of differential calculus in noncommutative geometry on algebras and quantum groups. We show in the symplectic case that the infinitesimal data for a differential calculus is a symplectic connection, and interpret its curvature as lowest order nonassociativity of the exterior algebra. Semiclassicalisation of the noncommutative torus provides an example with zero curvature. In the Poisson-Lie group case we study left-covariant infinitesimal data in terms of partially defined preconnections. We show that the moduli space of bicovariant infinitesimal data for quasitriangular Poisson-Lie groups has a canonical reference point which is flat in the triangular case. Using a theorem of Kostant, we completely determine the moduli space when the Lie algebra is simple: the canonical preconnection is the unique point for other than sl_n, n>2, when the moduli space is 1-dimensional. We relate the canonical preconnection to Drinfeld twists and thereby quantise it to a super coquasi-Hopf exterior algebra. We also discuss links with Fedosov quantisation.

math.QA↗