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Let $R$ be a ring. In \cite{MD4} Mao and Ding defined an special class of $R$-modules that they called \( FP_n \)-projective $R$-modules. In this paper, we give some new characterizations of \( FP_n \)-projective $R$-modules and strong $n$-coherent rings. Some known results are extended and some new characterizations of the \( FP_n \)-injective global dimension in terms of \( FP_n \)-projective $R$-modules are obtained. Using the \( FP_n \)-projective dimension of an $R$-module defined by Ouyang, Duan and Li in \cite{Ouy} we introduce a slightly different \( FP_n \)-projective global dimension over the ring $R$ which measures how far away the ring is from being Noetherian. This dimension agrees with the $(n,0)$-projective global dimension of \cite{Ouy} when the ring in question is strong $n$-coherent.

The unification of quantum mechanics and general relativity has long been elusive. Only recently have empirical predictions of various possible theories of quantum gravity been put to test. The dawn of multi-messenger high-energy astrophysics has been tremendously beneficial, as it allows us to study particles with much higher energies and travelling much longer distances than possible in terrestrial experiments, but more progress is needed on several fronts. A thorough appraisal of current strategies and experimental frameworks, regarding quantum gravity phenomenology, is provided here. Our aim is twofold: a description of tentative multimessenger explorations, plus a focus on future detection experiments. As the outlook of the network of researchers that formed through the COST Action CA18108 "Quantum gravity phenomenology in the multi-messenger approach (QG-MM)", in this work we give an overview of the desiderata that future theoretical frameworks, observational facilities, and data-sharing policies should satisfy in order to advance the cause of quantum gravity phenomenology., Comment: Submitted to CQG for the Focus Issue on "Quantum Gravity Phenomenology in the Multi-Messenger Era: Challenges and Perspectives". Please contact us to express interesst of endorsement of this white paper

The aim of this contribution is twofold. First, we show that when two (or more) different quantum groups share the same noncommutative spacetime, such an 'ambiguity' can be resolved by considering together their corresponding noncommutative spaces of geodesics. In any case, the latter play a mathematical/physical role by themselves and, in some cases, they can be interpreted as deformed phase spaces. Second, we explicitly show that noncommutative spacetimes can be reproduced from 'extended' noncommutative spaces of geodesics which are those enlarged by the time translation generator. These general ideas are described in detail for the $\kappa$-Poincar\'e and $\kappa$-Galilei algebras., Comment: 11 pages. Based on the contribution presented at "The XII International Symposium on Quantum Theory and Symmetries" (QTS12), July 24-28, 2023, Czech Technical University in Prague, Czech Republic. To appear in Journal of Physics: Conference Series

We study the dynamics of the homogeneous and isotropic cosmological background in the recently proposed ``quantum phenomenological gravitational dynamics'', characterised by logarithmic corrections to the Bekenstein entropy. We show that the model admits a family of solutions that are self-accelerating both at early and late times: they approach de Sitter in the future and admit a past attractor corresponding to an inflationary acceleration era. On the other hand, there are no solutions corresponding to a primordial bounce. We also show that asking scalar perturbations to be unaffected by instabilities on observable scales puts stringent constraints on the deviations from general relativity encoded by the model., Comment: 14 pages, 2 figures

Studies of in-vacuo dispersion are the most active area of quantum-gravity phenomenology. The way in which in-vacuo dispersion produces redshift-dependent corrections to the time of flight of astrophysics particles depends on the model-dependent interplay between Planck-scale effects and spacetime curvature/expansion, and we here derive the most general formula for the leading order redshift-dependent correction to the time of flight for the scenario in which relativistic symmetries are deformed at the Planck scale (DSR) for the constant-curvature case. We find that, contrary to the broken symmetries scenario (LIV), where in principle any arbitrary form of redshift dependence could be allowed, for the DSR scenario only linear combinations of three possible forms of redshift dependence are allowed. We also derive a generalization of our results to the FRW case, and discuss some specific combinations of the three forms of redshift dependence whose investigation might deserve priority from the quantum-gravity perspective., Comment: In this updated version we extended the part devoted to the quantitative analysis of our results, and added an appendix with an explanatory application of the slicing procedure

We investigate the implications of decoherence induced by quantum spacetime properties on neutrino oscillation phenomena. We develop a general formalism where the evolution of neutrinos is governed by a Lindblad-type equation and we compute the oscillation damping factor for various models that have been proposed in the literature. Furthermore, we discuss the sensitivity to these effects of different types of neutrino oscillation experiments, encompassing astrophysical, atmospheric, solar, and reactor neutrino experiments. By using neutrino oscillation data from long-baseline reactors and atmospheric neutrino observations, we establish stringent constraints on the energy scale governing the strength of the decoherence induced by stochastic metric fluctuations, amounting to, respectively, $E_{QG}\geq 2.6 \cdot 10^{34}\; \text{GeV}$ and $E_{QG}\geq 2.5\cdot 10^{55}\;\text{GeV}$., Comment: 18 pages, 4 figures; v3 matches the version accepted for publication in PRD

In this paper we give an explicit and pure combinatorial description of the $m$-coloured quivers that appears in the $m$-coloured mutation class of a quiver of type $\mathbb{A}_n$. The $m$-coloured mutation defined by Buan and Thomas in \cite{BT} generalizes the well-known quiver mutation of Fomin and Zelevinsky \cite{FZ}. In particular, our description generalizes a result of Buan and Vatne, \cite{BV}, which we recover when $m=1$.

We discuss the quantum Poincar\'e symmetries of the $\varrho$-Minkowski spacetime, a space characterised by an angular form of noncommutativity. We show that it is possible to give them both a bicrossproduct and a Drinfel'd twist structure. We also obtain a new noncommutative $\star$-product, which is cyclic with respect to the standard integral measure., Comment: 38 pages, including appendices. Minor corrections. Two references added

In this paper we consider codes in $\mathbb{F}_q^{s\times r}$ with packing radius $R$ regarding the NRT-metric (i.e. when the underlying poset is a disjoint union of chains with the same length) and we establish necessary condition on the parameters $s,r$ and $R$ for the existence of perfect codes. More explicitly, for $r,s\geq 2$ and $R\geq 1$ we prove that if there is a non-trivial perfect code then $(r+1)(R+1)\leq rs$. We also explore a connection to the knapsack problem and establish a correspondence between perfect codes with $r>R$ and those with $r=R$. Using this correspondence we prove the non-existence of non-trivial perfect codes also for $s=R+2$., Comment: This new version contains a correction of Proposition 2 of Section 4. We also added relevant references

We perform a comprehensive study of the signatures of Lorentz violation in electrodynamics on the Cosmic Microwave Background (CMB) anisotropies. In the framework of the minimal Standard Model Extension (SME), we consider effects generated by renormalizable operators, both CPT-odd and CPT-even. These operators are responsible for sourcing, respectively, cosmic birefringence and circular polarization. We propagate jointly the effects of all the relevant Lorentz-violating parameters to CMB observables and provide constraints with the most recent CMB datasets. We bound the CPT-even coefficient to $k_{F,E+B} < 2.31 \times 10^{-31}$ at 95\% CL. This improves previous CMB bounds by one order of magnitude. The limits we obtain on the CPT-odd coefficients, i.e. $|k_{(V)00}^{(3)}| < 1.54 \times 10^{-44} \; {\rm GeV}$ and $|\mathbf{k_{AF}}| < 0.74 \times 10^{-44} \; {\rm GeV}$ at 95\% CL, are respectively one and two orders of magnitude stronger than previous CMB-based limits, superseding also bounds from non-CMB searches. This analysis provides the strongest constraints to date on CPT-violating coefficients in the minimal SME from CMB searches.

The noncommutative spacetimes associated to the $\kappa$-Poincar\'e relativistic symmetries and their "non-relativistic" (Galilei) and "ultra-relativistic" (Carroll) limits are indistinguishable, since their coordinates satisfy the same algebra. In this work, we show that the three quantum kinematical models can be differentiated when looking at the associated spaces of time-like worldlines. Specifically, we construct the noncommutative spaces of time-like geodesics with $\kappa$-Galilei and $\kappa$-Carroll symmetries as contractions of the corresponding $\kappa$-Poincar\'e space and we show that these three spaces are defined by different algebras. In particular, the $\kappa$-Galilei space of worldlines resembles the so-called Euclidean Snyder model, while the $\kappa$-Carroll space turns out to be commutative. Furthermore, we identify the map between quantum spaces of geodesics and the corresponding noncommutative spacetimes, which requires to extend the space of geodesics by adding the noncommutative time coordinate.

We review the main features of models where relativistic symmetries are deformed at the Planck scale. We cover the motivations and links to other quantum gravity approaches. We describe in some detail the most studied theoretical frameworks, including Hopf algebras, relative locality, and other scenarios with deformed momentum space geometry. We discuss possible phenomenological consequences, and point out current open questions., Comment: Invited contribution as a chapter to the book "Modified and Quantum Gravity - From theory to experimental searches on all scales" (C. L\"ammerzahl and C. Pfeifer (eds.), Springer)

In addition to its implications for astrophysics, the hunt for GRB neutrinos could also be significant in quantum-gravity research, since they are excellent probes of the microscopic fabric of spacetime. Some previous studies based on IceCube neutrinos had found intriguing preliminary evidence that some of them might be GRB neutrinos with travel times affected by quantum properties of spacetime, with the noticeable feature that quantum spacetime would slow down some of the neutrinos while others would be sped up. Recently the IceCube collaboration revised significantly the estimates of the direction of observation of their neutrinos, and we here investigate how the corrected directional information affects the results of the previous quantum-spacetime-inspired analyses. We find that there is now no evidence for neutrinos sped up by quantum-spacetime properties, whereas the evidence for neutrinos slowed down by quantum spacetime is even stronger than previously found. Our most conservative estimates find a false alarm probability of less than 1% for these "slow neutrinos", providing motivation for future studies on larger data samples., Comment: Latex, 5 pages

We show that quantum properties of spacetime, encoded by noncommutativity at the Planck scale, lead to a generalized time evolution of quantum systems in which pure states can evolve into mixed states. Specifically, a decoherence mechanism is obtained in the form of a Lindblad-like time evolution for the density operator when the action of time translations generator is deformed by the effects of spacetime noncommutativity. The decoherence time for the evolution of a free particle is used to show that the Planck mass is the maximum allowed mass for elementary quantum systems., Comment: v2 matches the version accepted for publication in Communications Physics

Let $G$ be a finite group acting on a ring $R$ and $H$ a subgroup of $G$. In this paper we compare some homological dimensions over the skew group rings $RG$ and $RH$. Moreover, under the assumption that $RG$ is a separable extension over $RH$, we show that the skew group rings $RG$ and $RH$ share some properties such as being $n$-Gorenstein, $n$-perfect, $n$-coherent, $(n,d)$, Ding-Chen or IF-rings.

In this paper, we compute the Frobenius dimension of any cluster tilted algebra of finite type. Moreover, we give conditions on the bounded quiver of a cluster tilted algebra $\Lambda$ such that $\Lambda$ has non-trivial open Frobenius structures., Comment: arXiv admin note: text overlap with arXiv:0906.3422 by other authors

In a quantum gravity theory, it is expected that the classical notion of spacetime disappears, leading to a quantum structure with new properties. A possible way to take into account these quantum effects is through a noncommutativity of spacetime coordinates. In the literature, there is not a clear way to describe at the same time a noncommutativity of spacetime and the phase-space noncommutativity of quantum mechanics. In this paper we address this issue by constructing a Drinfel'd twist in phase space which deals with both quantizations. This method can be applied to a noncommutativity which involves only space, leaving time aside. We apply our construction to the so-called $\lambda$-Minkwoski and $\mathbb{R}^3_\lambda$ noncommutative spaces., Comment: v2 matches the version accepted for publication on Phys. Rev. D. It includes additional clarifications and references. v3 includes some missing terms in a couple of equations. The conclusions are essentially unaffected

It has been pointed out that different choices of momenta can be associated to the same noncommutative spacetime model. The question of whether these momentum spaces, related by diffeomorphisms, produce the same physical predictions is still debated. In this work, we focus our attention on a few different momentum spaces that can be associated to the Galilean Snyder noncommutative spacetime model and show that they produce different predictions for the energy spectrum of the harmonic oscillator.

The exploration of the universe has recently entered a new era thanks to the multi-messenger paradigm, characterized by a continuous increase in the quantity and quality of experimental data that is obtained by the detection of the various cosmic messengers (photons, neutrinos, cosmic rays and gravitational waves) from numerous origins. They give us information about their sources in the universe and the properties of the intergalactic medium. Moreover, multi-messenger astronomy opens up the possibility to search for phenomenological signatures of quantum gravity. On the one hand, the most energetic events allow us to test our physical theories at energy regimes which are not directly accessible in accelerators; on the other hand, tiny effects in the propagation of very high energy particles could be amplified by cosmological distances. After decades of merely theoretical investigations, the possibility of obtaining phenomenological indications of Planck-scale effects is a revolutionary step in the quest for a quantum theory of gravity, but it requires cooperation between different communities of physicists (both theoretical and experimental). This review is aimed at promoting this cooperation by giving a state-of-the art account of the interdisciplinary expertise that is needed in the effective search of quantum gravity footprints in the production, propagation and detection of cosmic messengers., Comment: This review was written by participants of the COST Action CA18108. Further information on the review can be found at https://qg-mm.unizar.es/review/, updated to published version

Recent work showed that $\kappa$-deformations can describe the quantum deformation of several relativistic models that have been proposed in the context of quantum gravity phenomenology. Starting from the Poincar\'e algebra of special-relativistic symmetries, one can toggle the curvature parameter $\Lambda$, the Planck scale quantum deformation parameter $\kappa$ and the speed of light parameter $c$ to move to the well-studied $\kappa$-Poincar\'e algebra, the (quantum) (A)dS algebra, the (quantum) Galilei and Carroll algebras and their curved versions. In this review, we survey the properties and relations of these algebras of relativistic symmetries and their associated noncommutative spacetimes, emphasizing the nontrivial effects of interplay between curvature, quantum deformation and speed of light parameters., Comment: v2 matches the version accepted for publication. We added a few references and comments, plus fixed some typos. v3 includes a few additional references

A novel approach to study the properties of models with quantum-deformed relativistic symmetries relies on a noncommutative space of worldlines rather than the usual noncommutative spacetime. In this setting, spacetime can be reconstructed as the set of events, that are identified as the crossing of different worldlines. We lay down the basis for this construction for the $\kappa$-Poincar\'e model, analyzing the fuzzy properties of $\kappa$-deformed time-like worldlines and the resulting fuzziness of the reconstructed events., Comment: 26 pages, 8 figures, v2 matches version accepted by JHEP

Recently, it was shown that when reference frames are associated to quantum systems, the transformation laws between such quantum reference frames need to be modified to take into account the quantum and dynamical features of the reference frames. This led to a relational description of the phase space variables of the quantum system of which the quantum reference frames are part of. While such transformations were shown to be symmetries of the system's Hamiltonian, the question remained unanswered as to whether they enjoy a group structure, similar to that of the Galilei group relating classical reference frames in quantum mechanics. In this work, we identify the canonical transformations on the phase space of the quantum systems comprising the quantum reference frames, and show that these transformations close a group structure defined by a Lie algebra, which is different from the usual Galilei algebra of quantum mechanics. We further find that the elements of this new algebra are in fact the building blocks of the quantum reference frames transformations previously identified, which we recover. Finally, we show how the transformations between classical reference frames described by the standard Galilei group symmetries can be obtained from the group of transformations between quantum reference frames by taking the zero limit of the parameter that governs the additional noncommutativity introduced by the quantum nature of inertial transformations., Comment: Version accepted in Quantum. 15 pages main text; 3 pages Appendix; 1 table

Given a group of kinematical symmetry generators, one can construct a compatible noncommutative spacetime and deformed phase space by means of projective geometry. This was the main idea behind the very first model of noncommutative spacetime, proposed by H.S. Snyder in 1947. In this framework, spacetime coordinates are the translation generators over a manifold that is symmetric under the required generators, while momenta are projective coordinates on such a manifold. In these proceedings we review the construction of Euclidean and Lorentzian noncommutative Snyder spaces and investigate the freedom left by this construction in the choice of the physical momenta, because of different available choices of projective coordinates. In particular, we derive a quasi-canonical structure for both the Euclidean and Lorentzian Snyder noncommutative models such that their phase space algebra is diagonal although no longer quadratic., Comment: Contribution to the proceedings of the Corfu Summer Institute 2019 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2019), 31 August-25 September 2019, Corfu, Greece. Contains previously unpublished material. V2: references added

In this paper, we show that the repetitive cluster category of type $D_n$, defined as the orbit category $\mathcal{D}^b(\mathrm{mod}K D_n)/(\tau^{-1}[1])^p$, is equivalent to a category defined on a subset of tagged edges in a regular punctured polygon. This generalizes the construction of Schiffler, \cite{S}, which we recover when $p=1$., Comment: arXiv admin note: substantial text overlap with arXiv:1201.1767, arXiv:math/0608264 by other authors

We derive the non-relativistic $c\to\infty$ and ultra-relativistic $c\to 0$ limits of the $\kappa$-deformed symmetries and corresponding spacetime in (3+1) dimensions, with and without a cosmological constant. We apply the theory of Lie bialgebra contractions to the Poisson version of the $\kappa$-(A)dS quantum algebra, and quantize the resulting contracted Poisson-Hopf algebras, thus giving rise to the $\kappa$-deformation of the Newtonian (Newton-Hooke and Galilei) and Carrollian (Para-Poincar\'e, Para-Euclidean and Carroll) quantum symmetries, including their deformed quadratic Casimir operators. The corresponding $\kappa$-Newtonian and $\kappa$-Carrollian noncommutative spacetimes are also obtained as the non-relativistic and ultra-relativistic limits of the $\kappa$-(A)dS noncommutative spacetime. These constructions allow us to analyze the non-trivial interplay between the quantum deformation parameter $\kappa$, the curvature parameter $\eta$ and the speed of light parameter $c$., Comment: 19 pages. v2 matches version accepted for publication by Phys. Lett. B; some references added

We show that the Lorentzian Snyder models, together with their Galilei and Carroll limiting cases, can be rigorously constructed through the projective geometry description of Lorentzian, Galilean and Carrollian spaces with nonvanishing constant curvature. The projective coordinates of such curved spaces take the role of momenta, while translation generators over the same spaces are identified with noncommutative spacetime coordinates. In this way, one obtains a deformed phase space algebra, which fully characterizes the Snyder model and is invariant under boosts and rotations of the relevant kinematical symmetries. While the momentum space of the Lorentzian Snyder models is given by certain projective coordinates on (Anti-)de Sitter spaces, we discover that the momentum space of the Galilean (Carrollian) Snyder models is given by certain projective coordinates on curved Carroll (Newton--Hooke) spaces. This exchange between the Galilei and Carroll limits emerging in the transition from the geometric picture to the phase space picture is traced back to an interchange of the role of coordinates and translation operators. As a physically relevant feature, we find that in Galilean Snyder spacetimes the time coordinate does not commute with space coordinates, in contrast with previous proposals for non-relativistic Snyder models, which assume that time and space decouple in the non-relativistic limit $c\to \infty$. This remnant mixing between space and time in the non-relativistic limit is a quite general Planck-scale effect found in several quantum spacetime models., Comment: v2 matches version accepted by Class. Quant. Grav

The essential features of a quantum group deformation of classical symmetries of General Relativity in the case with non-vanishing cosmological constant $\Lambda$ are presented. We fully describe (anti-)de Sitter non-commutative spacetimes and curved momentum spaces in (1+1) and (2+1) dimensions arising from the $\kappa$-deformed quantum group symmetries. These non-commutative spacetimes are introduced semiclassically by means of a canonical Poisson structure, the Sklyanin bracket, depending on the classical $r$-matrix defining the $\kappa$-deformation, while curved momentum spaces are defined as orbits generated by the $\kappa$-dual of the Hopf algebra of quantum symmetries. Throughout this construction we use kinematical coordinates, in terms of which the physical interpretation becomes more transparent, and the cosmological constant $\Lambda$ is included as an explicit parameter whose $\Lambda \rightarrow 0$ limit provides the Minkowskian case. The generalization of these results to the physically relevant (3+1)-dimensional deformation is also commented., Comment: Based on the contribution presented at the "First Hermann Minkowski Meeting on the Foundations of Spacetime Physics" held in Albena, Bulgaria, May 15-18, 2017

We follow the life of a generic primordial perturbation mode (scalar or tensor) subject to modified dispersion relations (MDR), as its proper wavelength is stretched by expansion. A necessary condition ensuring that travelling waves can be converted into standing waves is that the mode starts its life deep inside the horizon and in the trans-Planckian regime, then leaves the horizon as the speed of light corresponding to its growing wavelength drops, to eventually become cis-Planckian whilst still outside the horizon, and finally re-enter the horizon at late times. We find that scalar modes in the observable range satisfy this condition, thus ensuring the viability of MDR models in this respect. For tensor modes we find a regime in which this does not occur, but in practice it can only be realised for wavelengths in the range probed by future gravity wave experiments if the quantum gravity scale experienced by gravity waves goes down to the PeV range. In this case travelling---rather than standing---primordial gravity waves could be the tell-tale signature of MDR scenarios., Comment: 9 pages, 1 figure. v2 matches version accepted for publication

We investigate the relativistic properties under boost transformations of the $\kappa$-Poincar\'e model with multiple causally connected interactions, both at the level of its formulation in momentum space only and when it is endowed with a full phase space construction, provided by the relative locality framework. Previous studies focussing on the momentum space picture showed that in presence of just one interaction vertex the model is relativistic, provided that the boost parameter acting on each given particle receives a "backreaction" from the momenta of the other particles that participate in the interaction. Here we show that in presence of multiple causally-connected vertices the model is relativistic if the boost parameter acting on each given particle receives a backreaction from the total momentum of all the particles that are causally connected, even those that do not directly enter the vertex. The relative locality framework constructs spacetime by defining a set of dual coordinates to the momentum of each particle and interaction coordinates as Lagrange multipliers that enforce momentum conservation at interaction events. We show that the picture is Lorentz invariant if one uses an appropriate "total boost" to act on the particles' worldlines and on the interaction coordinates. The picture we develop also allows for a reinterpretation of the backreaction as the manifestation of the "total boost" action. Our findings provide the basis to consistently define distant relatively boosted observers in the relative locality framework., Comment: 23 pages, 3 figures