Search

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links. [ABSTRACT FROM AUTHOR]

We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that analogous results hold for the self-dual almost Mathieu maps, but they seem presently beyond reach. [ABSTRACT FROM AUTHOR]

We prove a functorial correspondence between a category of logarithmic sl 2 -connections on a curve X with fixed generic residues and a category of abelian logarithmic connections on an appropriate spectral double cover. The proof is by constructing a pair of inverse functors π ab , π ab , and the key is the construction of a certain canonical cocycle valued in the automorphisms of the direct image functor π ∗ . [ABSTRACT FROM AUTHOR]

Abstract This paper concerns the long term behavior of the stochastic two-dimensional g-Navier–Stokes equations with additive noise defined on a sequence of expanding domains, where the ultimate domain is unbounded and of Poincaré type. We prove that the weak continuity is uniform with respect to all expanding cocycles, which yields the equi-asymptotic compactness by using an energy equation method. Finally, we show the existence of a random attractor for the equation on each domain and the upper semi-continuity of random attractors as the bounded domain is expanded to the unbounded ultimate domain.

When analytic quasi-periodic cocycles are defined on a high dimension torus, their Lyapunov exponents have perturbative positivity and continuity. In this article, we study a class of analytic quasi-periodic Jacobi cocycles defined on a two dimension torus. We show that in the non-perturbative large coupling regimes, the Lyapunov exponent is positive for any frequency and weak Holder continuous for the full-measured frequency.

We consider involutive, non-degenerate, finite set-theoretic solutions of the Yang–Baxter equation (YBE). Such solutions can be always obtained using certain algebraic structures that generalize nilpotent rings called braces. Our main aim here is to express such solutions in terms of admissible Drinfeld twists substantially extending recent preliminary results. We first identify the generic form of the twists associated to set-theoretic solutions and we show that these twists are admissible, i.e. they satisfy a certain co-cycle condition. These findings are also valid for Baxterized solutions of the YBE constructed from the set-theoretical ones. [ABSTRACT FROM AUTHOR]

We show that the Quot scheme QLn=QuotA3(IL,n) parameterising length n quotients of the ideal sheaf of a line in A3 is a global critical locus, and calculate the resulting motivic partition function (varying n), in the ring of relative motives over the configuration space of points in A3. As in the work of Behrend–Bryan–Szendrői, this enables us to define a virtual motive for the Quot scheme of n points of the ideal sheaf IC⊂OY, where C⊂Y is a smooth curve embedded in a smooth 3‐fold Y, and we compute the associated motivic partition function. The result fits into a motivic wall‐crossing type formula, refining the relation between Behrend's virtual Euler characteristic of QuotY(IC,n) and of the symmetric product SymnC. Our 'relative' analysis leads to results and conjectures regarding the pushforward of the sheaf of vanishing cycles along the Hilbert–Chow map QLn→Symn(A3), and connections with cohomological Hall algebra representations. [ABSTRACT FROM AUTHOR]

Copyright of Journal of Southeast University (English Edition) is the property of Journal of Southeast University Editorial Office and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

This is an overview of ideas related to brackets in early homotopy theory, crossed modules, the obstruction 3-cocycle for the nonabelian extension problem, the Teichmüller cocycle, Lie-Rinehart algebras, Lie algebroids, and differential algebra. [ABSTRACT FROM AUTHOR]

Toward the complete classification of poly- |${\mathbb{Z}}$| group actions on Kirchberg algebras, we prove several fundamental theorems that are used in the classification. In addition, as an application of them, we classify outer actions of poly- |${\mathbb{Z}}$| groups of Hirsch length not greater than three on unital Kirchberg algebras up to |$KK$| -trivial cocycle conjugacy. [ABSTRACT FROM AUTHOR]

In this paper, we shall generalize the concept of equivariance in statistics to "weak equivariance". Then, we summarize the properties of weakly equivariant estimators and their applications in statistics. At first we characterize the class of all weakly equivariant estimators. Then, we shall consider the concept of cocycles and isovariance, and so we find their connection with weakly equivariant functions. It is natural to restrict attention to the class of weakly equivariant estimator to find minimum risk weakly equivariant estimators. If the group acts in two different ways, we shall find a relation between the minimum risk equivariant and minimum risk weakly equivariant estimator under the old and new group actions. Also we shall introduce a necessary and sufficient condition for the invariance of the loss function under the new action. [ABSTRACT FROM AUTHOR]

We extend the notions of joint and generalized spectral radii to cocycles acting on Banach spaces and obtain a version of Berger–Wang's formula when restricted to the space of cocycles taking values in the space of compact operators. Moreover, we observe that the previous quantities depends continuously on the underlying cocycle. [ABSTRACT FROM AUTHOR]

We establish a new robustness theorem of delayed random attractors at zero-memory and the criteria are given by part convergence of cocycles along with regularity, recurrence and eventual compactness of attractors, where we relax the convergence condition of cocycles in all known robustness theorem of attractors, especially by Wang et al.(Siam-jads, 2015). As an application, we consider the stochastic non-autonomous 2D-Ginzburg-Landau delay equation, whose solutions seem not to be convergent for all initial data as the memory time goes to zero, but we can show the convergence of solutions toward zero-memory for part initial data in the lower-regular space. As a further result, we show that, for each memory time, the delay equation has a pullback random attractor such that it is upper semi-continuous at zero-memory. [ABSTRACT FROM AUTHOR]

We prove a reducibility result for sl(2,R) quasi-periodic cocycles close to a constant elliptic matrix in ultra-differentiable classes, under an adapted arithmetic condition which extends the Brjuno-Rüssmann condition in the analytic case. The proof is based on an elementary property of the fibered rotation number and deals with ultra-differentiable functions with a weighted Fourier norm. We also show that a weaker arithmetic condition is necessary for reducibility, and that it can be compared to a sufficient arithmetic condition. [ABSTRACT FROM AUTHOR]

Given a finite set of quasi-periodic cocycles the random product of them is defined as the random composition according to some probability measure. We prove that the set of Cr, 0 ≤ r ≤ ∞ (or analytic) κ + 1-tuples of quasi-periodic cocycles taking values in SL2(R) such that the random product of them has positive Lyapunov exponent contains a C0 open and Cr dense subset which is formed by C0 continuity point of the Lyapunov exponent. For κ + 1-tuples of quasi-periodic cocycles taking values in GLd(R) for d > 2, we prove that if one of them is diagonal, then there exists a Cr dense set of such κ + 1-tuples which have simple Lyapunov spectrum and are C0 continuity point of the Lyapunov exponent. [ABSTRACT FROM AUTHOR]

We prove that given a pseudo- z-embedding two Knapp-Stein R-groups are isomorphic and their 2-cocycles are identical. [ABSTRACT FROM AUTHOR]

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example, square roots) although the final result may be expressed without algebraic extensions. In this article, I present an improvement of this algorithm, which avoids algebraic extensions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a class of 2-cocycles on monoidal Hom–Hopf algebras, study their properties, and extend neat lazy 2-cocycles to a Radford (m , n) -biproduct monoidal Hom–Hopf algebra. [ABSTRACT FROM AUTHOR]

We investigate constructions of higher arity self-distributive operations, and give relations between cohomology groups corresponding to operations of different arities. For this purpose we introduce the notion of mutually distributive n -ary operations generalizing those for the binary case, and define a cohomology theory labeled by these operations. A geometric interpretation in terms of framed links is described, with the scope of providing algebraic background of constructing 2 -cocycles for framed link invariants. This theory is also studied in the context of symmetric monoidal categories. Examples from Lie algebras, coalgebras and Hopf algebras are given. [ABSTRACT FROM AUTHOR]

We prove the joint continuity of Lyapunov exponent in the energy and the Diophantine frequency for quasi-periodic Schrödinger cocycles with the C2 cos-type potentials. In particular, the Lyapunov exponent is log-Hölder continuous at each Diophantine frequency. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of uniform pullback asymptotic compactness of a non-autonomous random dynamical system, and prove the existence of random uniform and cocycle attractor with autonomous attraction universes for non-autonomous stochastic FitzHugh–Nagumo system with multiplicative noise defined on whole space. Some properties of random uniform and cocycle attractor are shown. The method of tail estimates plays an important role. [ABSTRACT FROM AUTHOR]

We show that, up to strong cocycle conjugacy, every countable exact group admits a unique equivariantly $\mathcal {O}_{2}$ -absorbing, pointwise outer action on the Cuntz algebra $\mathcal {O}_{2}$ with the quasi-central approximation property (QAP). In particular, we establish the equivariant analogue of the Kirchberg $\mathcal {O}_{2}$ -absorption theorem for these groups. [ABSTRACT FROM AUTHOR]

Rigid meromorphic cocycles were introduced by Darmon and Vonk as a conjectural p-adic extension of the theory of singular moduli to real quadratic base fields. They are certain cohomology classes of SL 2 (Z [ 1 / p ]) which can be evaluated at real quadratic irrationalities, and the values thus obtained are conjectured to lie in algebraic extensions of the base field. In this article, we present a construction of cohomology classes inspired by that of Darmon–Vonk, in which SL 2 (Z [ 1 / p ]) is replaced by an order in an indefinite quaternion algebra over a totally real number field F. These quaternionic cohomology classes can be evaluated at elements in almost totally complex extensions K of F, and we conjecture that the corresponding values lie in algebraic extensions of K. We also report on extensive numerical evidence for this algebraicity conjecture. [ABSTRACT FROM AUTHOR]

The classical Multiplicative Ergodic Theorem of Oseledets is generalized here to cocycles taking values in a semi-finite von Neumann algebra. This allows for a continuous Lyapunov distribution. [ABSTRACT FROM AUTHOR]

We characterize the existence of an exponential dichotomy for a nonautonomous linear delay equation via the hyperbolicity of an appropriate cocycle. An important advantage of this approach is that the base is compact under mild additional assumptions. Moreover, we give a few applications of the equivalence of the two notions of hyperbolicity. In particular, we consider the robustness and the admissibility of the equation, and we obtain stable and unstable invariant manifolds. [ABSTRACT FROM AUTHOR]

We prove a Poincaré duality for the Chow rings of smooth fans whose support are tropical linear spaces. As a consequence, we show that cycles and cocycles on tropical manifolds are Poincaré dual to each other. This allows us to define pull-backs of tropical cycles along arbitrary morphisms with smooth target. [ABSTRACT FROM AUTHOR]

In this paper we compute the intersection number of two double ramification (DR) cycles (with different ramification profiles) and the top Chern class of the Hodge bundle on the moduli space of stable curves of any genus. These quadratic DR integrals are the main ingredients for the computation of the DR hierarchy associated to the infinite‐dimensional partial cohomological field theory given by exp(μ2Θ), where μ is a parameter and Θ is Hain's theta class, appearing in Hain's formula for the DR cycle on the moduli space of curves of compact type. This infinite rank DR hierarchy can be seen as a rank 1 integrable system in two space and one time dimensions. We prove that it coincides with a natural analogue of the Korteweg‐de‐Vries (KdV) hierarchy on a noncommutative Moyal torus. [ABSTRACT FROM AUTHOR]

We study the norm potentials of Hölder continuous GL2(R)-cocycles over hyperbolic systems whose canonical holonomies converge and are Hölder continuous. Such cocycles include locally constant GL2(R)-cocycles as well as fiber-bunched GL2(R)-cocycles. We show that the norm potentials of irreducible such cocycles have unique equilibrium states. Among the reducible cocycles, we provide a characterization for cocycles whose norm potentials have more than one equilibrium states. [ABSTRACT FROM AUTHOR]

We consider cocycles obtained by composing sequences of transfer operators with positive weights, associated with uniformly expanding maps (possibly having countably many branches) and depending upon parameters. Assuming C k regularity with respect to coordinates and parameters, we show that when the sequence is picked within a certain uniform family the top characteristic exponent and generator of top Oseledets space of the cocycle are C k - 1 in parameters. As applications, we obtain a linear response formula for the equivariant measure associated with random products of uniformly expanding maps, and we study the regularity of the Hausdorff dimension of a repeller associated with random compositions of one-dimensional cookie-cutters. [ABSTRACT FROM AUTHOR]

Let M reg be the topological moduli space of long knots up to regular isotopy, and for any natural number n > 1 let M n reg be the moduli space of all n -cables of framed long knots which are twisted by a string link to a knot in the solid torus V 3 . We upgrade the Vassiliev invariant v 2 of a knot to an integer valued combinatorial 1-cocycle for M n reg by a very simple formula. This 1-cocycle depends on a natural number a ∈ ℤ ≅ H 1 (V 3 ; ℤ) with 0 < a < n as a parameter and we obtain a polynomial-valued 1-cocycle by taking the Lagrange interpolation polynomial with respect to the parameter. We show that it induces a non-trivial pairing on H 0 (M n reg ) × H 0 (M reg ) already for n = 2. [ABSTRACT FROM AUTHOR]

With applications in the Kudla program in mind we employ singular theta lifts for the reductive dual pair U(p,q) × U(1,1) to construct two different kinds of Green forms for codimension q-cycles in Shimura varieties associated to unitary groups. We establish an adjointness result between our singular theta lift and the Kudla-Millson lift. Further, we compare the two Greens forms and obtain modularity for the generating function of the difference of the two Green forms. Finally, we show that the Green forms obtained by the singular theta lift satisfy an eigenvalue equation for the Laplace operator and conclude that our Green forms coincide with the ones constructed by Oda and Tsuzuki by different means. [ABSTRACT FROM AUTHOR]

We will show a centrally free action of an amenable rigid C ∗ -tensor category on a properly infinite von Neumann algebra has the Rohlin property. Our main result is the classification of centrally free cocycle actions of an amenable rigid C ∗ -tensor category up to approximate inner conjugacy on properly infinite von Neumann algebras. This is regarded as a generalization of classification of amenable discrete groups due to A. Connes, V. Jones and A. Ocneanu. We have the following two applications: a classification of centrally free actions of amenable discrete quantum groups of Kac type on von Neumann algebras and another proof of S. Popa's celebrated classification result of amenable subfactors. As another application of the Rohlin property, we will prove the fullness of the crossed product of a full factor by a minimal action of a compact group. [ABSTRACT FROM AUTHOR]

Quantum stochastic cocycles provide a basic model for time-homogeneous Markovian evolutions in a quantum setting, and a direct counterpart in continuous time to quantum random walks, in both the Schrödinger and Heisenberg pictures. This paper is a sequel to one in which correspondences were established between classes of quantum stochastic cocycle on an operator space or C ∗ -algebra, and classes of Schur-action 'global' semigroup on associated matrix spaces over the operator space. In this paper we investigate the stochastic generation of cocycles via the generation of their corresponding global semigroups, with the primary purpose of strengthening the scope of applicability of semigroup theory to the analysis and construction of quantum stochastic cocycles. An explicit description is given of the affine relationship between the stochastic generator of a completely bounded cocycle and the generator of any one of its associated global semigroups. Using this, the structure of the stochastic generator of a completely positive quasicontractive quantum stochastic cocycle on a C ∗ -algebra whose expectation semigroup is norm continuous is derived, giving a comprehensive stochastic generalisation of the Christensen–Evans extension of the GKS&L theorem of Gorini, Kossakowski and Sudarshan, and Lindblad. The transformation also provides a new existence theorem for cocycles with unbounded structure map as stochastic generator. The latter is applied to a model of interacting particles known as the quantum exclusion Markov process, in particular on integer lattices in dimensions one and two. [ABSTRACT FROM AUTHOR]

Given a discrete-time random dynamical system represented by a cocycle of non-singular measurable maps, we may obtain information on dynamical quantities by studying the cocycle of Perron–Frobenius operators associated to the maps. Of particular interest is the second-largest Lyapunov exponent for the cocycle of operators, λ2, which can tell us about mixing rates and decay of correlations in the system. We prove a generalized Perron–Frobenius theorem for cocycles of bounded linear operators on Banach spaces that preserve and occasionally contract a cone; this theorem shows that the top Oseledets space for the cocycle is one-dimensional, and there is a lower bound for the gap between the largest Lyapunov exponents λ1 and λ2 (that is, an upper bound for λ2 which is strictly less than λ1) explicitly in terms of quantities related to cone contraction. We then apply this theorem to the case of cocycles of Perron–Frobenius operators arising from a parametrized family of maps to obtain an upper bound on λ2; to the best of our knowledge, this work is the first time λ2 has been upper-bounded for a family of maps. In doing so, we utilize a new balanced Lasota–Yorke inequality. We also examine random perturbations of a fixed map within the family with two invariant densities and show that as the perturbation is scaled back down to the unperturbed map, λ2 is at least asymptotically linear in the scale parameter. Our estimates are sharp, in the sense that there is a sequence of scaled perturbations of the fixed map that are all Markov, such that λ2 is asymptotic to −2 times the scale parameter. [ABSTRACT FROM AUTHOR]

Codes from generalized Hadamard matrices have already been introduced. Here we deal with these codes when the generalized Hadamard matrices are cocyclic. As a consequence, a new class of codes that we call generalized Hadamard full propelinear codes turns out. We prove that their existence is equivalent to the existence of central relative (v, w, v, v/w)-difference sets. Moreover, some structural properties of these codes are studied and examples are provided. Some of the propelinear codes constructed for the examples perform better than any comparable linear code. [ABSTRACT FROM AUTHOR]

We bring cocycle enhancement theory to the case of psyquandles. Analogously to our previous work on virtual biquandle cocycle enhancements, we define enhancements of the psyquandle counting invariant via pairs of a biquandle 2-cocycle and a new function satisfying some conditions. As an application we define new single-variable and two-variable polynomial invariants of oriented pseudoknots and singular knots and links. We provide examples to show that the new invariants are proper enhancements of the counting invariant and are not determined by the Jablan polynomial. [ABSTRACT FROM AUTHOR]

We study flows on C ∗ -algebras with the Rokhlin property. We show that every Kirchberg algebra carries a unique Rokhlin flow up to cocycle conjugacy, which confirms a long-standing conjecture of Kishimoto. We moreover present a classification theory for Rokhlin flows on C ∗ -algebras satisfying certain technical properties, which hold for many C ∗ -algebras covered by the Elliott program. As a consequence, we obtain the following further classification theorems for Rokhlin flows. Firstly, we extend the statement of Kishimoto's conjecture to the non-simple case: Up to cocycle conjugacy, a Rokhlin flow on a separable, nuclear, O ∞ -absorbing C ∗ -algebra is uniquely determined by its induced action on the prime ideal space. Secondly, we give a complete classification of Rokhlin flows on simple classifiable KK-contractible C ∗ -algebras: Two Rokhlin flows on such a C ∗ -algebra are cocycle conjugate if and only if their induced actions on the cone of lower-semicontinuous traces are affinely conjugate. [ABSTRACT FROM AUTHOR]

The homotopy category of the bordism category hBordd has as objects closed oriented (d-1)-manifolds and as morphisms diffeomorphism classes of d-dimensional bordisms. Using a new fiber sequence for bordism categories, we compute the classifying space of hBordd for d=1, exhibiting it as a circle bundle over CP∞-1. As part of our proof we construct a quotient Bordred1 of the cobordism category where circles are deleted. We show that this category has classifying space Ω∞-2CP∞-1 and moreover that, if one equips these bordisms with a map to a simply connected space X, the resulting Bordred1(X) can be thought of as a cobordism model for the topological cyclic homology TC(S[ΩX]). In the second part of the paper we construct an infinite loop space map B(hBordred1)→Q(Σ²CP∞+) in this model and use it to derive combinatorial formulas for rational cocycles on Bordred1 representing the Miller-Morita-Mumford classes κiH2i+2((B(hC1);Q). [ABSTRACT FROM AUTHOR]

We propose a general framework for quantum walks on d-dimensional spaces. We investigate asymptotic behavior of these walks. We prove that every homogeneous walks with finite degree of freedom have limit distribution. This theorem can also be applied to every crystal lattice. In this theorem, it is not necessary to assume that the support of the initial unit vector is finite. We also pay attention on 1-cocycles, which is related to Heisenberg representation of time evolution of observables. For homogeneous walks with finite degree of freedom, convergence of averages of 1-cocycles associated with the position observable is also proved. [ABSTRACT FROM AUTHOR]

In this survey we prove the sharpest results on the loss of Sobolev regularity for solutions of the cohomological equation for translation flows on translation surfaces, available to the methods developed by the author in Forni [Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. of Math. (2)146(2) (1997), 295–344] and Forni [Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2)155(1) (2002), 1–103]. The paper was mostly written between 2005 and 2006 while the author was at the University of Toronto, Canada, and was posted on arXiv in July 2007 [Forni. Sobolev regularity of solutions of the cohomological equation. Preprint, 2007, arXiv:0707.0940v2]. In an updated introduction we describe our results, taking into account later work on the problem and relevant recent progress in the field of Teichmüller dynamics, interval exchange transformations and translation flows. [ABSTRACT FROM AUTHOR]

By generalizing Ledrappier's criterion [Mesures d'èquilibre d'entropie complètement positive, in Systèmes dynamiques II – Varsovie, number 50 in Astérisque, Société mathématique de France, 1977, pp. 251–272] for the K-property of equilibrium states, we extend the criterion to subadditive potentials. In particular, supposing that the unique equilibrium state for a subadditive potential with quasi-multiplicativity and bounded distortion is totally ergodic, we show that it has the K-property. We apply this result to subadditive potentials arising from certain classes of matrix cocycles; for the norm potentials of irreducible locally constant cocycles and the singular value potentials of typical cocycles, we show that their unique equilibrium states have the K-property. This partly generalizes the work of Morris [Ergodic properties of matrix equilibrium states, Ergodic Theory Dyn. Syst. 38(6), 2018, pp. 2295-2320] on irreducible locally constant cocycles and their subadditive equilibrium states. [ABSTRACT FROM AUTHOR]

We prove that any uniformly exponentially stable linear cocycle of matrices defined over a topological dynamical system can be reduced via suitable change of variables to a linear cocycle whose generator has a spectral norm less than 1 at each point. Moreover, we establish an analogous result for nonuniformly exponentially stable linear cocycles, i.e. cocycles with negative Lyapunov exponents. In addition, by using tools from ergodic theory, we obtain converse results that give conditions for (non)uniform exponential stability of linear cocyles under the assumption that they admit a suitable reduction to a linear cocycle whose generator has a spectral norm less than 1 at points that form a 'large' subset of our topological dynamical system in the appropriate sense. Finally, we discuss the versions of our results in the case of (non)uniformly hyperbolic linear cocyles. [ABSTRACT FROM AUTHOR]

In this paper, we give explicit rates in the central limit theorem and in the almost sure invariance principle for general Rd-valued cocycles that appear in the study of the left random walk on linear groups. Our method of proof lies on a suitable martingale approximation and on a careful estimation of some coupling coefficients linked with the underlying Markov structure. Concerning the martingale part, the available results in the literature are not accurate enough to give almost optimal rates either in the central limit theorem for the Wasserstein distance, or in the strong approximation. A part of this paper is devoted to circumvent this issue. We then exhibit near optimal rates both in the central limit theorem in terms of the Wasserstein distance and in the almost sure invariance principle for Rd-valued martingales with stationary increments having moments of order p ∈ (2, 3] (the case of sequences of reversed martingale differences is also considered). Note also that, as an application of our results for general Rd-valued cocycles, a special attention is paid to the Iwasawa cocycle and the Cartan projection for reductive Lie groups (like for instance GLd(R)). [ABSTRACT FROM AUTHOR]

We define two-cocycles and cleft extensions in categories that are not necessarily braided, but where specific objects braid from one direction, like for a Hopf algebra H a Yetter–Drinfeld module braids from the left with H -modules. We will generalize classical results to this context and give some application for the categories of Yetter–Drinfeld modules and H -modules. In particular, we will describe liftings of coradically graded Hopf algebras in the category of Yetter–Drinfeld modules with these techniques. [ABSTRACT FROM AUTHOR]

HIV-1 employs a rich arsenal of viral factors throughout its life cycle and co-opts intracellular trafficking pathways. This exquisitely coordinated process requires precise manipulation of the host microenvironment, most often within defined subcellular compartments. The virus capitalizes on the host by modulating cell-surface proteins and cleverly exploiting nuclear import pathways for post entry events, among other key processes. Successful virus–cell interactions are indeed crucial in determining the extent of infection. By evolving defenses against host restriction factors, while simultaneously exploiting host dependency factors, the life cycle of HIV-1 presents a fascinating montage of an ongoing host–virus arms race. Herein, we provide an overview of how HIV-1 exploits native functions of the host cell and discuss recent findings that fundamentally change our understanding of the post-entry replication events. [ABSTRACT FROM AUTHOR]