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We discuss the growth of the singular values of symplectic transfer matrices associated with ergodic discrete Schrödinger operators in one dimension, with scalar and matrix-valued potentials. While for an individual value of the spectral parameter the rate of exponential growth is almost surely governed by the Lyapunov exponents, this is not, in general, true simultaneously for all the values of the parameter. The structure of the exceptional sets is interesting in its own right, and is also of importance in the spectral analysis of the operators. We present new results along with amplifications and generalisations of several older ones, and also list a few open questions. Here are two sample results. On the negative side, for any square-summable sequence |$p_{n}$| there is a residual set of energies in the spectrum on which the middle singular value (the |$W$| -th out of |$2W$|⁠) grows no faster than |$p_{n}^{-1}$|⁠. On the positive side, for a large class of cocycles including the i.i.d. ones, the set of energies at which the growth of the singular values is not as given by the Lyapunov exponents has zero Hausdorff measure with respect to any gauge function |$\rho (t)$| such that |$\rho (t)/t$| is integrable at zero. The employed arguments from the theory of subharmonic functions also yield a generalisation of the Thouless formula, possibly of independent interest: for each |$k$|⁠ , the average of the first |$k$| Lyapunov exponents is the logarithmic potential of a probability measure. [ABSTRACT FROM AUTHOR]

We show that braided Cherednik algebras introduced by Bazlov and Berenstein are cocycle twists of rational Cherednik algebras of the imprimitive complex reflection groups |$G(m,p,n)$|⁠ , when m is even. This gives a new construction of mystic reflection groups which have Artin–Schelter regular rings of quantum polynomial invariants. As an application of this result, we show that a braided Cherednik algebra has a finite-dimensional representation if and only if its rational counterpart has one. [ABSTRACT FROM AUTHOR]

We construct a non-trivial |$U(1)/\mathbb {Z}_q$| principal bundle on  T 4 from the compact U (1) lattice gauge field by generalizing Lüscher's constriction so that the cocycle condition contains |$\mathbb {Z}_q$| elements (the 't Hooft flux). The construction requires an admissibility condition on lattice gauge field configurations. From the transition function so constructed, we have the fractional topological charge that is |$\mathbb {Z}_q$| one-form gauge invariant and odd under the lattice time reversal transformation. Assuming a rescaling of the vacuum angle θ → q θ suggested from the Witten effect, our construction provides a lattice implementation of the mixed 't Hooft anomaly between the |$\mathbb {Z}_q$| one-form symmetry and the time reversal symmetry in the U (1) gauge theory with matter fields of charge  |$q\in 2\mathbb {Z}$| when θ = π, which was studied by Honda and Tanizaki [J. High Energy Phys. 12 , 154 (2020)] in the continuum framework. [ABSTRACT FROM AUTHOR]

When the reduced twisted -algebra of a non-principal groupoid admits a Cartan subalgebra, Renault's work on Cartan subalgebras implies the existence of another groupoid description of. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid of. In this paper, we study the relationship between the original groupoids and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum of the Cartan subalgebra. We then show that the quotient groupoid acts on , and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map admits a continuous section, then the Weyl twist is also given by an explicit continuous -cocycle on. [ABSTRACT FROM AUTHOR]

This article concerns the locus of locally constant |$\textrm{SL}(2,\mathbb{R})$| -valued cocycles that have a dominated splitting, called the hyperbolic locus. By developing the theory of Möbius semigroups we show that cocycles on the boundary of the hyperbolic locus, apart from a few exceptions, exhibit some form of hyperbolic behaviour. This behaviour is used to answer a question posed by Avila, Bochi and Yoccoz. Our approach introduces a new locus of cocycles, closely related to the hyperbolic locus, and motivates a line of investigation on the subject. [ABSTRACT FROM AUTHOR]

We use [ 11 ] to study the algebra structure of twisted cotriangular Hopf algebras |${}_J\mathcal{O}(G)_{J}$|⁠ , where |$J$| is a Hopf |$2$| -cocycle for a connected nilpotent algebraic group |$G$| over |$\mathbb{C}$|⁠. In particular, we show that |${}_J\mathcal{O}(G)_{J}$| is an affine Noetherian domain with Gelfand–Kirillov dimension |$\dim (G)$|⁠ , and that if |$G$| is unipotent and |$J$| is supported on |$G$|⁠ , then |${}_J\mathcal{O}(G)_{J}\cong U({\mathfrak{g}})$| as algebras, where |${\mathfrak{g}}={\textrm{Lie}}(G)$|⁠. We also determine the finite dimensional irreducible representations of |${}_J\mathcal{O}(G)_{J}$|⁠ , by analyzing twisted function algebras on |$(H,H)$| -double cosets of the support |$H\subset G$| of |$J$|⁠. Finally, we work out several examples to illustrate our results. [ABSTRACT FROM AUTHOR]

We consider one-dimensional quasi-periodic Schrödinger operators with analytic potentials. In the positive Lyapunov exponent regime, we prove large deviation estimates, which lead to refined Hölder continuity of the Lyapunov exponents and the integrated density of states, in both small Lyapunov exponent and large coupling regimes. Our results cover all the Diophantine frequencies and some Liouville frequencies. [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]

We wish to point out errors in the paper "Abelian Arithmetic Chern–Simons Theory and Arithmetic Linking Numbers", International Mathematics Research Notices, Vol. 2017, No. 00, pp. 1–29. The main error concerns the symmetry of the "ramified case" of the height pairing, which relies on the vanishing of the Bockstein map in Proposition 3.5. The surjectivity claimed in the 1st line of the proof of Proposition 3.5 is incorrect. The specific results that are affected are Proposition 3.5; Lemmas 3.6, 3.7, 3.8, and 3.9; and Corollary 3.11. The definition of the (S,n)-height pairing following Lemma 3.9 is also invalid, since the symmetry of the pairing was required for it to be well defined. The results of Section 3 before Proposition 3.5 as well as those of the other Sections are unaffected. Proposition 3.10 is correct, but the proof is unclear and has some sign errors. So we include here a correction. As in the paper, let I be an ideal such that In is principal in OF,S⁠. Write In = (ƒ−1)⁠. Then the Kummer cocycles kn(ƒ) will be in Z1(U, Z/nZ)⁠. For any a ∈ F⁠ , denote by aS its image in ∏v∈S Fv⁠. Thus, we get an element   [ƒ]S,n := [(kn(&#402), kn2(ƒS), 0)] ∈ Z1(U, Z/nZ × SZ/n2Z), which is well defined in cohomology independently of the choice of roots used to define the Kummer cocycles. (We have also trivialized both μn2 and μn⁠.) [ABSTRACT FROM AUTHOR]

In recent work of the authors, it was shown how to use any finite quantum stop time to stop the CCR flow and its strongly continuous isometric cocycles (Q. J. Math. 65:1145-1164, 2014). The stopped cocycle was shown to satisfy a stopped form of the cocycle identity, valid for deterministic increments of the time used for stopping. Here, a generalization of this identity is obtained, where both cocycle parameters are replaced with finite quantum stop times. [ABSTRACT FROM AUTHOR]

After the classification of the finite-dimensional Nichols algebras of diagonal type [17, 18], the determination of its defining relations [6, 7], and the verification of the generation in degree-s1 conjecture [6], there is still one step missing in the classification of complex finite-dimensional Hopf algebras with abelian group, without restrictions on the order of the latter: the computation of all deformations or liftings. A technique towards solving this question was developed in [5], built on cocycle deformations. In this paper, we elaborate further and present an explicit algorithm to compute liftings. In our main result we classify all liftings of finite-dimensional Nichols algebras of Cartan type A, over a cosemisimple Hopf algebra H. This extends [2], where it was assumed that the parameter is a root of unity of order >3 and that H is a commmutative group algebra. When the parameter is a root of unity of order 2 or 3, new phenomena appear: the quantum Serre relations can be deformed; this allows in turn the power root vectors to be deformed to elements in lower terms of the coradical filtration, but not necessarily in the group algebra. These phenomena are already present in the calculation of the liftings in type A2 at a parameter of order 2 or 3 over an abelian group [11, 19], done by a different method using a computer program. As a byproduct of our calculations, we present new infinite families of finite-dimensional pointed Hopf algebras. [ABSTRACT FROM AUTHOR]

Let k be an algebraically closed field of characteristic zero. Let D be a division algebra of degree d over its center Z(D). Assume that k ⊂ Z(D). We show that a finite group G faithfully grades D if and only if G contains a normal abelian subgroup of index dividing d. We also prove that if a finite dimensional Hopf algebra coacts on D defining a Hopf- Galois extension, then its PI degree is at most d². Finally, we construct Hopf-Galois actions on division algebras of twisted group algebras attached to bijective cocycles. [ABSTRACT FROM AUTHOR]

We establish localization-type dynamical bounds as a corollary of positive Lyapunov exponents for general operators with one-frequency quasiperiodic potentials defined by piecewise Hölder functions. This, in particular, extends some results previously known only for trigonometric polynomials [9] to the case of surprisingly low regularity. On the technical level, an important part of the argument is an extension of uniform uppersemicontinuity to cocycles with discontinuities, a result of independent interest. [ABSTRACT FROM AUTHOR]

We show that the orthogonal free quantum groups are not inner amenable and we construct an explicit proper cocycle weakly contained in the regular representation. This strengthens the result of Vaes and the second author, showing that the associated von Neumann algebras are full II1-factors and Brannan's result showing that the orthogonal free quantum groups have Haagerup's approximation property. We also deduce Ozawa-Popa's property strong (HH) and give a new proof of Isono's result about strong solidity. [ABSTRACT FROM AUTHOR]

We consider the problem of approximating a linear cocycle (or, more generally, a vector bundle automorphism) over a fixed base dynamics by another cocycle admitting a dominated splitting. We prove that the possibility of doing so depends only on the homotopy class of the cocycle, provided that the base dynamics is a minimal diffeomorphism and the fiber dimension is at least 3. This result is obtained by means of a general theorem on the existence of almost invariant sections for fiberwise isometries of bundles with compact fibers and finite fundamental group. As another consequence of that theorem, we characterize almost coboundaries on compact Lie groups with finite center. The main novelty of the proofs is the use of a quantitative homotopy result due to Calder, Siegel and Williams. [ABSTRACT FROM AUTHOR]

A CP-semigroup is aligned if its set of trivially maximal subordinates is totally ordered by subordination. We prove that aligned spatial E0-semigroups are prime: they have no nontrivial tensor product decompositions up to cocycle conjugacy. As a consequence, we establish the existence of uncountably many noncocycle conjugate E0-semigroups of type II0 which are prime. [ABSTRACT FROM AUTHOR]

We construct flows on an amalgamated free product factor and characterize the Rohlin property for them by faithfulness of R × Z actions on a measured space. As a corollary, we find Rohlin flows on this factor. This is the first example of Rohlin flows on non- McDuff factors. We also apply Masuda-Tomatsu's theorem to our Rohlin flows, and obtain a classification up to strong cocycle conjugacy. However, it also turns out that usual cocycle conjugacy is different from strong cocycle conjugacy for our flows. Consequently, even though Masuda-Tomatsu's analysis is applicable for flows on non-McDuff factors, the effect of their analysis on classification up to cocycle conjugacy is limited. [ABSTRACT FROM AUTHOR]

It is shown how to use non-commutative stopping times in order to stop the CCR flow of arbitrary index and also its isometric cocycles, i.e. left operator Markovian cocycles on Boson Fock space. Stopping the CCR flow yields a homomorphism from the semigroup of stopping times, equipped with the convolution product, into the semigroup of unital endomorphisms of the von Neumann algebra of bounded operators on the ambient Fock space. The operators produced by stopping cocycles themselves satisfy a cocycle relation. [ABSTRACT FROM PUBLISHER]

An operator space analysis of quantum stochastic cocycles is undertaken. These are cocycles with respect to an ampliated CCR flow, adapted to the associated filtration of subspaces, or subalgebras. They form a noncommutative analog of stochastic semigroups in the sense of Skorohod. One-to-one correspondences are established between classes of cocycle of interest and corresponding classes of one-parameter semigroups on associated matrix spaces. Each of these “global” semigroups may be viewed as the expectation semigroup of an associated quantum stochastic cocycle on the corresponding matrix space. Proof of the two key characterizations, namely that of completely positive contraction cocycles on a C*-algebra, and contraction cocycles on a Hilbert space, involves all of the analysis undertaken here. As indicated by Accardi and Kozyrev, the Schur-action matrix semigroup viewpoint circumvents technical (domain) limitations inherent in the theory of quantum stochastic differential equations. [ABSTRACT FROM PUBLISHER]

We prove that for not a nontrivial root of unity the cohomology group defined by invariant 2-cocycles in a completion of is isomorphic to , where P and Q are the weight and root lattices of , respectively. This implies that the group of autoequivalences of the tensor category of -modules is the semidirect product of and the automorphism group of the based root datum of . For q=1 we also obtain similar results for all compact connected separable groups. [ABSTRACT FROM AUTHOR]

The gauge group is computed explicitly for a family of E0-semigroups of type II0 arising from the boundary weight double construction introduced earlier by Jankowski. This family contains many E0-semigroups that are not cocycle conjugate to any examples whose gauge groups have been computed earlier. Further results are obtained regarding the classification up to cocycle conjugacy and up to conjugacy for boundary weight doubles (φ,ν) in two separate cases: first in the case when φ is unital, invertible and q-pure, and ν is any type II Powers weight, and secondly when φ is a unital q-positive map whose range has dimension one and ν(A)=(f,Af) for some function f such that . All E0-semigroups in the former case are cocycle conjugate to the one arising simply from ν, and any two E0-semigroups in the latter case are cocycle conjugate if and only if they are conjugate. [ABSTRACT FROM PUBLISHER]

We define families of invariants for elements of the mapping class group of Σ, a compact-orientable surface. For any characteristic subgroup , let J(H) denote the subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H, we associate a higher-order ρψ-invariant and a signature 2-cocycle σψ. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular, each ρψ is a quasimorphism and each σψ is a bounded 2-cocycle on J(H). In one of the simplest nontrivial cases, by varying ψ, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the ρψ restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on Σ. [ABSTRACT FROM PUBLISHER]

We describe all fusion subcategories of the representation category of a twisted quantum double , where G is a finite group and is a 3-cocycle on G. In view of the fact that every group-theoretical braided fusion category can be embedded into some , this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of . We also give a characterization of group-theoretical braided fusion categories as equivariantizations of pointed categories. [ABSTRACT FROM PUBLISHER]

The long‐time behaviour of Runge–Kunge discretizations is investigated when applied to a smooth nonautonomous index 2 differential algebraic equation (DAE) with a cocycle structure, i.e. a DAE driven by an autonomous dynamical system, which is assumed to have a uniform attractor. It is shown that the cocycle structure of the continuous dynamics is preserved under discretization and that a uniform forward or pullback attractor of the DAE persists under discretization by a Runge–Kutta scheme with the component subsets of the numerical attractor converging upper semicontinuously to their continuous time counterparts. [ABSTRACT FROM PUBLISHER]

We introduce Virasoro‐type extensions for the Neretin spheromorphism groups Np (for each integer p ≥ 2), the combinatorial or p‐adic (when p is prime) analogues of the diffeomorphism group of the circle. The proof of their non‐triviality is based on the computation of the Schur multiplier H2 (Vp, Z) of the Higman–Thompson groups Vp, which are countable subgroups of Np. [ABSTRACT FROM PUBLISHER]

The article focuses on the generalization of the Schwarzian derivative through the use of the three nontrivial cocycles on diffeomorphisms. It discusses the modules of linear differential operators on the space of tensor-densities. The exceptional modules of differential operators is tackled as well as the property of the Schwarzian derivative.

The article focuses on the alternative construction of associative algebra semi-infinite cohomology with regards to graded associative algebras' bar duality. The semi-infinite cohomology in the general enveloping algebra case which coincides with Lie algebra semi-infinite cohomology is cited. The direct calculation of the critical 2-cocycle for affine Lie algebras and for different triangular decompositions is inferred.

The article discusses the construction of geometrically meaningful 1- and 2-cocycles of Lie algebras and infinite-dimensional Lie groups. It mentions that the groups are groups of smooth manifolds' diffeomorphisms, such as the Lie algebras as algebras of vector fields and the group of symplectomorphisms. It states that coefficient spaces of the cocytes are spaces of different tensor fields and the constructions are closely associated with the deformation quantization.

The article presents a report on finite-dimensional Kac algebras arising from certain group actions on a factor. The authors deal with very special 3-cocycles if G-kernels must be actions when restricted to the subgroups N and H because M=L x H and N=LN. They also investigates the cocycles and corresponding finite-dimensional Kac algebras.

These notes contain an Ergodic-theoretic account of the Cocycle Superrigidity Theorem recently discovered by Sorin Popa. We state and prove a relative version of the result, discuss some applications to measurable equivalence relations, and point out that Gaussian actions (of “rigid” groups) satisfy the assumptions of Popa's theorem. [ABSTRACT FROM PUBLISHER]