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Beni Yoshida - One of the best experts on this subject based on the ideXlab platform.
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gapped boundaries group cohomology and fault tolerant logical gates
Annals of Physics, 2017Co-Authors: Beni YoshidaAbstract:This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the d-dimensional quantum double model by using d-Cocycles functions (d≥2). We point out that the system supports mm-dimensional excitations (m
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gapped boundaries group cohomology and fault tolerant logical gates
arXiv: Strongly Correlated Electrons, 2015Co-Authors: Beni YoshidaAbstract:This paper attempts to establish the connection among classifications of gapped boundaries in topological phases of matter, bosonic symmetry-protected topological (SPT) phases and fault-tolerantly implementable logical gates in quantum error-correcting codes. We begin by presenting constructions of gapped boundaries for the $d$-dimensional quantum double model by using $d$-Cocycles functions ($d\geq 2$). We point out that the system supports $m$-dimensional excitations ($m
Takefumi Nosaka - One of the best experts on this subject based on the ideXlab platform.
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on third homologies of groups and of quandles via the dijkgraaf witten invariant and inoue kabaya map
Algebraic & Geometric Topology, 2014Co-Authors: Takefumi NosakaAbstract:We propose a simple method for producing quandle Cocycles from group Cocycles by a modification of the Inoue‐Kabaya chain map. Further, we show that, with respect to “universal extension of quandles”, the chain map induces an isomorphism between third homologies (modulo some torsion). For example, all Mochizuki’s quandle 3‐Cocycles are shown to be derived from group Cocycles. As an application, we calculate some Z‐equivariant parts of the Dijkgraaf‐Witten invariants of some cyclic branched covering spaces, via some Cocycle invariant of links. 20J06, 57M12; 57M27, 57N65
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quandle Cocycles from invariant theory
Advances in Mathematics, 2013Co-Authors: Takefumi NosakaAbstract:Abstract Let G be a group. Any G -module M has an algebraic structure called a G -family of Alexander quandles. Given a 2-Cocycle of a cohomology associated with this G -family, topological invariants of (handlebody) knots in the 3-sphere are defined. We develop a simple algorithm to algebraically construct n -Cocycles of this G -family from G -invariant group n -Cocycles of the abelian group M . We present many examples of 2-Cocycles of these G -families using facts from (modular) invariant theory.
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on third homologies of groups and of quandles via the dijkgraaf witten invariant and inoue kabaya map
arXiv: Geometric Topology, 2012Co-Authors: Takefumi NosakaAbstract:We propose a simple method to produce quandle Cocycles from group Cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-Cocycles are shown to be derived from group Cocycles of some non-abelian group. As an application, we calculate some $\Z$-equivariant parts of the Dijkgraaf-Witten invariants of some cyclic branched covering spaces, via some Cocycle invariant of links.
Nina J Rutten - One of the best experts on this subject based on the ideXlab platform.
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poisson brackets symmetry from the pentagon wheel Cocycle in the graph complex
Physics of Particles and Nuclei, 2018Co-Authors: Ricardo Buring, Arthemy V Kiselev, Nina J RuttenAbstract:Kontsevich designed a scheme to generate infinitesimal symmetries $$\dot {\mathcal{P}} = \mathcal{Q}(\mathcal{P})$$ of Poisson brackets $$\mathcal{P}$$ on all affine manifolds $${{M}^{r}};$$ every such deformation is encoded by oriented graphs on $$n + 2$$ vertices and $$2n$$ edges. In particular, these symmetries can be obtained by orienting sums of non-oriented graphs γ on n vertices and $$2n - 2$$ edges. The bi-vector flow $$\dot {\mathcal{P}} = {{\text{O}\vec{\text{r}}}}(\gamma )(\mathcal{P})$$ preserves the space of Poisson structures if γ is a Cocycle with respect to the vertex-expanding differential d in the graph complex. A class of such Cocycles $${{\gamma }_{{2\ell + 1}}}$$ is known to exist: marked by $$\ell \in \mathbb{N},$$ each of them contains a $$(2\ell + 1)$$ -gon wheel with a nonzero coefficient. At $$\ell = 1$$ the tetrahedron $${{\gamma }_{3}}$$ itself is a Cocycle; at $$\ell = 2$$ the Kontsevich–Willwacher pentagon-wheel Cocycle $${{\gamma }_{5}}$$ consists of two graphs. We reconstruct the symmetry $${{\mathcal{Q}}_{5}}(\mathcal{P}) = {{\text{O}\vec{\text{r}}}}({{\gamma }_{5}})(\mathcal{P})$$ and verify that $${{\mathcal{Q}}_{5}}$$ is a Poisson Cocycle indeed: $$\left[\kern-0.15em\left[ {\mathcal{P},{{\mathcal{Q}}_{5}}(\mathcal{P})} \right]\kern-0.15em\right] \doteq 0$$ via $$\left[\kern-0.15em\left[ {\mathcal{P},\mathcal{P}} \right]\kern-0.15em\right] = 0.$$
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the heptagon wheel Cocycle in the kontsevich graph complex
Journal of Nonlinear Mathematical Physics, 2017Co-Authors: Ricardo Buring, Arthemy V Kiselev, Nina J RuttenAbstract:The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on n vertices and 2n − 2 edges, induce–under the orientation mapping–infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial Cocycle that contains the (2l + 1)-wheel graph with a nonzero coefficient at every l∈ℕ. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel Cocycles, consisting at l = 1 and l = 2 of one and two graphs respectively, the Cocycle condition d(γ) = 0 is verified by hand. For the next, heptagonwheel Cocycle (known to exist at l = 3), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.
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poisson brackets symmetry from the pentagon wheel Cocycle in the graph complex
arXiv: Mathematical Physics, 2017Co-Authors: Ricardo Buring, Arthemy V Kiselev, Nina J RuttenAbstract:Kontsevich designed a scheme to generate infinitesimal symmetries $\dot{\mathcal{P}} = \mathcal{Q}(\mathcal{P})$ of Poisson brackets $\mathcal{P}$ on all affine manifolds $M^r$; every such deformation is encoded by oriented graphs on $n+2$ vertices and $2n$ edges. In particular, these symmetries can be obtained by orienting sums of non-oriented graphs $\gamma$ on $n$ vertices and $2n-2$ edges. The bi-vector flow $\dot{\mathcal{P}} = \text{Or}(\gamma)(\mathcal{P})$ preserves the space of Poisson structures if $\gamma$ is a Cocycle with respect to the vertex-expanding differential in the graph complex. A class of such Cocycles $\boldsymbol{\gamma}_{2\ell+1}$ is known to exist: marked by $\ell \in \mathbb{N}$, each of them contains a $(2\ell+1)$-gon wheel with a nonzero coefficient. At $\ell=1$ the tetrahedron $\boldsymbol{\gamma}_3$ itself is a Cocycle; at $\ell=2$ the Kontsevich--Willwacher pentagon-wheel Cocycle $\boldsymbol{\gamma}_5$ consists of two graphs. We reconstruct the symmetry $\mathcal{Q}_5(\mathcal{P}) = \text{Or}(\boldsymbol{\gamma}_5)(\mathcal{P})$ and verify that $\mathcal{Q}_5$ is a Poisson Cocycle indeed: $[\![\mathcal{P},\mathcal{Q}_5(\mathcal{P})]\!]\doteq 0$ via $[\![\mathcal{P},\mathcal{P}]\!]=0$.
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the heptagon wheel Cocycle in the kontsevich graph complex
arXiv: Combinatorics, 2017Co-Authors: Ricardo Buring, Arthemy V Kiselev, Nina J RuttenAbstract:The real vector space of non-oriented graphs is known to carry a differential graded Lie algebra structure. Cocycles in the Kontsevich graph complex, expressed using formal sums of graphs on $n$ vertices and $2n-2$ edges, induce -- under the orientation mapping -- infinitesimal symmetries of classical Poisson structures on arbitrary finite-dimensional affine real manifolds. Willwacher has stated the existence of a nontrivial Cocycle that contains the $(2\ell+1)$-wheel graph with a nonzero coefficient at every $\ell\in\mathbb{N}$. We present detailed calculations of the differential of graphs; for the tetrahedron and pentagon-wheel Cocycles, consisting at $\ell = 1$ and $\ell = 2$ of one and two graphs respectively, the Cocycle condition $d(\gamma) = 0$ is verified by hand. For the next, heptagon-wheel Cocycle (known to exist at $\ell = 3$), we provide an explicit representative: it consists of 46 graphs on 8 vertices and 14 edges.
Hans Henrik Rugh - One of the best experts on this subject based on the ideXlab platform.
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Regularity of Characteristic Exponents and Linear Response for Transfer Operator Cocycles
Communications in Mathematical Physics, 2021Co-Authors: Julien Sedro, Hans Henrik RughAbstract: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 $$C^k$$ 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 $$C^{k-1}$$ 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.
Stephen J Wills - One of the best experts on this subject based on the ideXlab platform.
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HOMOMORPHIC FELLER CocycleS ON A C∗-ALGEBRA
2015Co-Authors: J Martin, Stephen J WillsAbstract:Abstract. When a Fock-adapted Feller Cocycle on a C∗-algebra is regular, completely positive and contractive it possesses a stochastic generator that is necessarily completely bounded. Here necessary and sufficient conditions are given, in the form of a sequence of identities, for a completely bounded map to generate a weakly multiplicative Cocycle. These are derived from a product formula for iterated quantum stochastic integrals. Under two alterna-tive assumptions, one of which covers all previously considered cases, the first identity in the sequence is shown to imply the rest. To appear in Journal of the London Mathematical Society In a previous paper we showed that completely bounded mapping matrices on a C∗-algebra (or indeed any operator space) stochastically generate Markovian cocy-cles on the algebra. Since stochastic generators of (Fock-adapted Markov-regular) completely positive contraction Cocycles on a C∗-algebra are necessarily completel
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quantum stochastic Cocycles and completely bounded semigroups on operator spaces
International Mathematics Research Notices, 2014Co-Authors: Martin J Lindsay, Stephen J WillsAbstract: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 analogue 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. The classes of Cocycle covered include completely positive contraction Cocycles on an operator system, or C*-algebra; completely contractive Cocycles on an operator space; and contraction operator Cocycles on a Hilbert space. 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. An infinitesimal analysis of quantum stochastic Cocycles from the present wider perspective is given in a sister paper.
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dilation of markovian Cocycles on a von neumann algebra
Pacific Journal of Mathematics, 2003Co-Authors: Debashish Goswami, Martin J Lindsay, Kalyan B. Sinha, Stephen J WillsAbstract:We consider normal Markovian Cocycles on a von Neumann algebra which are adapted to a Fock filtration. Every such Cocycle k which is Markov-regular and consists of completely positive contractions is realised as a conditioned ∗ -homomorphic Cocycle. This amounts to a stochastic generalisation of a recent dilation result for norm-continuous normal completely positive contraction semigroups. To achieve this stochastic dilation we use the fact that k is governed by a quantum stochastic differential equation whose coefficient matrix has a specific structure, and extend a technique for obtaining stochastic flow generators from Markov semigroup generators, to the context of Cocycles. Number/exchange-free dilatability is seen to be related to locality in the case where the Cocycle is a Markovian semigroup. In the same spirit unitary dilations of Markov-regular contraction Cocycles on a Hilbert space are also described. The paper ends with a discussion of connections with measure valued diffusion.