Deformations associated with rigid algebras

Gerstenhaber, Murray; Giaquinto, Anthony

doi:10.1007/s40062-013-0068-x

Cited by 5 publications

(6 citation statements)

References 33 publications

(43 reference statements)

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“…Therefore, the complex C KV (A, ) is the solution to the conjecture of Muray Gerstenhaber in the category of locally flat manifolds [27].…”

Section: Anomaly Functions Of Algebroids and Of Modulesmentioning

confidence: 99%

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Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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Let us begin by considering two book titles: A provocative title, What Is a Statistical Model? McCullagh (2002) and an alternative title, In a Search for Structure. The Fisher Information. Gromov (2012). It is the richness in open problems and the links with other research domains that make a research topic exciting. Information geometry has both properties. Differential information geometry is the differential geometry of statistical models. The topology of information is the topology of statistical models. This highlights the importance of both questions raised by Peter McCullagh and Misha Gromov. The title of this paper looks like a list of key words. However, the aim is to emphasize the links between those topics. The theory of homology of Koszul-Vinberg algebroids and their modules (KV homology in short) is a useful key for exploring those links. In Part A we overview three constructions of the KV homology. The first construction is based on the pioneering brute formula of the coboundary operator. The second construction is based on the theory of semi-simplicial objects. The third construction is based on the anomaly functions of abstract algebras and their abstract modules. We use the KV homology for investigating links between differential information geometry and differential topology. For instance, "dualistic relation of Amari" and "Riemannian or symplectic Foliations"; "Koszul geometry" and "linearization of webs"; "KV homology" and "complexity of models". Regarding the complexity of a model, the challenge is to measure how far from being an exponential family is a given model. In Part A we deal with the classical theory of models. Part B is devoted to answering both questions raised by McCullagh and B Gromov. A few criticisms and examples are used to support our criticisms and to motivate a new approach. In a given category an outstanding challenge is to find an invariant which encodes the points of a moduli space. In Part B we face four challenges. (1) The introduction of a new theory of statistical models. This re-establishment must answer both questions of McCullagh and Gromov; (2) The search for an characteristic invariant which encodes the points of the moduli space of isomorphism class of models; (3) The introduction of the theory of homological statistical models. This is a pioneering notion. We address its links with Hessian geometry; (4) We emphasize the links between the classical theory of models, the new theory and Vanishing Theorems in the theory of homological statistical models. Subsequently, the differential information geometry has a homological nature. That is another notable feature of our approach. This paper is dedicated to our friend and colleague Alexander Grothendieck.

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“…Therefore, the complex C KV (A, ) is the solution to the conjecture of Muray Gerstenhaber in the category of locally flat manifolds [27].…”

Section: Anomaly Functions Of Algebroids and Of Modulesmentioning

confidence: 99%

“…About relationships between the theory of deformation and the theory of cohomology, the readers are referred to [9,27,56].…”

Section: Theorem 14 a Necessary Condition For (M D) Being Hyperbolimentioning

confidence: 99%

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Boyom

2016

Entropy

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“…There are a few routes one could take, such as examining A q m (k) for m ≥ 2, or more generally, addressing Problem 2 for multiparameter quantum Weyl algebras as in [28, Why care? One reason is that quantum Weyl algebras have appeared in numerous works in mathematics and physics, including Deformation Theory [27,28,37,39], Knot Theory [24], Category Theory [48], Quantum mechanics and Hypergeometric Functions [65] to name a few. Therefore, any (partial) resolution to Problem 2 would be a welcomed addition to the literature.…”

Section: Take Another Representationmentioning

confidence: 99%

An Invitation to Noncommutative Algebra

Walton

2019

Association for Women in Mathematics Series

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“…This need not be the case when one has a perturbation which is not a deformation, for in that case, some 'fragile' cohomology classes may be lost at a given specialization. The cohomology of the specialized algebra consists of those original 'resilient' classes that survive, together with others which may arise as a result of the specialization, cf [11]. The family of q-Weyl algebras provides an extreme example.…”

Section: Semigroup Algebras Coherent Cochains and Twistsmentioning

confidence: 99%

Path algebras, wave-particle duality, and quantization of phase space

Gerstenhaber

2016

Lett Math Phys

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Semigroup algebras admit certain `coherent' deformations which, in the special case of a path algebra, may associate a periodic function to an evolving path; for a particle moving freely on a straight line after an initial impulse, the wave length is that hypothesized by de Broglie's wave-particle duality. This theory leads to a model of "physical" phase space of which mathematical phase space, the cotangent bundle of configuration space, is a projection. This space is singular, quantized at the Planck level, its structure implies the existence of spin, and the spread of a packet can be described as a random walk. The wavelength associated to a particle moving in this space need not be constant and its phase can change discontinuously.Comment: 15 pages, replaces arXiv1505.0589; to appear Letters in Mathematical Physic

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Deformations associated with rigid algebras

Cited by 5 publications

References 33 publications

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

Foliations-Webs-Hessian Geometry-Information Geometry-Entropy and Cohomology

An Invitation to Noncommutative Algebra

Path algebras, wave-particle duality, and quantization of phase space

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