Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise

Luigi Accardi; Andreas Boukas

Displaying similar documents to “Central extensions and stochastic processes associated with the Lie algebra of the renormalized higher powers of white noise”

Integrating central extensions of Lie algebras via Lie 2-groups

Christoph Wockel, Chenchang Zhu (2016)

Journal of the European Mathematical Society

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The purpose of this paper is to show how central extensions of (possibly infinite-dimensional) Lie algebras integrate to central extensions of étale Lie 2-groups in the sense of [Get09, Hen08]. In finite dimensions, central extensions of Lie algebras integrate to central extensions of Lie groups, a fact which is due to the vanishing of $π_{2}$ for each finite-dimensional Lie group. This fact was used by Cartan (in a slightly other guise) to construct the simply connected Lie group associated...

Extensions of hom-Lie algebras in terms of cohomology

Abdoreza R. Armakan, Mohammed Reza Farhangdoost (2017)

Czechoslovak Mathematical Journal

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We study (non-abelian) extensions of a given hom-Lie algebra and provide a geometrical interpretation of extensions, in particular, we characterize an extension of a hom-Lie algebra $𝔤$ by another hom-Lie algebra $𝔥$ and discuss the case where $𝔥$ has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, Maurer-Cartan formula, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological...

Universal central extension of direct limits of Hom-Lie algebras

Valiollah Khalili (2019)

Czechoslovak Mathematical Journal

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We prove that the universal central extension of a direct limit of perfect Hom-Lie algebras $(ℒ_{i}, α_{ℒ_{i}})$ is (isomorphic to) the direct limit of universal central extensions of $(ℒ_{i}, α_{ℒ_{i}})$ . As an application we provide the universal central extensions of some multiplicative Hom-Lie algebras. More precisely, we consider a family of multiplicative Hom-Lie algebras ${({sl}_{k} (å), α_{k})}_{k \in I}$ and describe the universal central extension of its direct limit.

On the relative fundamental solutions for a second order differential operator on the Heisenberg group

T. Godoy, L. Saal (2001)

Studia Mathematica

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Let Hₙ be the (2n+1)-dimensional Heisenberg group, let p,q ≥ 1 be integers satisfying p+q=n, and let $L = \sum_{j = 1}^{p} (X ²_{j} + Y ²_{j}) - \sum_{j = p + 1}^{n} (X ²_{j} + Y ²_{j})$ , where X₁,Y₁,...,Xₙ,Yₙ,T denotes the standard basis of the Lie algebra of Hₙ. We compute explicitly a relative fundamental solution for L.

A note on central extensions of Lie groups.

Neeb, Karl-Hermann (1996)

Journal of Lie Theory

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Normal integral bases and tameness conditions for Kummer extensions

Ilaria Del Corso, Lorenzo Paolo Rossi (2013)

Acta Arithmetica

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We present a detailed analysis of some properties of a general tamely ramified Kummer extension of number fields L/K. Our main achievement is a criterion for the existence of a normal integral basis for a general Kummer extension, which generalizes the existing results. Our approach also allows us to explicitly describe the Steinitz class of L/K and we get an easy criterion for this class to be trivial. In the second part of the paper we restrict to the particular case of tame Kummer...

Ergodicity of ℤ² extensions of irrational rotations

Yuqing Zhang (2011)

Studia Mathematica

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Let = [0,1) be the additive group of real numbers modulo 1, α ∈ be an irrational number and t ∈ . We study ergodicity of skew product extensions T : × ℤ² → × ℤ², $T (x, s ₁, s ₂) = (x + α, s ₁ + 2 χ_{[0, 1 / 2)} (x) - 1, s ₂ + 2 χ_{[0, 1 / 2)} (x + t) - 1)$ .

Relative Bogomolov extensions

Robert Grizzard (2015)

Acta Arithmetica

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A subfield K ⊆ ℚ̅ has the Bogomolov property if there exists a positive ε such that no non-torsion point of $K^{\times}$ has absolute logarithmic height below ε. We define a relative extension L/K to be Bogomolov if this holds for points of $L^{\times} ∖ K^{\times}$ . We construct various examples of extensions which are and are not Bogomolov. We prove a ramification criterion for this property, and use it to show that such extensions can always be constructed if some rational prime has bounded ramification index in K. ...

White noise distribution theory and its application

Yoshihito Shimada (2007)

Banach Center Publications

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The paper gives a new application of the white noise distribution theory via a proof of irreducibility of the energy representation of a group of $C^{\infty}$ -maps from a compact Riemann manifold to a semi-simple compact Lie group.

The Wells map for abelian extensions of 3-Lie algebras

Youjun Tan, Senrong Xu (2019)

Czechoslovak Mathematical Journal

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The Wells map relates automorphisms with cohomology in the setting of extensions of groups and Lie algebras. We construct the Wells map for some abelian extensions $0 \to A ↪ L \overset{π}{\to} B \to 0$ of 3-Lie algebras to obtain obstruction classes in $H^{1} (B, A)$ for a pair of automorphisms in $Aut (A) \times Aut (B)$ to be inducible from an automorphism of $L$ . Application to free nilpotent 3-Lie algebras is discussed.

The variety of dual mock-Lie algebras

Luisa M. Camacho, Ivan Kaygorodov, Viktor Lopatkin, Mohamed A. Salim (2020)

Communications in Mathematics

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We classify all complex $7$ - and $8$ -dimensional dual mock-Lie algebras by the algebraic and geometric way. Also, we find all non-trivial complex $9$ -dimensional dual mock-Lie algebras.

Class group of a cyclotomic $ℤ_{p} \times ℤ_{ℓ}$ -extension

Humio Ichimura (2011)

Acta Arithmetica

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Group Extensions with Infinite Conjugacy Classes

Jean-Philippe Préaux (2013)

Confluentes Mathematici

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We characterize the group property of being with infinite conjugacy classes (or , infinite and of which all conjugacy classes except ${1}$ are infinite) for groups which are extensions of groups. We prove a general result for extensions of groups, then deduce characterizations in semi-direct products, wreath products, finite extensions, among others examples we also deduce a characterization for amalgamated products and HNN extensions. The icc property is correlated to the Theory of von...

Symbolic extensions in intermediate smoothness on surfaces

David Burguet (2012)

Annales scientifiques de l'École Normale Supérieure

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We prove that $𝒞^{r}$ maps with $r > 1$ on a compact surface have symbolic extensions, i.e., topological extensions which are subshifts over a finite alphabet. More precisely we give a sharp upper bound on the so-called symbolic extension entropy, which is the infimum of the topological entropies of all the symbolic extensions. This answers positively a conjecture of S. Newhouse and T. Downarowicz in dimension two and improves a previous result of the author [11].

When unit groups of continuous inverse algebras are regular Lie groups

Helge Glöckner, Karl-Hermann Neeb (2012)

Studia Mathematica

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It is a basic fact in infinite-dimensional Lie theory that the unit group $A^{\times}$ of a continuous inverse algebra A is a Lie group. We describe criteria ensuring that the Lie group $A^{\times}$ is regular in Milnor’s sense. Notably, $A^{\times}$ is regular if A is Mackey-complete and locally m-convex.

Crystallographic actions on Lie groups and post-Lie algebra structures

Dietrich Burde (2021)

Communications in Mathematics

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This survey on crystallographic groups, geometric structures on Lie groups and associated algebraic structures is based on a lecture given in the Ostrava research seminar in $2017$ .

On a noncommutative Iwasawa main conjecture for varieties over finite fields

Malte Witte (2014)

Journal of the European Mathematical Society

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We formulate and prove an analogue of the noncommutative Iwasawa main conjecture for $ℓ$ -adic Lie extensions of a separated scheme $X$ of finite type over a finite field of characteristic prime to $ℓ$ .

Algorithmic computations of Lie algebras cohomologies

Šilhan, Josef

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From the text: The aim of this work is to advertise an algorithmic treatment of the computation of the cohomologies of semisimple Lie algebras. The base is Kostant’s result which describes the representation of the proper reductive subalgebra on the cohomologies space. We show how to (algorithmically) compute the highest weights of irreducible components of this representation using the Dynkin diagrams. The software package $L i E$ offers the data structures and corresponding procedures for...