Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups

K. H. Neeb

Displaying similar documents to “Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups”

SCAP-subalgebras of Lie algebras

Sara Chehrazi, Ali Reza Salemkar (2016)

Czechoslovak Mathematical Journal

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A subalgebra $H$ of a finite dimensional Lie algebra $L$ is said to be a $SCAP$ -subalgebra if there is a chief series $0 = L_{0} \subset L_{1} \subset ... \subset L_{t} = L$ of $L$ such that for every $i = 1, 2, ..., t$ , we have $H + L_{i} = H + L_{i - 1}$ or $H \cap L_{i} = H \cap L_{i - 1}$ . This is analogous to the concept of $SCAP$ -subgroup, which has been studied by a number of authors. In this article, we investigate the connection between the structure of a Lie algebra and its $SCAP$ -subalgebras and give some sufficient conditions for a Lie algebra to be solvable or supersolvable.

$𝔤$ -quasi-Frobenius Lie algebras

David N. Pham (2016)

Archivum Mathematicum

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A Lie version of Turaev’s $\bar{G}$ -Frobenius algebras from 2-dimensional homotopy quantum field theory is proposed. The foundation for this Lie version is a structure we call a $𝔤$ -quasi-Frobenius Lie algebra for $𝔤$ a finite dimensional Lie algebra. The latter consists of a quasi-Frobenius Lie algebra $(𝔮, β)$ together with a left $𝔤$ -module structure which acts on $𝔮$ via derivations and for which $β$ is $𝔤$ -invariant. Geometrically, $𝔤$ -quasi-Frobenius Lie algebras are the Lie algebra structures associated to...

Hall algebra of morphism category

QingHua Chen, Liwang Zhang (2024)

Czechoslovak Mathematical Journal

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This paper investigates a universal PBW-basis and a minimal set of generators for the Hall algebra $ℋ (C_{2} (𝒫))$ , where $C_{2} (𝒫)$ is the category of morphisms between projective objects in a finitary hereditary exact category $𝒜$ . When $𝒜$ is the representation category of a Dynkin quiver, we develop multiplication formulas for the degenerate Hall Lie algebra $ℒ$ , which is spanned by isoclasses of indecomposable objects in $C_{2} (𝒫)$ . As applications, we demonstrate that $ℒ$ contains a Lie subalgebra isomorphic to the central...

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Si studiano gli omomorfismi reticolari ( $\mathcal{L}$ -omomorfismi) di algebre di Lie sopra anelli commutativi con unità. Le algebre di Lie sopra un campo e le $p$ -algebre di Lie non ammettono $\mathcal{L}$ -omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un « $\mathcal{L}$ -omomorfismo su una catena di lunghezza $n$ .

Tempered reductive homogeneous spaces

Yves Benoist, Toshiyuki Kobayashi (2015)

Journal of the European Mathematical Society

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Let $G$ be a semisimple algebraic Lie group and $H$ a reductive subgroup. We find geometrically the best even integer $p$ for which the representation of $G$ in $L^{2} (G / H)$ is almost $L^{p}$ . As an application, we give a criterion which detects whether this representation is tempered.

The covering semigroup of invariant control systems on Lie groups

Víctor Ayala, Eyüp Kizil (2016)

Kybernetika

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It is well known that the class of invariant control systems is really relevant both from theoretical and practical point of view. This work was an attempt to connect an invariant systems on a Lie group $G$ with its covering space. Furthermore, to obtain algebraic properties of this set. Let $G$ be a Lie group with identity $e$ and $Σ \subset 𝔤$ a cone in the Lie algebra $𝔤$ of $G$ that satisfies the Lie algebra rank condition. We use a formalism developed by Sussmann, to obtain an algebraic structure on...

Nonlinear $*$ -Lie higher derivations of standard operator algebras

Mohammad Ashraf, Shakir Ali, Bilal Ahmad Wani (2018)

Communications in Mathematics

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Let $ℋ$ be an infinite-dimensional complex Hilbert space and $𝔄$ be a standard operator algebra on $ℋ$ which is closed under the adjoint operation. It is shown that every nonlinear $*$ -Lie higher derivation $𝒟 = {δ_{n}}_{n \in ℕ}$ of $𝔄$ is automatically an additive higher derivation on $𝔄$ . Moreover, $𝒟 = {δ_{n}}_{n \in ℕ}$ is an inner $*$ -higher derivation.

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang, Feng Wei, Ajda Fošner (2019)

Czechoslovak Mathematical Journal

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Let $ℛ$ be a commutative ring, $𝒢$ be a generalized matrix algebra over $ℛ$ with weakly loyal bimodule and $𝒵 (𝒢)$ be the center of $𝒢$ . Suppose that $𝔮 : 𝒢 \times 𝒢 \to 𝒢$ is an $ℛ$ -bilinear mapping and that $𝔗_{𝔮} : 𝒢 \to 𝒢$ is a trace of $𝔮$ . The aim of this article is to describe the form of $𝔗_{𝔮}$ satisfying the centralizing condition $[𝔗_{𝔮} (x), x] \in 𝒵 (𝒢)$ (and commuting condition $[𝔗_{𝔮} (x), x] = 0$ ) for all $x \in 𝒢$ . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) $𝔗_{𝔮}$ has the so-called proper form from a new perspective. Using the aforementioned...

Local superderivations on Lie superalgebra $𝔮 (n)$

Haixian Chen, Ying Wang (2018)

Czechoslovak Mathematical Journal

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Let $𝔮 (n)$ be a simple strange Lie superalgebra over the complex field $ℂ$ . In a paper by A. Ayupov, K. Kudaybergenov (2016), the authors studied the local derivations on semi-simple Lie algebras over $ℂ$ and showed the difference between the properties of local derivations on semi-simple and nilpotent Lie algebras. We know that Lie superalgebras are a generalization of Lie algebras and the properties of some Lie superalgebras are similar to those of semi-simple Lie algebras, but $𝔭 (n)$ is an exception....

$\mathcal{L}$ -homomorphisms of Lie algebras

Aleksander A. Lashkhi (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Invariant symbolic calculus for semidirect products

Benjamin Cahen (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be the semidirect product $V ⋊ K$ where $K$ is a connected semisimple non-compact Lie group acting linearly on a finite-dimensional real vector space $V$ . Let $π$ be a unitary irreducible representation of $G$ which is associated by the Kirillov-Kostant method of orbits with a coadjoint orbit of $G$ whose little group is a maximal compact subgroup of $K$ . We construct an invariant symbolic calculus for $π$ , under some technical hypothesis. We give some examples including the Poincaré group. ...

Multiplicative Lie triple derivations on standard operator algebras

Bilal Ahmad Wani (2021)

Communications in Mathematics

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Let $𝒳$ be a Banach space of dimension $n > 1$ and $𝔄 \subset ℬ (𝒳)$ be a standard operator algebra. In the present paper it is shown that if a mapping $d : 𝔄 \to 𝔄$ (not necessarily linear) satisfies $d ([[U, V], W]) = [[d (U), V], W] + [[U, d (V)], W] + [[U, V], d (W)]$ for all $U, V, W \in 𝔄$ , then $d = ψ + τ$ , where $ψ$ is an additive derivation of $𝔄$ and $τ : 𝔄 \to 𝔽 I$ vanishes at second commutator $[[U, V], W]$ for all $U, V, W \in 𝔄$ . Moreover, if $d$ is linear and satisfies the above relation, then there exists an operator $S \in 𝔄$ and a linear mapping $τ$ from $𝔄$ into $𝔽 I$ satisfying $τ ([[U, V], W]) = 0$ for all $U, V, W \in 𝔄$ , such that $d (U) = S U - U S + τ (U)$ for all $U \in 𝔄$ .

Generalized Higher Derivations on Lie Ideals of Triangular Algebras

Mohammad Ashraf, Nazia Parveen, Bilal Ahmad Wani (2017)

Communications in Mathematics

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Let $𝔄 = (\begin{matrix} 𝒜 & ℳ \\ ℬ \end{matrix})$ be the triangular algebra consisting of unital algebras $𝒜$ and $ℬ$ over a commutative ring $R$ with identity $1$ and $ℳ$ be a unital $(𝒜, ℬ)$ -bimodule. An additive subgroup $𝔏$ of $𝔄$ is said to be a Lie ideal of $𝔄$ if $[𝔏, 𝔄] \subseteq 𝔏$ . A non-central square closed Lie ideal $𝔏$ of $𝔄$ is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on $𝔄$ , every generalized Jordan triple higher derivation of $𝔏$ into $𝔄$ is a generalized higher derivation of $𝔏$ into $𝔄$ . ...

Generating varieties for the triple loop space of classical Lie groups

Yasuhiko Kamiyama (2003)

Fundamenta Mathematicae

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For G = SU(n), Sp(n) or Spin(n), let $C_{G} (S U (2))$ be the centralizer of a certain SU(2) in G. We have a natural map $J : G / C_{G} (S U (2)) \to Ω ₀ ³ G$ . For a generator α of $H ⁎ (G / C_{G} (S U (2)); ℤ / 2)$ , we describe J⁎(α). In particular, it is proved that $J ⁎ : H ⁎ (G / C_{G} (S U (2)); ℤ / 2) \to H ⁎ (Ω ₀ ³ G; ℤ / 2)$ is injective.

On the nilpotent residuals of all subalgebras of Lie algebras

Wei Meng, Hailou Yao (2018)

Czechoslovak Mathematical Journal

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Let $𝒩$ denote the class of nilpotent Lie algebras. For any finite-dimensional Lie algebra $L$ over an arbitrary field $𝔽$ , there exists a smallest ideal $I$ of $L$ such that $L / I \in 𝒩$ . This uniquely determined ideal of $L$ is called the nilpotent residual of $L$ and is denoted by $L^{𝒩}$ . In this paper, we define the subalgebra $S (L) = ⋂_{H \leq L} I_{L} (H^{𝒩})$ . Set $S_{0} (L) = 0$ . Define $S_{i + 1} (L) / S_{i} (L) = S (L / S_{i} (L))$ for $i \geq 1$ . By $S_{\infty} (L)$ denote the terminal term of the ascending series. It is proved that $L = S_{\infty} (L)$ if and only if $L^{𝒩}$ is nilpotent. In addition, we investigate the basic properties of a...

Subalgebra of $L_{1} (G)$ associated with Laplacian on a Lie group

A. Hulanicki (1974)

Colloquium Mathematicae

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Quantized semisimple Lie groups

Rita Fioresi, Robert Yuncken (2024)

Archivum Mathematicum

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The goal of this expository paper is to give a quick introduction to $q$ -deformations of semisimple Lie groups. We discuss principally the rank one examples of $𝒰_{q} ({𝔰𝔩}_{2})$ , $𝒪 ({SU}_{q} (2))$ , $𝒟 ({SL}_{q} (2, ℂ))$ and related algebras. We treat quantized enveloping algebras, representations of $𝒰_{q} ({𝔰𝔩}_{2})$ , generalities on Hopf algebras and quantum groups, $*$ -structures, quantized algebras of functions on $q$ -deformed compact semisimple groups, the Peter-Weyl theorem, $*$ -Hopf algebras associated to complex semisimple Lie groups and the Drinfeld...

Dual Blobs and Plancherel Formulas

Ju-Lee Kim (2004)

Bulletin de la Société Mathématique de France

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Let $k$ be a $p$ -adic field. Let $G$ be the group of $k$ -rational points of a connected reductive group $𝖦$ defined over $k$ , and let $𝔤$ be its Lie algebra. Under certain hypotheses on $𝖦$ and $k$ , wethe tempered dual $\hat{G}$ of $G$ via the Plancherel formula on $𝔤$ , using some character expansions. This involves matching spectral decomposition factors of the Plancherel formulas on $𝔤$ and $G$ . As a consequence, we prove that any tempered representation contains a good minimal $𝖪$ -type; we extend this result to irreducible...

The groups of automorphisms of the Witt $W_{n}$ and Virasoro Lie algebras

Vladimir V. Bavula (2016)

Czechoslovak Mathematical Journal

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Let $L_{n} = K [x_{1}^{\pm 1}, ..., x_{n}^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_{n} : = {Der}_{K} (L_{n})$ the Lie algebra of $K$ -derivations of the algebra $L_{n}$ , the so-called Witt Lie algebra, and let $Vir$ be the Virasoro Lie algebra which is a $1$ -dimensional central extension of the Witt Lie algebra. The Lie algebras $W_{n}$ and $Vir$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${Aut}_{Lie} (Vir) ≃ {Aut}_{Lie} (W_{1}) ≃ {\pm 1} ⋉ K^{*}$ , and give a short proof that ${Aut}_{Lie} (W_{n}) ≃ {Aut}_{K - alg} (L_{n}) ≃ {GL}_{n} (ℤ) ⋉ K^{* n}$ .