Algebras whose groups of units are Lie groups

Helge Glöckner

Displaying similar documents to “Algebras whose groups of units are Lie 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.

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...

Group algebras whose groups of normalized units have exponent 4

Victor Bovdi, Mohammed Salim (2018)

Czechoslovak Mathematical Journal

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We give a full description of locally finite $2$ -groups $G$ such that the normalized group of units of the group algebra $F G$ over a field $F$ of characteristic $2$ has exponent $4$ .

Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups

K. H. Neeb (2014)

Annales de l’institut Fourier

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A unitary representation $π$ of a, possibly infinite dimensional, Lie group $G$ is called semibounded if the corresponding operators $i d π (x)$ from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra $𝔤$ of $G$ . We classify all irreducible semibounded representations of the groups ${\hat{ℒ}}_{φ} (K)$ which are double extensions of the twisted loop group $ℒ_{φ} (K)$ , where $K$ is a simple Hilbert–Lie group (in the sense that the scalar product on its Lie algebra is invariant)...

The unit groups of semisimple group algebras of some non-metabelian groups of order $144$

Gaurav Mittal, Rajendra K. Sharma (2023)

Mathematica Bohemica

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We consider all the non-metabelian groups $G$ of order $144$ that have exponent either $36$ or $72$ and deduce the unit group $U (𝔽_{q} G)$ of semisimple group algebra $𝔽_{q} G$ . Here, $q$ denotes the power of a prime, i.e., $q = p^{r}$ for $p$ prime and a positive integer $r$ . Up to isomorphism, there are $6$ groups of order $144$ that have exponent either $36$ or $72$ . Additionally, we also discuss how to simply obtain the unit groups of the semisimple group algebras of those non-metabelian groups of order $144$ that are a direct product of two...

$𝔤$ -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...

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...

Divisors in global analytic sets

Francesca Acquistapace, A. Díaz-Cano (2011)

Journal of the European Mathematical Society

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We prove that any divisor $Y$ of a global analytic set $X \subset ℝ^{n}$ has a generic equation, that is, there is an analytic function vanishing on $Y$ with multiplicity one along each irreducible component of $Y$ . We also prove that there are functions with arbitrary multiplicities along $Y$ . The main result states that if $X$ is pure dimensional, $Y$ is locally principal, $X / Y$ is not connected and $Y$ represents the zero class in $H_{q - 1}^{\infty} (X, ℤ_{2})$ then the divisor $Y$ is globally principal.

$\mathcal{L}$ -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 $n$ .

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 $U \in 𝔄$ .

A pure smoothness condition for Radó’s theorem for $α$ -analytic functions

Abtin Daghighi, Frank Wikström (2016)

Czechoslovak Mathematical Journal

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Let $Ω \subset ℂ^{n}$ be a bounded, simply connected $ℂ$ -convex domain. Let $α \in ℤ_{+}^{n}$ and let $f$ be a function on $Ω$ which is separately $C^{2 α_{j} - 1}$ -smooth with respect to $z_{j}$ (by which we mean jointly $C^{2 α_{j} - 1}$ -smooth with respect to $Re z_{j}$ , $Im z_{j}$ ). If $f$ is $α$ -analytic on $Ω ∖ f^{- 1} (0)$ , then $f$ is $α$ -analytic on $Ω$ . The result is well-known for the case $α_{i} = 1$ , $1 \leq i \leq n$ , even when $f$ a priori is only known to be continuous.

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.