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Displaying similar documents to “An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds”

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 ± 1 , ... , x n ± 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 ) { ± 1 } K * , and give a short proof that Aut Lie ( W n ) Aut K - alg ( L n ) GL n ( ) K * n .

Universal lifting theorem and quasi-Poisson groupoids

David Inglesias-Ponte, Camille Laurent-Gengoux, Ping Xu (2012)

Journal of the European Mathematical Society

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We prove the universal lifting theorem: for an α -simply connected and α -connected Lie groupoid Γ with Lie algebroid A , the graded Lie algebra of multi-differentials on A is isomorphic to that of multiplicative multi-vector fields on Γ . As a consequence, we obtain the integration theorem for a quasi-Lie bialgebroid, which generalizes various integration theorems in the literature in special cases. The second goal of the paper is the study of basic properties of quasi-Poisson groupoids....

Leibniz's rule on two-step nilpotent Lie groups

Krystian Bekała (2016)

Colloquium Mathematicae

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Let be a nilpotent Lie algebra which is also regarded as a homogeneous Lie group with the Campbell-Hausdorff multiplication. This allows us to define a generalized multiplication f g = ( f g ) of two functions in the Schwartz class (*), where and are the Abelian Fourier transforms on the Lie algebra and on the dual * and ∗ is the convolution on the group . In the operator analysis on nilpotent Lie groups an important notion is the one of symbolic calculus which can be viewed as a higher order...

On the stratification of the orbit space for the action of automorphisms on connections

Witold Kondracki, Jan Rogulski

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CONTENTSIntroduction..................................................................................................................................................5§1. Basic notions and notation.....................................................................................................................7  1.1. Automorphisms of principal bundles....................................................................................................7  1.2. Connections and parallel...

The Banach-Lie Group of Lie Automorphisms of an H * -Algebra

Antonio J. Calderón Martín, Candido Martín González (2007)

Bollettino dell'Unione Matematica Italiana

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We study the Banach-Lie group Aut ( A - ) of Lie automorphisms of a complex associative H * -algebra. Also some consequences about its Lie algebra, the algebra of Lie derivations of A , are obtained. For a topologically simple A , in the infinite-dimensional case we have Aut ( A - ) 0 = Aut ( A ) implying Der ( A ) = Der ( A - ) . In the finite dimensional case Aut ( A - ) 0 is a direct product of Aut ( A ) and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.

Antiassociative groupoids

Milton Braitt, David Hobby, Donald Silberger (2017)

Mathematica Bohemica

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Given a groupoid G , , and k 3 , we say that G is antiassociative if an only if for all x 1 , x 2 , x 3 G , ( x 1 x 2 ) x 3 and x 1 ( x 2 x 3 ) are never equal. Generalizing this, G , is k -antiassociative if and only if for all x 1 , x 2 , ... , x k G , any two distinct expressions made by putting parentheses in x 1 x 2 x 3 x k are never equal. We prove that for every k 3 , there exist finite groupoids that are k -antiassociative. We then generalize this, investigating when other pairs of groupoid terms can be made never equal.

𝔤 -quasi-Frobenius Lie algebras

David N. Pham (2016)

Archivum Mathematicum

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A Lie version of Turaev’s 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...

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 L 1 ... L t = L of L such that for every i = 1 , 2 , ... , t , we have H + L i = H + L i - 1 or H L i = H 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.

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 . In a paper by A. Ayupov, K. Kudaybergenov (2016), the authors studied the local derivations on semi-simple Lie algebras over 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....

On the nilpotent residuals of all subalgebras of Lie algebras

Wei Meng, Hailou Yao (2018)

Czechoslovak Mathematical Journal

Similarity:

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 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 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 1 . By S ( L ) denote the terminal term of the ascending series. It is proved that L = S ( L ) if and only if L 𝒩 is nilpotent. In addition, we investigate the basic properties of a...

Classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center

Bin Ren, Lin Sheng Zhu (2017)

Czechoslovak Mathematical Journal

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A Lie algebra L is called 2-step nilpotent if L is not abelian and [ L , L ] lies in the center of L . 2-step nilpotent Lie algebras are useful in the study of some geometric problems, and their classification has been an important problem in Lie theory. In this paper, we give a classification of 2-step nilpotent Lie algebras of dimension 9 with 2-dimensional center.

Two-spinor tetrad and Lie derivatives of Einstein-Cartan-Dirac fields

Daniel Canarutto (2018)

Archivum Mathematicum

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An integrated approach to Lie derivatives of spinors, spinor connections and the gravitational field is presented, in the context of a previously proposed, partly original formulation of a theory of Einstein-Cartan-Maxwell-Dirac fields based on “minimal geometric data”: the needed underlying structure is determined, via geometric constructions, from the unique assumption of a complex vector bundle S M with 2-dimensional fibers, called a 2 -spinor bundle. Any further considered object is...

-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 ( -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 -omomorfismi propri. Si assegnano condizioni necessarie e sufficienti affinchè un'algebra di Lie periodica o mista possieda un « -omomorfismo su una catena di lunghezza n .

The evolution and Poisson kernels on nilpotent meta-abelian groups

Richard Penney, Roman Urban (2013)

Studia Mathematica

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Let S be a semidirect product S = N⋊ A where N is a connected and simply connected, non-abelian, nilpotent meta-abelian Lie group and A is isomorphic to k , k>1. We consider a class of second order left-invariant differential operators on S of the form α = L a + Δ α , where α k , and for each a k , L a is left-invariant second order differential operator on N and Δ α = Δ - α , , where Δ is the usual Laplacian on k . Using some probabilistic techniques (e.g., skew-product formulas for diffusions on S and N respectively)...