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Cohomology of Hom-Lie superalgebras and q -deformed Witt superalgebra

Faouzi Ammar, Abdenacer Makhlouf, Nejib Saadaoui (2013)

Czechoslovak Mathematical Journal

Hom-Lie algebra (superalgebra) structure appeared naturally in q -deformations, based on σ -derivations of Witt and Virasoro algebras (superalgebras). They are a twisted version of Lie algebras (superalgebras), obtained by deforming the Jacobi identity by a homomorphism. In this paper, we discuss the concept of α k -derivation, a representation theory, and provide a cohomology complex of Hom-Lie superalgebras. Moreover, we study central extensions. As application, we compute derivations and the second...

Extremal projectors in the semi-classical case

Sophie Chemla (1997)

Annales de l'institut Fourier

Using extremal projectors, Zhelobenko solved extremal equations in a generic Verma module of a complex semi-simple Lie algebra. We will solve similar equations in the semi-classical case. Our proof will be geometric. In the appendix, we give a factorization for the extremal projector of the Virasoro algebra in the semi-classical case.

Highest Weight Modules of W1+∞, Darboux Transformations and the Bispectral Problem

Bakalov, B., Horozov, E., Yakimov, M. (1997)

Serdica Mathematical Journal

This paper is a survey of our recent results on the bispectral problem. We describe a new method for constructing bispectral algebras of any rank and illustrate the method by a series of new examples as well as by all previously known ones. Next we exhibit a close connection of the bispectral problem to the representation theory of W1+∞–algerba. This connection allows us to explain and generalise to any rank the result of Magri and Zubelli on the symmetries of the manifold of the bispectral operators...

Nombre de rotation, structures géométriques sur un cercle et groupe de Bott-Virasoro

Laurent Guieu (1996)

Annales de l'institut Fourier

Une classification complète des stabilisateurs coadjoints du groupe de Bott-Virasoro est obtenue par une méthode essentiellement géométrique. L’outil de base est le nombre de rotation d’un difféomorphisme du cercle. En particulier, nous mettons en évidence la présence de groupes d’isotropie non-connexes et montrons que la transformation de Miura des opérateurs de Hill peut s’interpréter comme une application moment sur l’espace des structures affines du cercle.

On Griess algebras.

Roitman, Michael (2008)

SIGMA. Symmetry, Integrability and Geometry: Methods and Applications [electronic only]

On the cohomology of vector fields on parallelizable manifolds

Yuly Billig, Karl-Hermann Neeb (2008)

Annales de l’institut Fourier

In the present paper we determine for each parallelizable smooth compact manifold M the second cohomology spaces of the Lie algebra 𝒱 M of smooth vector fields on M with values in the module Ω ¯ M p = Ω M p / d Ω M p - 1 . The case of p = 1 is of particular interest since the gauge algebra of functions on M with values in a finite-dimensional simple Lie algebra has the universal central extension with center Ω ¯ M 1 , generalizing affine Kac-Moody algebras. The second cohomology H 2 ( 𝒱 M , Ω ¯ M 1 ) classifies twists of the semidirect product of 𝒱 M with the...

Semi-infinite cohomology and superconformal algebras

Elena Poletaeva (2001)

Annales de l’institut Fourier

We describe representations of certain superconformal algebras in the semi-infinite Weil complex related to the loop algebra of a complex finite-dimensional Lie algebra and in the semi-infinite cohomology. We show that in the case where the Lie algebra is endowed with a non-degenerate invariant symmetric bilinear form, the relative semi-infinite cohomology of the loop algebra has a structure, which is analogous to the classical structure of the de Rham cohomology in Kähler...

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