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Cohomology of G / P for classical complex Lie supergroups G and characters of some atypical G -modules

Ivan Penkov, Vera Serganova (1989)

Annales de l'institut Fourier

We compute the unique nonzero cohomology group of a generic G 0 - linearized locally free 𝒪 -module, where G 0 is the identity component of a complex classical Lie supergroup G and P ↪ G 0 is an arbitrary parabolic subsupergroup. In particular we prove that for G ≠ ℙ ( m ) , S ℙ ( m ) this cohomology group is an irreducible G 0 -module. As an application we generalize the character formula of typical irreducible G 0 -modules to a natural class of atypical modules arising in this way.

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...

Cohomology of tensor product of quantum planes

Paolo Papi (1992)

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

We consider the Lie algebra of inner derivations of the n -fold tensor product of Manin quantum planes and compute its second cohomology group with trivial coefficients.

Cohomology ring of n-Lie algebras.

Mikolaj Rotkiewicz (2005)

Extracta Mathematicae

Natural graded Lie brackets on the space of cochains of n-Leibniz and n-Lie algebras are introduced. It turns out that these brackets agree under the natural embedding introduced by Gautheron. Moreover, n-Leibniz and n-Lie algebras turn to be canonical structures for these brackets in a similar way in which associative algebras (respectively, Lie algebras) are canonical structures for the Gerstenhaber bracket (respectively, Nijenhuis-Richardson bracket).

Combinatorial bases of modules for affine Lie algebra B 2(1)

Mirko Primc (2013)

Open Mathematics

We construct bases of standard (i.e. integrable highest weight) modules L(Λ) for affine Lie algebra of type B 2(1) consisting of semi-infinite monomials. The main technical ingredient is a construction of monomial bases for Feigin-Stoyanovsky type subspaces W(Λ) of L(Λ) by using simple currents and intertwining operators in vertex operator algebra theory. By coincidence W(kΛ0) for B 2(1) and the integrable highest weight module L(kΛ0) for A 1(1) have the same parametrization of combinatorial bases...

Commutator algebras arising from splicing operations

Sergei Sverchkov (2014)

Open Mathematics

We prove that the variety of Lie algebras arising from splicing operation coincides with the variety CM of centreby-metabelian Lie algebras. Using these Lie algebras we find the minimal dimension algebras generated the variety CM and the variety of its associative envelope algebras. We study the splicing n-ary operation. We show that all n-ary (n > 2) commutator algebras arising from this operation are nilpotent of index 3. We investigate the generalization of the splicing n-ary operation, and...

Comparaison des homologies du groupe linéaire et de son algèbre de Lie

Jean-Louis Loday (1987)

Annales de l'institut Fourier

Pour un anneau local R l’homologie du groupe discret G L n ( R ) a un comportement tout à fait analogue à l’homologie de l’algèbre de Lie g l n ( A ) lorsque A est une algèbre associative sur un corps de caractéristique zéro. L’objet de cet article est de faire une synthèse (sans démonstration) des résultats connus sur ces groupes d’homologie en exhibant leurs liens avec la K -théorie algébrique, l’homologie cyclique et la cohomologie motivique. On y pose un certain nombre de questions et on propose une définition pour...

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