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-graded Lie superalgebras of infinite depth and finite growth

Nicoletta Cantarini (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In 1998 Victor Kac classified infinite-dimensional -graded Lie superalgebras of finite depth. We construct new examples of infinite-dimensional Lie superalgebras with a -gradation of infinite depth and finite growth and classify -graded Lie superalgebras of infinite depth and finite growth under suitable hypotheses.

A 4 3 -grading on a 56 -dimensional simple structurable algebra and related fine gradings on the simple Lie algebras of type E

Diego Aranda-Orna, Alberto Elduque, Mikhail Kochetov (2014)

Commentationes Mathematicae Universitatis Carolinae

We describe two constructions of a certain 4 3 -grading on the so-called Brown algebra (a simple structurable algebra of dimension 56 and skew-dimension 1 ) over an algebraically closed field of characteristic different from 2 . The Weyl group of this grading is computed. We also show how this grading gives rise to several interesting fine gradings on exceptional simple Lie algebras of types E 6 , E 7 and E 8 .

An extension of Rais' theorem and seaweed subalgebras of simple Lie algebras

Dmitri I. Panyushev (2005)

Annales de l’institut Fourier

We prove an extension of Rais' theorem on the coadjoint representation of certain graded Lie algebras. As an application, we prove that, for the coadjoint representation of any seaweed subalgebra in a general linear or symplectic Lie algebra, there is a generic stabiliser and the field of invariants is rational. It is also shown that if the highest root of a simple Lie algerba is not fundamental, then there is a parabolic subalgebra whose coadjoint representation do not...

Characteristic zero loop space homology for certain two-cones

Calin Popescu (1999)

Commentationes Mathematicae Universitatis Carolinae

Given a principal ideal domain R of characteristic zero, containing 1/2, and a two-cone X of appropriate connectedness and dimension, we present a sufficient algebraic condition, in terms of Adams-Hilton models, for the Hopf algebra F H ( Ω X ; R ) to be isomorphic with the universal enveloping algebra of some R -free graded Lie algebra; as usual, F stands for free part, H for homology, and Ω for the Moore loop space functor.

Coalgebraic Approach to the Loday Infinity Category, Stem Differential for 2 n -ary Graded and Homotopy Algebras

Mourad Ammar, Norbert Poncin (2010)

Annales de l’institut Fourier

We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space V . The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on V (resp. graded Loday structures on V , sequences that we call Loday infinity structures on V ). We prove a minimal model theorem for Loday infinity algebras and observe that the Lod category contains the L category as...

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 ( m ) , S ( 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 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).

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