On the cohomology based on the generalized representations of nnn-Lie Algebras (original) (raw)

2000

If L is a Lie algebra over R and Z its centre, the natural inclu- sion Z, ! (L) extends to a representation i :Z! EndH(L;R )o f the exterior algebra of Z in the cohomology of L. We begin a study of this rep- resentation by examining its Poincar e duality properties, its associated higher cohomology operations and its

A cohomology theory for Lie 2-algebras

2018

In this article, we introduce a new cohomology theory associated to a Lie 2-algebras. This cohomology theory is shown to extend the classical cohomology theory of Lie algebras; in particular, we show that the second cohomology group classifies an appropriate type of extensions.

A cohomology theory for Lie 2-algebras and Lie 2-groups

arXiv: Differential Geometry, 2018

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the classical cohomology theories of Lie algebras and Lie groups in that their second groups classify extensions. We use this fact together with an adapted van Est map to prove the integrability of Lie 2-algebras anew.

Cohomology of graded Lie algebras of maximal class

Journal of Algebra, 2006

It was shown by A. Fialowski that an arbitrary infinite-dimensional N-graded "filiform type" Lie algebra g= ∞ i=1 g i with one-dimensional homogeneous components g i such that [g 1 , g i ] = g i+1 , ∀i ≥ 2 over a field of zero characteristic is isomorphic to one (and only one) Lie algebra from three given ones: m 0 , m 2 , L 1 , where the Lie algebras m 0 and m 2 are defined by their structure relations: m 0 : [e 1 , e i ] = e i+1 , ∀i ≥ 2 and m 2 : [e 1 , e i ] = e i+1 , ∀i ≥ 2, [e 2 , e j ] = e j+2 , ∀j ≥ 3 and L 1 is the "positive" part of the Witt algebra. In the present article we compute the cohomology H * (m 0) and H * (m 2) with trivial coefficients, give explicit formulas for their representative cocycles and describe the multiplicative structure in the cohomology. Also we discuss the relations with combinatorics and representation theory. The cohomology H * (L 1

Representations of the generalized Lie algebra

Journal of Physics A: Mathematical and General, 1998

We construct finite-dimensional irreducible representations of two quantum algebras related to the generalized Lie algebra sl(2) q introduced by Lyubashenko and the second named author. We consider separately the cases of q generic and q at roots of unity. Some of the representations have no classical analog even for generic q. Some of the representations have no analog to the finite-dimensional representations of the quantised enveloping algebra U q (sl(2)), while in those that do there are different matrix elements.

Representations and cohomology of n-ary multiplicative Hom–Nambu–Lie algebras

Journal of Geometry and Physics, 2011

The aim of this paper is to provide cohomologies of n-ary Hom-Nambu-Lie algebras governing central extensions and one parameter formal deformations. We generalize to n-ary algebras the notions of derivations and representation introduced by Sheng for Hom-Lie algebras. Also we show that a cohomology of n-ary Hom-Nambu-Lie algebras could be derived from the cohomology of Hom-Leibniz algebras.

On the Cohomology of Infinite Dimensional Nilpotent Lie Algebras

Advances in Mathematics, 1993

In the paper one-and two-dimensional cohomology is compared for finite and infinite nilpotent Lie algebras, with coefficients in the adjoint representation. It turns out that, because the adjoint representation is not a highest weight representation in infinite dimension, the considered cohomology shows basic differences. On my visit to M.I.T., B. Kostant asked the following question: What is the main difference between the cohomology of finite and infinite dimensional nilpotent Lie algebras with coefficients in the adjoint representation, at what points does the generalization of the finite dimensional situation fail? Understanding this difference is especially important as the nilpotent Lie algebra cohomology is very hard to compute and in both finite and infinite dimensional cases only the one-and two-dimensional cohomology is known so far.

Three contributions in representation theory

The aim of the current dissertation is to address certain problems in the representation theory of simple Lie algebras and associated quantum algebras. In Part I, we study the simple Lie group of type G 2 from the cluster point of view. We prove a conjecture of Geiss, Leclerc and Schröer [25], relating the geometry of the partial flag varieties to cluster algebras in the case of G 2. In Part II, we establish a concrete relationship between certain infinite dimensional quantum groups, namely the quantum loop algebra U (Lg) and the Yangian Y (g), associated with a simple Lie algebra g. The main result of this part gives a construction of an explicit isomorphism between completions of these algebras, thus strengthening the well known Drinfeld's degeneration homomorphism [10, 27]. In Part III, we give a proof of the monodromy conjecture of Toledano Laredo [50] for the case of g = sl 2. The monodromy conjecture relates two classes of representations of the affine braid group, the first arising from the quantum Weyl group operators of the quantum loop algebra and the second coming from the monodromy of the trigonometric Casimir connection, a flat connection introduced by Toledano Laredo [50]. To Prof. Valerio Toledano Laredo, for teaching me the monodromy side of quantum groups, for your constant availability and for having great patience with me during all our joint projects.

Representations and cohomology of -ary multiplicative Hom–Nambu–Lie algebras

Journal of Geometry and Physics, 2011

On the cohomology based on the generalized representations of nnn-Lie Algebras (original) (raw)

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