Algebra structures on hom ( C , L ) (original) (raw)

Coalgebra, Cocomposition and Cohomology

Non-Associative Algebra and Its Applications, 1994

The (co)homology theory of n-ary (co)compositions is a functor associating to n-ary (co)composition a complex. We present unified approach to the cohomology theory of coassociative and Lie coalgebras and for 2n-ary cocompositions. This approach points to a possible generalization.

The Cohomology Structure of an Algebra Entwined with a Coalgebra

Journal of Algebra, 2001

Two cochain complexes are constructed for an algebra A and a coalgebra C entwined with each other via the map ψ : C ⊗ A → A ⊗ C. One complex is associated to an A-bimodule, the other to a C-bicomodule. In the former case the resulting complex can be considered as a ψ-twisted Hochschild complex of A, while for the latter one obtains a ψ-twist of the Cartier complex of C. The notion of a weak comp algebra is introduced by weakening the axioms of the Gerstenhaber comp algebra. It is shown that such a weak comp algebra is a cochain complex with two cup products that descend to the cohomology. It is also shown that the complexes associated to an entwining structure and A or C are examples of a weak comp algebra. Finally both complexes are combined in a double complex whose role in the deformation theory of entwining structures is outlined.

Α-Type Chevalley–Eilenberg Cohomology of Hom-Lie Algebras and Bialgebras

Glasgow Mathematical Journal

The purpose of this paper is to define an α-type cohomology, which we call α-type Chevalley–Eilenberg cohomology, for Hom-Lie algebras. We relate it to the known Chevalley–Eilenberg cohomology and provide explicit computations for some examples. Moreover, using this cohomology, we study formal deformations of Hom-Lie algebras, where the bracket as well as the structure map α are deformed. Furthermore, we provide a generalization of the grand crochet and study, in a particular case, the α-type cohomology for Hom-Lie bialgebras.

Hom-algebras and Hom-coalgebras

2010

The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Homalgebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom-Hopf algebras, and prove some fundamental properties and give examples. Finally, we define the concept of Hom-Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group. 1 2 ABDENACER MAKHLOUF AND SERGEI SILVESTROV six terms in Jacobi identity of the quasi-Lie or of the quasi-Hom-Lie algebras can be combined pairwise in a suitable way. That possibility depends deeply on how the twisting maps interact with each other and with the bracket multiplication.

Twisted cocycles of lie algebras and corresponding invariant functions

Linear Algebra and Its Applications, 2009

We consider finite-dimensional complex Lie algebras. Using certain complex parameters we generalize the concept of cohomology cocycles of Lie algebras. A special case is generalization of 1-cocycles with respect to the adjoint representation -so called (α, β, γ)-derivations. Parametric sets of spaces of cocycles allow us to define complex functions which are invariant under Lie isomorphisms. Such complex functions thus represent useful invariants -we show how they classify three and four-dimensional Lie algebras as well as how they apply to some eight-dimensional one-parametric nilpotent continua of Lie algebras. These functions also provide necessary criteria for existence of 1-parametric continuous contraction.

Cyclic cohomology of Lie algebras

Arxiv preprint arXiv:1108.2806, 2011

In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module over it. This complex is quasi-isomorphic to the Hopf cyclic complex of the enveloping algebra of the Lie algebra with coefficients in the corresponding module. It is shown that the (truncated) Weil algebra, the Weil algebra with generalized coefficients defined by Alekseev-Meinrenken, and the perturbed Koszul complex introduced by Kumar-Vergne are examples of such a mixed complex.

Twisted bialgebras, cofreeness and cointeraction

arXiv (Cornell University), 2019

We study twisted bialgebras and double twisted bialgebras, that is to say bialgebras in the category of linear species, or in the category of species in the category of coalgebras. We define the notion of cofree twisted coalgebra and generalize Hoffman's quasi-shuffle product, obtaining in particular a twisted bialgebra of set compositions Comp. Given a special character, this twisted bialgebra satisfies a terminal property, generalizing the one of the Hopf algebra of quasisymmetric functions proved by Aguiar, Bergeron and Sottile. We give Comp a second coproduct, making it a double twisted bialgebra, and prove that it is a terminal object in the category of double twisted bialgebras. Actions of characters on morphisms allow to obtain every twisted bialgebra morphisms from a connected double twisted bialgebra to Comp. These results are applied to examples based on graphs and on finite topologies, obtaining species versions of the chromatic symmetric series and chromatic polynomials, or of the Ehrhart polynomials. Moreover, through actions of monoids of characters, we obtain a twisted bialgebraic interpretation of the duality principle.

COHOMOLOGY OPERATIONS FOR LIE ALGEBRAS

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 ] 19 N ov 2 00 3 A concept of 23 PROP and deformation theory of ( co ) associative bialgebras

2003

We introduce a concept of 2 3 PROP generalizing the Kontsevich concept of 1 2 PROP. We prove that some Stasheff-type compactification of the Kontsevich spaces K(m, n) defines a topological 2 3 PROP structure. The corresponding chain complex is a minimal model for its cohomology (both are considered as 2 3 PROPs). We construct a 2 3 PROP End(V ) for a vector space V . Finally, we construct a dg Lie algebra controlling the deformations of a (co)associative bialgebra. Philosophically, this construction is a version of the Markl’s operadic construction from [M1] applied to minimal models of 2 3 PROPs.