Results 1 
9 of
9
The classical HomYangBaxter equation and HomLie bialgebras
, 905
"... Abstract. Motivated by recent work on HomLie algebras and the HomYangBaxter equation, we introduce a twisted generalization of the classical YangBaxter equation (CYBE), called the classical HomYangBaxter equation (CHYBE). We show how an arbitrary solution of the CYBE induces multiple infinite ..."
Abstract

Cited by 6 (4 self)
 Add to MetaCart
Abstract. Motivated by recent work on HomLie algebras and the HomYangBaxter equation, we introduce a twisted generalization of the classical YangBaxter equation (CYBE), called the classical HomYangBaxter equation (CHYBE). We show how an arbitrary solution of the CYBE induces multiple infinite families of solutions of the CHYBE. We also introduce the closely related structure of HomLie bialgebras, which generalize Drinfel’d’s Lie bialgebras. In particular, we study the questions of duality and cobracket perturbation and the subclasses of coboundary and quasitriangular HomLie bialgebras. 1.
The HomYangBaxter equation and HomLie algebras
, 2009
"... Abstract. Motivated by recent work on HomLie algebras, a twisted version of the YangBaxter equation, called the HomYangBaxter equation (HYBE), was introduced by the author in [62]. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE a ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
Abstract. Motivated by recent work on HomLie algebras, a twisted version of the YangBaxter equation, called the HomYangBaxter equation (HYBE), was introduced by the author in [62]. In this paper, several more classes of solutions of the HYBE are constructed. Some of these solutions of the HYBE are closely related to the quantum enveloping algebra of sl(2), the JonesConway polynomial, and YetterDrinfel’d modules. We also construct a new infinite sequence of solutions of the HYBE from a given one. Along the way, we compute all the Lie algebra endomorphisms on the (1 + 1)Poincaré algebra and sl(2). 1.
Homquantum groups II:cobraided Hombialgebras and Homquantum geometry, arXiv:0907.1880v1
 UNIVERSITÉ DE HAUTEALSACE, LABORATOIRE DE MATHÉMATIQUES, INFORMATIQUE ET APPLICATIONS, 4 RUE DES FRÈRES LUMIÈRE, 68093
, 2009
"... Abstract. A class of nonassociative and noncoassociative generalizations of cobraided bialgebras, called cobraided Hombialgebras, is introduced. The non(co)associativity in a cobraided Hombialgebra is controlled by a twisting map. Several methods for constructing cobraided Hombialgebras are giv ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
Abstract. A class of nonassociative and noncoassociative generalizations of cobraided bialgebras, called cobraided Hombialgebras, is introduced. The non(co)associativity in a cobraided Hombialgebra is controlled by a twisting map. Several methods for constructing cobraided Hombialgebras are given. In particular, Homtype generalizations of FRT quantum groups, including quantum matrices and related quantum groups, are obtained. Each cobraided Hombialgebra comes with solutions of the operator quantum HomYangBaxter equations, which are twisted analogues of the operator form of the quantum YangBaxter equation. Solutions of the HomYangBaxter equation can be obtained from comodules of suitable cobraided Hombialgebras. Homtype generalizations of the usual quantum matrices coactions on the quantum planes give rise to nonassociative and noncoassociative analogues of quantum geometry.
HomHopf algebras
"... Abstract. Homstructures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Homstructures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Homalgebras coincide with algebra ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
Abstract. Homstructures (Lie algebras, algebras, coalgebras, Hopf algebras) have been investigated in the literature recently. We study Homstructures from the point of view of monoidal categories; in particular, we introduce a symmetric monoidal category such that Homalgebras coincide with algebras in this monoidal category, and similar properties for coalgebras, Hopf algebras and Lie algebras.
Homquantum groups I: quasitriangular Hombialgebras
"... Abstract. We introduce a Homtype generalization of quantum groups, called quasitriangular Hombialgebras. They are nonassociative and noncoassociative analogues of Drinfel’d’s quasitriangular bialgebras, in which the non(co)associativity is controlled by a twisting map. A family of quasitriang ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Abstract. We introduce a Homtype generalization of quantum groups, called quasitriangular Hombialgebras. They are nonassociative and noncoassociative analogues of Drinfel’d’s quasitriangular bialgebras, in which the non(co)associativity is controlled by a twisting map. A family of quasitriangular Hombialgebras can be constructed from any quasitriangular bialgebra, such as Drinfel’d’s quantum enveloping algebras. Each quasitriangular Hombialgebra comes with a solution of the quantum HomYangBaxter equation, which is a nonassociative version of the quantum YangBaxter equation. Solutions of the HomYangBaxter equation can be obtained from modules of suitable quasitriangular Hombialgebras. 1.
Homalternative algebras and HomJordan algebras
"... ABSTRACT. The purpose of this paper is to introduce Homalternative algebras and HomJordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Homassociative algebra leads to ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
ABSTRACT. The purpose of this paper is to introduce Homalternative algebras and HomJordan algebras. We discuss some of their properties and provide construction procedures using ordinary alternative algebras or Jordan algebras. Also, we show that a polarization of Homassociative algebra leads to HomJordan algebra.
HOMNOVIKOV ALGEBRAS
, 909
"... Abstract. We study a twisted generalization of Novikov algebras, called HomNovikov algebras, in which the two defining identities are twisted by a linear map. It is shown that HomNovikov algebras can be obtained from Novikov algebras by twisting along any algebra endomorphism. All algebra endomorp ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We study a twisted generalization of Novikov algebras, called HomNovikov algebras, in which the two defining identities are twisted by a linear map. It is shown that HomNovikov algebras can be obtained from Novikov algebras by twisting along any algebra endomorphism. All algebra endomorphisms on complex Novikov algebras of dimensions two or three are computed, and their associated HomNovikov algebras are described explicitly. Another class of HomNovikov algebras is constructed from Homcommutative algebras together with a derivation, generalizing a construction due to Dorfman and Gel’fand. Two other classes of HomNovikov algebras are constructed from HomLie algebras together with a suitable linear endomorphism, generalizing a
Abstract
, 906
"... The purpose of this paper is to study HomLie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of Γgraded quasiLie algebras introduced by Larsson and Silvestrov. In this paper, we characterize HomLi ..."
Abstract
 Add to MetaCart
The purpose of this paper is to study HomLie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of Γgraded quasiLie algebras introduced by Larsson and Silvestrov. In this paper, we characterize HomLie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of HomLie superalgebras deforming the orthosymplectic Lie superalgebra. Also, we prove a Z2graded version of a HartwigLarssonSilvestrov Theorem which leads us to a construction of a qdeformed Witt superalgebra.