Results 1  10
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15
Generalization of nary Nambu algebras and beyond
, 2008
"... Abstract. The aim of this paper is to introduce nary Homalgebra structures generalizing the nary algebras of Lie type enclosing nary Nambu algebras, nary NambuLie algebras, nary Lie algebras, and nary algebras of associative type enclosing nary totally associative and nary partially associ ..."
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Cited by 11 (1 self)
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Abstract. The aim of this paper is to introduce nary Homalgebra structures generalizing the nary algebras of Lie type enclosing nary Nambu algebras, nary NambuLie algebras, nary Lie algebras, and nary algebras of associative type enclosing nary totally associative and nary partially associative algebras. Also, we provide a way to construct examples starting from an nary algebra and an nary algebras endomorphism. Several examples could be derived using this process.
HOMBIALGEBRAS AND COMODULE ALGEBRAS
, 810
"... Abstract. We construct a Hombialgebra M(2) representing the functor of 2×2matrices on Homassociative algebras. We also construct a Homalgebra analogue of the affine plane and show that it is a comodule Homalgebra over M(2) in a suitable sense. 1. ..."
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Cited by 10 (5 self)
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Abstract. We construct a Hombialgebra M(2) representing the functor of 2×2matrices on Homassociative algebras. We also construct a Homalgebra analogue of the affine plane and show that it is a comodule Homalgebra over M(2) in a suitable sense. 1.
HOMYANGBAXTER EQUATION, HOMLIE ALGEBRAS, AND QUASITRIANGULAR BIALGEBRAS
, 903
"... Abstract. We study a twisted version of the YangBaxter Equation, called the HomYangBaxter Equation (HYBE), which is motivated by HomLie algebras. Three classes of solutions of the HYBE are constructed, one from HomLie algebras and the others from Drinfeld’s (dual) quasitriangular bialgebras. Ea ..."
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Cited by 9 (5 self)
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Abstract. We study a twisted version of the YangBaxter Equation, called the HomYangBaxter Equation (HYBE), which is motivated by HomLie algebras. Three classes of solutions of the HYBE are constructed, one from HomLie algebras and the others from Drinfeld’s (dual) quasitriangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group. 1.
Module Homalgebras
"... Abstract. We study a twisted version of module algebras called module Homalgebras. It is shown that module algebras deform into module Homalgebras via endomorphisms. As an example, we construct certain qdeformations of the usual sl(2)action on the affine plane. 1. ..."
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Cited by 6 (5 self)
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Abstract. We study a twisted version of module algebras called module Homalgebras. It is shown that module algebras deform into module Homalgebras via endomorphisms. As an example, we construct certain qdeformations of the usual sl(2)action on the affine plane. 1.
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 ..."
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Cited by 6 (4 self)
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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.
On Hom type algebras
, 903
"... Homalgebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist. We enumerate here all the possible choices in the Lie and associative categories and study the relations between the obtained algebras. The associative case is r ..."
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Cited by 5 (0 self)
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Homalgebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist. We enumerate here all the possible choices in the Lie and associative categories and study the relations between the obtained algebras. The associative case is richer since it admits the notion of unit element. We use this fact to find sufficient conditions for homassociative algebras to be associative and classify the implications between the homassociative types of unital algebras. Introduction. Homalgebras were first introduced in the Lie case by Hartwig, Larson and Silverstrov in [3]. This notion was then extended in the associative case by Makhlouf and Silverstrov in [7]. It turns out that many of the classical
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 ..."
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Cited by 5 (3 self)
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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 ..."
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Cited by 4 (1 self)
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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 ..."
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Cited by 3 (0 self)
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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 ..."
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Cited by 3 (2 self)
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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.