arXiv subjects
Publications and source records attributed to Florin Panaite.
Given two associative algebras A, C and a linear space V together with some linear maps R_1, R_2, R_3, E satisfying some conditions, we define an associative algebra structure on A\otimes V\otimes C called a two-sided crossed product. Particular cases of this construction are the iterated twisted tensor product of algebras and the two-sided crossed product over a quasi-bialgebra.
Given an associative algebra H, a linear space U and some linear maps J, T, γ, ηsatisfying some axioms, we define an associative algebra structure on U\otimes H, called an L-R-crossed product. This contains as particular cases some previous constructions, such as the (iterated) Brzezinski crossed product and the L-R-smash product over quasi-bialgebras.
We generalize to arbitrary categories of algebras the notion of an NS-algebra. We do this by using a bimodule property, as we did for defining the general notions of a dendriform and tridendriform algebra. We show that several types of operators lead to NS-algebras: Nijenhuis operators, twisted Rota-Baxter operators and relative Rota-Baxter operators of arbitrary weight.
The purpose of this paper is to introduce and study BiHom-NS-algebras, which are a generalization of NS-algebras using two homomorphisms. Moreover, we discuss their relationships with twisted Rota-Baxter operators in a BiHom-associative context. Furthermore, we introduce a generalization of Nijenhuis operators that lead to BiHom-NS-algebras along BiHom-associative algebras
We study BiHom-Novikov-Poisson algebras, which are twisted generalizations of Novikov-Poisson algebras and Hom-Novikov-Poisson algebras, and find that BiHom-Novikov-Poisson algebras are closed under tensor products and several kinds of perturbations. Necessary and sufficient conditions are given under which BiHom-Novikov-Poisson algebras give rise to BiHom-Poisson algebras.
We introduce and study infinitesimal BiHom-bialgebras, BiHom-Novikov algebras, BiHom-Novikov-Poisson algebras, and find some relations among these concepts. Our main result is to show how to obtain a left BiHom-pre-Lie algebra from an infinitesimal BiHom-bialgebra
We investigate some properties of Rota-Baxter operators on BiHom-Lie algebras. Along the way, we introduce BiHom analogues of pre-Lie and Leibniz algebras.
We generalize three results of M. Aguiar, which are valid for Loday's dendriform algebras, to arbitrary dendriform algebras, i.e., dendriform algebras associated to algebras satisfying any given set of relations. We define these dendriform algebras using a bimodule property and show how the dendriform relations are easily determined. An important concept which we use is the notion of polarization of an algebra, which we generalize here to (arbitrary) dendriform algebras: it leads to a generalization of two of Aguiar's results, dealing with deformations and filtrations of dendriform algebras. We also introduce weak Rota-Baxter operators for arbitrary algebras, which lead to the construction of generalized dendriform algebras and to a generalization of Aguiar's third result, which provides an interpretation of the natural relation between infinitesimal bialgebras and pre-Lie algebras in terms of dendriform algebras. Throughout the text, we give many examples and show how they are related.
The purpose of this paper is to study Rota-Baxter operators for BiHom-associative algebras. Moreover, we introduce and discuss the properties of the notions of BiHom-(tri)dendriform algebra, BiHom-Zinbiel algebra and BiHom-quadri-algebra. We construct the free Rota-Baxter BiHom-associative algebra and present some observations about categories and functors related to Rota-Baxter structures.
We construct an explicit isomorphism between the quasitriangular quasi-Hopf algebra $D^ω(H)$ defined in \cite{bp} and a certain quantum double quasi-Hopf algebra. We give also new characterizations for a quasitriangular quasi-Hopf algebra to be ribbon and use them to construct some ribbon elements for $D^ω(H)$.
We introduce the concept of {sigma, tau}-Rota-Baxter operator, as a twisted version of a Rota-Baxter operator of weight zero. We show how to obtain a certain {sigma, tau}-Rota-Baxter operator from a solution of the associative (Bi)Hom-Yang-Baxter equation, and, in a compatible way, a Hom-pre-Lie algebra from an infinitesimal Hom-bialgebra.
We introduce a new type of categorical object called a \emph{hom-tensor category} and show that it provides the appropriate setting for modules over an arbitrary hom-bialgebra. Next we introduce the notion of \emph{hom-braided category} and show that this is the right setting for modules over quasitriangular hom-bialgebras. We also show how the hom-Yang-Baxter equation fits into this framework and how the category of Yetter-Drinfeld modules over a hom-bialgebra with bijective structure map can be organized as a hom-braided category. Finally we prove that, under certain conditions, one can obtain a tensor category (respectively a braided tensor category) from a hom-tensor category (respectively a hom-braided category).
We define the concept of weak pseudotwistor for an algebra $(A, μ)$ in a monoidal category $\mathcal{C}$, as a morphism $T:A\otimes A\rightarrow A\otimes A$ in $\mathcal{C}$, satisfying some axioms ensuring that $(A, μ\circ T)$ is also an algebra in $\mathcal{C}$. This concept generalizes the previous proposal called pseudotwistor and covers a number of exemples of twisted algebras that cannot be covered by pseudotwistors, mainly examples provided by Rota-Baxter operators and some of their relatives (such as Leroux's TD-operators and Reynolds operators). By using weak pseudotwistors, we introduce an equivalence relation (called "twist equivalence") for algebras in a given monoidal category.
A BiHom-associative algebra is a (nonassociative) algebra $A$ endowed with two commuting multiplicative linear maps $α,β\colon A\rightarrow A$ such that $α(a)(bc)=(ab)β(c)$, for all $a, b, c\in A$. This concept arose in the study of algebras in so-called group Hom-categories. In this paper, we introduce as well BiHom-Lie algebras (also by using the categorical approach) and BiHom-bialgebras. We discuss these new structures by presenting some basic properties and constructions (representations, twisted tensor products, smash products etc).
If $A\otimes _{R, σ}V$ and $A\otimes _{P, ν}W$ are two Brzeziński crossed products and $Q:W\otimes V\rightarrow V\otimes W$ is a linear map satisfying certain properties, we construct a Brzeziński crossed product $A\otimes _{S, θ}(V\otimes W)$. This construction contains as a particular case the iterated twisted tensor product of algebras.
We introduce the Hom-analogue of the L-R-smash product and use it to define the Hom-analogue of the diagonal crossed product. When H is a finite dimensional Hom-Hopf algebra with bijective antipode and bijective structure map, we define the Drinfeld double of H; its algebra structure is a Hom-diagonal crossed product and it has all expected properties, namely it is quasitriangular and modules over it coincide with left-right Yetter-Drinfeld modules over H.
The purpose of this paper is to provide new constructions of Hom-associative algebras using Hom-analogues of certain operators called twistors and pseudotwistors, by deforming a given Hom-associative multiplication into a new Hom-associative multiplication. As examples, we introduce Hom-analogues of the twisted tensor product and smash product. Furthermore we show that the construction by the ''twisting principle'' introduced by Yau and the twisting of associative algebras using pseudotwistors admit a common generalization
The aim of this paper is to define and study Yetter-Drinfeld modules over Hom-bialgebras, a generalized version of bialgebras obtained by modifying the algebra and coalgebra structures by a homomorphism. Yetter-Drinfeld modules over a Hom-bialgebra with bijective structure map provide solutions of the Hom-Yang-Baxter equation. The category $_H^H{\mathcal YD}$ of Yetter-Drinfeld modules with bijective structure maps over a Hom-bialgebra H with bijective structure map can be organized, in two different ways, as a quasi-braided pre-tensor category. If H is quasitriangular (respectively coquasitriangular) the first (respectively second) quasi-braided pre-tensor category $_H^H{\mathcal YD}$ contains, as a quasi-braided pre-tensor subcategory, the category of modules (respectively comodules) with bijective structure maps over H.