We describe the general non-associative version of Lie theory that relates unital formal multiplications (formal loops), Sabinin algebras and non-associative bialgebras. Starting with a formal multiplication we construct a non-associative bialgebra, namely, the bialgebra of distributions with the convolution product. Considering the primitive elements in this bialgebra gives a functor from formal loops to Sabinin algebras. We compare this functor to that of Mikheev and Sabinin and show that although the brackets given by both constructions coincide, the multioperator does not. We also show how identities in loops produce identities in bialgebras. While associativity in loops translates into associativity in algebras, other loop identities (such as the Moufang identity) produce new algebra identities. Finally, we define a class of unital formal multiplications for which Ado's theorem holds and give examples of formal loops outside this class. A by-product of the constructions of this paper is a new identity on Bernoulli numbers. We give two proofs: one coming from the formula for the non-associative logarithm, and the other (due to D. Zagier) using generating functions.
From the definition and properties of unital hom-associative algebras, and the use of the Kaplansky's constructions, we develop new algebraic structures called 2-hom-associative bialgebras, 2-hom-bialgebras, and 2-2-hom-bialgebras. Besides, we devise a construction of hom-left symmetric dialgebras and discuss their main relevant properties.
The (co)associative, partially (co)associative and totally (co)associative $3$-ary (co) algebras and infinitesimal bialgebras are constructed and discussed. Their trimodules and matched pairs are defined and completely characterized. The main structural properties and relations are also deduced and analyzed.
This work is devoted to study new bialgebra structures related to 2-associative algebras. A 2-associative algebra is a vector space equipped with two associative multiplications. We discuss the notions of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras. The first structure was revealed by J.-L. Loday and M. Ronco in an analogue of a Cartier-Milnor-Moore theorem, the second was suggested by Loday and the third is a variation of the second one. The main results of this paper are the construction of 2-associative bialgebras, 2-bialgebras and 2-2-bialgebras starting from an associative a…
Rota-Baxter systems were introduced by Brzeziński as a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant bialgebras. In this paper, we define Rota-Baxter systems on Hom-associative algebras and show how they induce Hom-dendriform structures and weak pseudotwistors on the underlying Hom-associative algebra. We introduce Hom-Yang-Baxter pairs and a notion of covariant Hom-bialgebra. Given a Hom-Yang-Baxter pair, we construct a twisted Rota-Baxter system and a (quasitriangular) covariant Hom-bialgebra. Finally, we consid…
Rota-Baxter operators and bialgebras go hand in hand in their applications, such as in the Connes-Kreimer approach to renormalization and the operator approach to the classical Yang-Baxter equation. We establish a bialgebra structure that is compatible with the Rota-Baxter operator, called the Rota-Baxter antisymmetric infinitesimal (ASI) bialgebra. This bialgebra is characterized by generalizations of matched pairs of algebras and double constructions of Frobenius algebras to the context of Rota-Baxter algebras. The study of the coboundary case leads to an enrichment of the associative Yang-B…
Basic definitions and properties of nearly associative algebras are described. Nearly associative algebras are proved to be Lie-admissible algebras. Two-dimensional nearly associative algebras are classified, and its main classes are derived. The bimodules, matched pairs and Manin triple of a nearly associative algebras are derived and their equivalence with nearly associative bialgebras is proved. Basic definitions and properties of nearly Hom-associative algebras are described. Related bimodules and matched pairs are given, and associated identities are established.
In this paper we define a new cohomology for multiplicative Hom-associative algebras, which generalize Hochschild cohomology and fits with deformations of Hom-associative algebras including the structure map $α$. It is a generalization of the known Hochschild-type cohomology for Hom-associative algebras defined by Amar, Ejbehi and Makhlouf. Moreover, we provide various observations and similarly a new cohomology of Hom-bialgebras extending the Gerstenhaber-Schack cohomology for Hom-bialgebras given by Dekkar and Makhlouf.
The purpose of this paper is to study the structure and the algebraic varieties of BiHom-associative algebras. We provide a classication of n-dimensional BiHom-associative and BiHom-bialgebras and BiHom Hopf algebras for n $\le$ 3.
We equip a matrix algebra with a weighted infinitesimal unitary bialgebraic structure, via a construction of a suitable coproduct. Furthermore, an infinitesimal unitary Hopf algebra, under the view of Aguiar, is constructed on a matrix algebra. By exploring the relationship between weighted infinitesimal bialgebras and pre-Lie algebras, we construct a pre-Lie algebraic structure and then a new Lie algebraic structure on a matrix algebra. We also introduce the weighted associative Yang-Baxter equations (AYBEs) and obtain the relationship between solutions of weighted AYBEs and weighted infinite…
The aim of this paper is to establish a contravariant adjunction between the category of quasi-bialgebras and a suitable full subcategory of dual quasi-bialgebras, adapting the notion of finite dual to this framework. Various functorial constructions involving non-associative algebras and non-coassociative coalgebras are then carried out. Several examples illustrating our methods are expounded as well.
We introduce a non-symmetric operad $\mathcal{N}$, whose dimension in degree $n$ is given by the Catalan number $c_{n-1}$. It arises naturally in the study of coalgebra structures defined on compatible associative algebras. We prove that any free compatible associative algebra admits a compatible infinitesimal bialgebra structure, whose subspace of primitive elements is a $\mathcal{N}$-algebra. The data $({\rm As},{\rm As}^2, \mathcal{N})$ is a good triple of operads, in J.-L. Loday's sense. Our construction induces another triple of operads $({\rm As},{\rm As}_2,{\rm As})$, where ${\rm As}_2$…
We introduce a notion of split extension of (non-associative) bialgebras which generalizes the notion of split extension of magmas introduced by M. Gran, G. Janelidze and M. Sobral. We show that this definition is equivalent to the notion of action of (non-associative) bialgebras. We particularize this equivalence to (non-associative) Hopf algebras by defining split extensions of (non-associative) Hopf algebras and proving that they are equivalent to actions of (non-associative) Hopf algebras. Moreover, we prove the validity of the Split Short Five Lemma for these kinds of split extensions, an…
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).
In this paper, we construct a bialgebra theory for associative conformal algebras, namely antisymmetric infinitesimal conformal bialgebras. On the one hand, it is an attempt to give conformal structures for antisymmetric infinitesimal bialgebras. On the other hand, under certain conditions, such structures are equivalent to double constructions of Frobenius conformal algebras, which are associative conformal algebras that are decomposed into the direct sum of another associative conformal algebra and its conformal dual as $\mathbb{C}[\partial]$-modules such that both of them are subalgebras an…
We establish a bialgebra theory for anti-flexible algebras in this paper. We introduce the notion of an anti-flexible bialgebra which is equivalent to a Manin triple of anti-flexible algebras. The study of a special case of anti-flexible bialgebras leads to the introduction of anti-flexible Yang-Baxter equation in an anti-flexible algebra which is an analogue of the classical Yang-Baxter equation in a Lie algebra or the associative Yang-Baxter equation in an associative algebra. It is a unexpected consequence that both the anti-flexible Yang-Baxter equation and the associative Yang-Baxter equa…
We study Hom-bialgebras and objects admitting coactions by Hom-bialgebras. In particular, we construct a Hom-bialgebra M representing the functor of 2x2-matrices on Hom-associative algebras. Then we construct a Hom-algebra analogue of the affine plane and show that it is a comodule Hom-algebra over M in a suitable sense. It is also shown that the enveloping Hom-associative algebra of a Hom-Lie algebra is naturally a Hom-bialgebra.
We introduce a Hom-type generalization of quantum groups, called quasi-triangular Hom-bialgebras. They are non-associative and non-coassociative analogues of Drinfel'd's quasi-triangular bialgebras, in which the non-(co)associativity is controlled by a twisting map. A family of quasi-triangular Hom-bialgebras can be constructed from any quasi-triangular bialgebra, such as Drinfel'd's quantum enveloping algebras. Each quasi-triangular Hom-bialgebra comes with a solution of the quantum Hom-Yang-Baxter equation, which is a non-associative version of the quantum Yang-Baxter equation. Solutions of …
A class of non-associative and non-coassociative generalizations of cobraided bialgebras, called cobraided Hom-bialgebras, is introduced. The non-(co)associativity in a cobraided Hom-bialgebra is controlled by a twisting map. Several methods for constructing cobraided Hom-bialgebras are given. In particular, Hom-type generalizations of FRT quantum groups, including quantum matrices and related quantum groups, are obtained. Each cobraided Hom-bialgebra comes with solutions of the operator quantum Hom-Yang-Baxter equations, which are twisted analogues of the operator form of the quantum Yang-Bax…
We study the Hom-type generalization of infinitesimal bialgebras, called infinitesimal Hom-bialgebras. In particular, we consider infinitesimal Hom-bialgebras arising from quivers, the sub-classes of coboundary and quasi-triangular infinitesimal Hom-bialgebras, the associative Hom-Yang-Baxter equation, and homological perturbation of the comultiplications in infinitesimal Hom-bialgebras. The relationships between infinitesimal Hom-bialgebras, Hom-Lie bialgebras, and the classical Hom-Yang-Baxter equation are also studied.
Quasishuffle Hopf algebras, usually defined on a commutative monoid, can be more generally defined on any associative algebra V. If V is a commutative and cocommutative bialgebra, the associated quasishuffle bialgebra QSh(V) inherits a second coproduct $δ$ of contraction and extraction of words, cointeracting with the deconcatenation coproduct $Δ$, making QSh(V) a double bialgebra. In order to generalize the universal property of the Hopf algebra of quasisymmetric functions QSym (a particular case of quasishuffle Hopf algebra) as exposed by Aguiar, Bergeron and Sottile, we introduce the notion…
We consider a generalization of (pro)algebraic loops defined on general categories of algebras and the dual notion of a coloop bialgebra suitable to represent them as functors. Our main result is the proof that the natural loop of formal diffeomorphisms with associative coefficients is proalgebraic, and give a full description of the codivisions on its coloop bialgebra.This result provides a generalization of the Lagrange inversion formula to series with non-commutative coefficients, and a loop-theoretic explanation to the existence of the non-commutative F{à}a di Bruno Hopf algebra. MSC: 20N0…
Rota-Baxter systems of T. Brzeziński are a generalization of Rota-Baxter operators that are related to dendriform structures, associative Yang-Baxter pairs and covariant bialgebras. In this paper, we consider Rota-Baxter systems in the presence of bimodule, which we call generalized Rota-Baxter systems. We define a graded Lie algebra whose Maurer-Cartan elements are generalized Rota-Baxter systems. This allows us to define a cohomology theory for a generalized Rota-Baxter system. Formal one-parameter deformations of generalized Rota-Baxter systems are discussed from cohomological points of vie…
In this paper, we obtain a canonical factorization of basis elements in free Rota-Baxter algebras built on bracketed words. This canonical factorization is applied to give a coalgebra structure on the free Rota-Baxter algebras. Together with the Rota-Baxter algebra multiplication, this coproduct gives a bialgebra structure on the free Rota-Baxter algebra of rooted forests. When the weight of the Rota-Baxter algebra is zero, we further obtain a Hopf algebra structure.
We consider the three-dimensional associative algebra H consisting of the 3x3 strictly upper triangular matrices whose the commutator is the Heisenberg Lie algebra. We determine the solutions of the Yang-Baxter associative equation in H. For the antisymmetric solutions, the corresponding bialgebraic structures, double constructions of Frobenius algebras and properties are given explicitly.Besides, we determine some related compatible dendriform algebras and solutions of their D-equations. Using symmetric solutions of these equations, we build the double constructions of related Connes cocycles…
We extend the definition of tridendriform bialgebra by introducing a weight q. The subspace of primitive elements of a q-tridendriform bialgebra is equipped with an associative product and a natural structure of brace algebra, related by a distributive law. This data is called q-Gerstenhaber-Voronov algebras. We prove the equivalence between the categories of connected q-tridendriform bialgebras and of q-Gerstenhaber-Voronov algebras. The space spanned by surjective maps, as well as the space spanned by parking functions, have natural structures of q-tridendriform bialgebras, denoted ST(q) and…
Poisson superalgebras are known as a $\mathbb{Z}_2$-graded vector space with two operations, an associative supercommutative multiplication and a super bracket tied up by the super Leibniz relation. We show that we can consider a single nonassociative multiplication containing all these datas and then consider Poisson superalgebras as non associative algebras.
A commutative algebra is exact if its multiplication endomorphisms are trace-free and is Killing metrized if its Killing type trace-form is nondegenerate and invariant. A Killing metrized exact commutative algebra is necessarily neither unital nor associative. Such algebras can be viewed as commutative analogues of semisimple Lie algebras or, alternatively, as nonassociative generalizations of étale (associative) algebras. Some basic examples are described and there are introduced quantitative measures of nonassociativity, formally analogous to curvatures of connections, that serve to facilita…
A non-associative quantum mechanics is proposed in which the product of three and more operators can be non-associative one. The multiplication rules of the octonions define the multiplication rules of the corresponding operators with quantum corrections. The self-consistency of the operator algebra is proved for the product of three operators. Some properties of the non-associative quantum mechanics are considered. It is proposed that some generalization of the non-associative algebra of quantum operators can be helpful for understanding of the algebra of field operators with a strong interac…
We investigate Lie algebras whose Lie bracket is also an associative or cubic associative multiplication to characterize the class of nilpotent Lie algebras with a nilindex equal to 2 or 3. In particular we study the class of 2-step nilpotent Lie algebras, their deformations and we compute the cohomology which parametrize the deformations in this class.
Let $A$ be a Hopf algebra over a field $K$ of characteristic zero such that its coradical $H$ is a finite dimensional sub-Hopf algebra. Our main theorem shows that there is a gauge transformation $ζ$ on $A$ such that $A^ζ\cong Q#H$ where $A^ζ$ is the dual quasi-bialgebra obtained from $A$ by twisting its multiplication by $ζ$, $Q$ is a connected dual quasi-bialgebra in $^H_H\mathcal{YD}$ and $Q #H $ is a dual quasi-bialgebra called the bosonization of $Q$ by $H$.
We introduce the notion of quadri-algebras. These are associative algebras for which the multiplication can be decomposed as the sum of four operations in a certain coherent manner. We present several examples of quadri-algebras: the algebra of permutations, the shuffle algebra, tensor products of dendriform algebras. We show that a pair of commuting Baxter operators on an associative algebra gives rise to a canonical quadri-algebra structure on the underlying space of the algebra. The main example is provided by the algebra End(A) of linear endomorphisms of an infinitesimal bialgebra A. This …
A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic structures which generalize the well known associative, Leibniz and Lie admissible algebras. Also, we characterize the flexible Hom-algebras in this case. We also explain some connections between Hom-Lie algebras and Santilli…
Let $A$ be an $n$-dimensional algebra over a field $k$ and $a(A)$ its quantum symmetry semigroup. We prove that the automorphisms group ${\rm Aut}_{\rm Alg} (A)$ of $A$ is isomorphic to the group $U \bigl( G(a (A)^{\rm o} ) \bigl)$ of all invertible group-like elements of the finite dual $a (A)^{\rm o}$. For a group $G$, all $G$-gradings on $A$ are explicitly described and classified: the set of isomorphisms classes of all $G$-gradings on $A$ is in bijection with the quotient set $ {\rm Hom}_{\rm BiAlg} \, \bigl( a (A) , \, k[G] \bigl)/\approx$ of all bialgebra maps $a (A) \, \to k[G]$, via th…
We study the algebra of bilinear multiplications of an $n$-dimensional vector space. In particular, we study the Kantor product of some well-known (associative, Lie, alternative, Novikov and some other) multiplications.
We introduce hom-associative Ore extensions as non-unital, non-associative Ore extensions with a hom-associative multiplication, and give some necessary and sufficient conditions when such exist. Within this framework, we construct families of hom-associative quantum planes, universal enveloping algebras of a Lie algebra, and Weyl algebras, all being hom-associative generalizations of their classical counterparts, as well as prove that the latter are simple. We also provide a way of embedding any multiplicative hom-associative algebra into a multiplicative, weakly unital hom-associative algebr…
We consider algebras of $m\times m\times m$-cubic matrices (with $m=1,2,\dots$). Since there are several kinds of multiplications of cubic matrices, one has to specify a multiplication first and then define an algebra of cubic matrices (ACM) with respect to this multiplication. We mainly use the associative multiplications introduced by Maksimov. Such a multiplication depends on an associative binary operation on the set of size $m$. We introduce a notion of equivalent operations and show that such operations generate isomorphic ACMs. It is shown that an ACM is not baric. An ACM is commutative…
