Associative Dialgebra Research Articles

Multipliers of Nilpotent Diassociative Algebras

The paper concerns nilpotent diassociative algebras (also known as associative dialgebras) and their corresponding diassociative Schur multipliers. Using Lie (and group) theory as a guide, we first extend a classic five-term cohomological sequence under alternative conditions in the nilpotent setting. This main result is then applied to obtain a new proof for a previous extension of the same sequence. It also yields a different extension of the sequence that involves terms in the upper central series. Furthermore, we use the main result to obtain a collection of dimension bounds on the multiplier of a nilpotent diassociative algebra. These differ notably from the Lie case. Since diassociative algebras generalize associative algebras, we obtain an associative analogue of the results herein. We conclude by computing both the associative and diassociative multipliers of an associative algebra. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras.

Cohomology and deformations of oriented dialgebras

We introduce a notion of oriented dialgebra and develop a cohomology theory for oriented dialgebras by mixing the standard chain complexes computing group cohomology and associative dialgebra cohomology. We also introduce a formal deformation theory for oriented dialgebras and show that cohomology of oriented dialgebras controls such deformations.

No dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2

ABSTRACT The Gelfand-Kirillov dimension measures the asymptotic rate of growth of algebras. For every associative dialgebra , the quotient , where is the ideal of generated by the set , is called the associative algebra associated to . We show that . Moreover, we prove that no associative dialgebra has Gelfand-Kirillov dimension strictly between 1 and 2.

Equivariant associative dialgebras and its one-parameter formal deformations

Associative dialgebras generalize the notions of associative algebras. In this paper, we define a notion of finite group actions on associative dialgebras and define an equivariant dialgebra cohomology associated with such actions. We introduce a one-parameter formal deformation in the equivariant context and show that the equivariant dialgebra cohomology controls the equivariant one-parameter formal deformation of dialgebra equipped with an action of a finite group.

Augmented, free and tensor generalized digroups

Abstract The concept of generalized digroup was proposed by Salazar-Díaz, Velásquez and Wills-Toro in their paper “Generalized digroups” as a non trivial extension of groups. In this way, many concepts and results given in the category of groups can be extended in a natural form to the category of generalized digroups. The aim of this paper is to present the construction of the free generalized digroup and study its properties. Although this construction is vastly different from the one given for the case of groups, we will use this concept, the classical construction for groups and the semidirect product to construct the tensor generalized digroup as well as the semidirect product of generalized digroups. Additionally, we give a new structural result for generalized digroups using compatible actions of groups and an equivariant map from a group set to the group corresponding to notions of associative dialgebras and augmented racks.

Gröbner–Shirshov bases for commutative dialgebras

We establish Gröbner–Shirshov bases theory for commutative dialgebras. We show that for any ideal I of , I has a unique reduced Gröbner–Shirshov basis, where is the free commutative dialgebra generated by a set X, in particular, I has a finite Gröbner–Shirshov basis if X is finite. As applications, we give normal forms of elements of an arbitrary commutative disemigroup, prove that the word problem for finitely presented commutative dialgebras (disemigroups) is solvable, and show that if X is finite, then the problem whether two ideals of are identical is solvable. We construct a Gröbner–Shirshov basis in associative dialgebra by lifting a Gröbner–Shirshov basis in .

On Central Extensions of Associative Dialgebras

The concept of central extensions plays an important in constructing extensions of algebras. This technique has been successfully used in the classification problem of certain classes of algebras. In 1978 Skjelbred and Sund reduced the classification of nilpotent Lie algebras in a given dimension to the study of orbits under the action of automorphism group on the space of second degree cohomology of a smaller Lie algebra with coefficients in a trivial module. Then W. de Graaf applied the Skjelbred and Sund method to the classification problem of low-dimensional nilpotent Lie and associative algebras over some fields. The main purpose of this note is to establish elementary properties of central extensions of associative dialgebras and apply the above mentioned method to the classification of low dimensional nilpotent associative dialgebras.

Associative Dialgebras from a Structural Viewpoint

In this note we study associative dialgebras proving that the most interesting such structures arise precisely when the algebra is not semiprime. In fact the presence of some “perfection” property (simpleness, primitiveness, primeness, or semiprimeness) imply that the dialgebra comes from an associative algebra with both products ⊣ and ⊢ identified. We also describe the class of zero-cubed algebras and apply its study to that of dialgebras. Finally, we describe two-dimensional associative dialgebras.

Special identities for the pre-Jordan product in the free dendriform algebra

Pre-Jordan algebras were introduced recently in analogy with pre-Lie algebras. A pre-Jordan algebra is a vector space A with a bilinear multiplication x·y such that the product x∘y=x·y+y·x endows A with the structure of a Jordan algebra, and the left multiplications L·(x):y↦x·y define a representation of this Jordan algebra on A. Equivalently, x·y satisfies these multilinear identities:(x∘y)·(z·u)+(y∘z)·(x·u)+(z∘x)·(y·u)≡z·[(x∘y)·u]+x·[(y∘z)·u]+y·[(z∘x)·u],x·[y·(z·u)]+z·[y·(x·u)]+[(x∘z)∘y]·u≡z·[(x∘y)·u]+x·[(y∘z)·u]+y·[(z∘x)·u].The pre-Jordan product x·y=x≻y+y≺x in any dendriform algebra also satisfies these identities. We use computational linear algebra based on the representation theory of the symmetric group to show that every identity of degree ⩽7 for this product is implied by the identities of degree 4, but that there exist new identities of degree 8 which do not follow from those of lower degree. There is an isomorphism of S8-modules between these new identities and the special identities for the Jordan diproduct in an associative dialgebra.

Noncommutative Poisson brackets on Loday algebras and related deformation quantization

Given a Lie algebra, there uniquely exists a Poisson algebra which is called a Lie-Poisson algebra over the Lie algebra. We will prove that given a Loday/Leibniz algebra there exists uniquely a noncommutative Poisson algebra over the Loday algebra. The noncommutative Poisson algebras are called the Loday-Poisson algebras. In the super/graded cases, the Loday-Poisson bracket is regarded as a noncommutative version of classical (linear) Schouten-Nijenhuis bracket. It will be shown that the Loday-Poisson algebras form a special subclass of Aguiar's dual-prePoisson algebras. We also study a problem of deformation quantization over the Loday-Poisson algebra. It will be shown that the polynomial Loday-Poisson algebra is deformation quantizable and that the associated quantum algebra is Loday's associative dialgebra.

Jordan triple disystems

We take an algorithmic and computational approach to a basic problem in abstract algebra: determining the correct generalization to dialgebras of a given variety of nonassociative algebras. We give a simplified statement of the KP algorithm introduced by Kolesnikov and Pozhidaev for extending polynomial identities for algebras to corresponding identities for dialgebras. We apply the KP algorithm to the defining identities for Jordan triple systems to obtain a new variety of nonassociative triple systems, called Jordan triple disystems. We give a generalized statement of the BSO algorithm introduced by Bremner and Sánchez-Ortega for extending multilinear operations in an associative algebra to corresponding operations in an associative dialgebra. We apply the BSO algorithm to the Jordan triple product and use computer algebra to verify that the polynomial identities satisfied by the resulting operations coincide with the results of the KP algorithm; this provides a large class of examples of Jordan triple disystems. We formulate a general conjecture expressed by a commutative diagram relating the output of the KP and BSO algorithms. We conclude by generalizing the Jordan triple product in a Jordan algebra to operations in a Jordan dialgebra; we use computer algebra to verify that resulting structures provide further examples of Jordan triple disystems. For this last result, we also provide an independent theoretical proof using Jordan structure theory.

Special Identities for Quasi-Jordan Algebras

Semispecial quasi-Jordan algebras (also called Jordan dialgebras) are defined by the polynomial identities These identities are satisfied by the product ab = a ⊣ b + b ⊢ a in an associative dialgebra. We use computer algebra to show that every identity for this product in degree ≤7 is a consequence of the three identities in degree ≤4, but that six new identities exist in degree 8. Some but not all of these new identities are noncommutative preimages of the Glennie identity.

Restrictive, split and unital quasi-Jordan algebras

It is well known that by means of the right and left products of an associative dialgebra we can build a new product over the same vector space with respect to which it becomes a right version of a Jordan algebra (in fact, this new product is right commutative) called quasi-Jordan algebra. Recently, Kolesnikov and Bremner independently have discovered an interesting property of this new product. As the results of this paper indicate, when the said property is incorporated as an axiom in the definition of quasi-Jordan algebra then in a natural way one can introduce and study concepts in this new structure such as derivations (in particular inner derivations), quadratic representations, and the structure groups of a quasi-Jordan algebra.

On the Definition of Quasi-Jordan Algebra

Velásquez and Felipe recently introduced quasi-Jordan algebras based on the product in an associative dialgebra with operations ⊣ and ⊢. We determine the polynomial identities of degree ≤4 satisfied by this product. In addition to right commutativity and the right quasi-Jordan identity, we obtain a new associator-derivation identity.

The partially alternating ternary sum in an associative dialgebra

The alternating ternary sum in an associative algebra, gives rise to the partially alternating ternary sum in an associative dialgebra with products ⊣ and ⊢ by making the argument a the center of each term: We use computer algebra to determine the polynomial identities in degree ⩽9 satisfied by this new trilinear operation. In degrees 3 and 5, we obtain these identities define a new variety of partially alternating ternary algebras. We show that there is a 49-dimensional space of multilinear identities in degree 7, and we find equivalent nonlinear identities. We use the representation theory of the symmetric group to show that there are no new identities in degree 9.

Centralizer and faithful groupoid

We correlate existence and aspect of some properties of centralizers with the conceptual notion of Θ-faithful objects. We produce a large choice of examples from standard algebraic situations to topological models, going through less classical cases as the new notions of algebras introduced by Loday (Leibniz algebras, associative dialgebras and trialgebras), or the dual of any boolean topos.

F[x, y] as a Dialgebra and a Leibniz Algebra

In this article, we define a new associative dialgebra over a polynomial algebra F[x, y] with two indeterminates x and y. Left derivations, right derivations, derivations, and automorphisms of F[x, y] as associative dialgebra are determined. Meanwhile, we also determine all homogeneous derivations of F[x, y] as ℤ-graded Leibniz algebra, and automorphisms of Leibniz algebra F[x, y] preserving the standard filtration.

0-dialgebras with bar-unity and nonassociative Rota-Baxter algebras

We describe all homogeneous structures of Rota-Baxter algebras on a 0-dialgebra with associative bar-unity and give a corollary on the structure of a Rota-Baxter algebra on an arbitrary associative dialgebra with bar-unity as well as a unital associative conformal algebra. We prove that an arbitrary alternative dialgebra may be embedded into an alternative dialgebra with associative barunity. We suggest the definition of variety of dialgebras in the sense of Eilenberg which is equivalent to that introduced earlier by Kolesnikov.

Tiling the ( n 2 , 1)-De Bruijn Graph with n Coassociative Coalgebras#

We construct, via usual graph theory a class of associative dialgebras as well as a class of coassociative L-coalgebras, the two classes being related by a tool from graph theory called the line-extension. As a corollary, a tiling of the n 2-De Bruijn graph with n (geometric supports of) coassociative coalgebras is obtained. Via the tiling of the (3, 1)-De Bruijn graph, we also get an example of cubical trialgebra, notion introduced by Loday and Ronco. Other examples are obtained by letting M n (k) act on axioms defining such tilings. Examples of associative products which split into several associative ones are also given.

Dialgebra cohomology as a G-algebra

It is well known that the Hochschild cohomology H*(A, A) of an associative algebra A admits a G-algebra structure. In this paper we show that the dialgebra cohomology HY*(D, D) of an associative dialgebra D has a similar structure, which is induced from a homotopy G-algebra structure on the dialgebra cochain complex CY*(D, D).