Associative Algebra Structure Research Articles

Strolling through common meadows

This paper establishes a connection between rings, lattices and common meadows. Meadows are commutative and associative algebraic structures with two operations (addition and multiplication) with additive and multiplicative identities and for which inverses are total. Common meadows are meadows that introduce, as the inverse of zero, an error term a which is absorbent for addition. We show that common meadows are unions of rings which are ordered by a partial order that defines a lattice. These results allow us to extend some classical algebraic constructions to the setting of common meadows. We also briefly consider common meadows from a categorical perspective.

Using a q-shuffle algebra to describe the basic module V(Λ_0) for the quantized enveloping algebra Uq(sl^2)

We consider the quantized enveloping algebra Uq(sl^2). and its basic module V(Λ0). This module is infinite-dimensional, irreducible, integrable, and highest-weight. We describe V(Λ0) using a q-shuffle algebra in the following way. Start with the free associative algebra V on two generators x, y. The standard basis for V consists of the words in x, y. In 1995 M. Rosso introduced an associative algebra structure on V, called a q-shuffle algebra. For u, v ∈ {x, y} their q-shuffle product is u ⋆ v = uv + q(u,v)vu, where (u,v) = 2 (resp. (u,v) = − 2) if u = v (resp. u ≠ v). Let U denote the subalgebra of the q-shuffle algebra V that is generated by x, y. Rosso showed that the algebra U is isomorphic to the positive part of Uq(sl^2). In our first main result, we turn U into a Uq(sl^2).-module. Let U denote the Uq(sl^2).-submodule of U generated by the empty word. In our second main result, we show that the Uq(sl^2).-modules U and V(Λ0) are isomorphic. Let V denote the subspace of V spanned by the words that do not begin with y or xx. In our third main result, we show that U = U ∩ V.

Algebraic Constructions for Novikov–Poisson Algebras

A Novikov–Poisson algebra (A,∘,·) is a vector space with a Novikov algebra structure (A,∘) and a commutative associative algebra structure (A,·) satisfying some compatibility conditions. Give a Novikov–Poisson algebra (A,∘,·) and a vector space V. A natural problem is how to construct and classify all Novikov–Poisson algebra structures on the vector space E=A⊕V such that (A,∘,·) is a subalgebra of E up to isomorphism whose restriction on A is the identity map. This problem is called extending structures problem. In this paper, we introduce the definition of a unified product for Novikov–Poisson algebras, and then construct an object GH2(V,A) to answer the extending structures problem. Note that unified product includes many interesting products such as bicrossed product, crossed product and so on. Moreover, the special case when dim(V)=1 is investigated in detail.

On a family of vertex operator superalgebras

This paper is to study vertex operator superalgebras which are strongly generated by their weight-2 and weight-32 homogeneous subspaces. Among the main results, it is proved that if such a vertex operator superalgebra V is simple, then V(2) has a canonical commutative associative algebra structure equipped with a non-degenerate symmetric associative bilinear form and V(32) is naturally a V(2)-module equipped with a V(2)-valued symmetric bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying a set of conditions. On the other hand, assume that A is any commutative associative algebra equipped with a non-degenerate symmetric associative bilinear form and assume that U is an A-module equipped with a symmetric A-valued bilinear form and a non-degenerate (C-valued) symmetric bilinear form, satisfying the corresponding conditions. Then we construct a Lie superalgebra L(A,U) and a simple vertex operator superalgebra LL(A,U)(ℓ,0) for every nonzero number ℓ such that LL(A,U)(ℓ,0)(2)=A and LL(A,U)(ℓ,0)(32)=U.

Non-Associative Algebra Redesigning Block Cipher with Color Image Encryption

The substitution box (S-box) is a fundamentally important component of symmetric key cryptosystem. An S-box is a primary source of non-linearity in modern block ciphers, and it resists the linear attack. Various approaches have been adopted to construct S-boxes. S-boxes are commonly constructed over commutative and associative algebraic structures including Galois fields, unitary commutative rings and cyclic and non-cyclic finite groups. In this paper, first a non-associative ring of order 512 is obtained by using computational techniques, and then by this ring a triplet of 8 × 8 S-boxes is designed. The motivation behind the designing of these S-boxes is to upsurge the robustness and broaden the key space due to non-associative and non-commutative behavior of the algebraic structure under consideration. A novel color image encryption application is anticipated in which initially these 3 S-boxes are being used to produce confusion in three layers of a standard RGB image. However, for the sake of diffusion 3D Arnold chaotic map is used in the proposed encryption scheme. A comparison with some of existing chaos and S-box dependent color image encryption schemes specs the performance results of the anticipated RGB image encryption and observed as approaching the standard prime level.

Characterizing the Ordered AG-Groupoids Through the Properties of Their Different Classes of Ideals

In this article, we have presented some important charcterizations of the ordered non-associative semigroups in relation to their ideals. We have initially characterized the ordered AG-groupoid through the properties of the their ideals, then we characterized the two important classes of these AG-groupoids, namely the regular and intragregular non-associative AG-groupoids. Our aim is also to encourage the research and the maturity of the associative algebraic structures by studying a class of non-associative and non-commutative algebraic structures called the ordered AG-groupoid.

On geometric degenerations and Gerstenhaber formal deformations

We study the degeneration relations on the varieties of associative and Lie algebra structures on a fixed finite dimensional vector space and give a description of them in terms of Gerstenhaber formal deformations. We use this result to show how the orbit closure of the $3$-dimensional Lie algebras can be determined using homological algebra. For the case of finite dimensional associative algebras, we prove that the $N$-Koszul property is preserved under the degeneration relation for all $N\geq2$.

Species substitution, graph suspension, and graded Hopf algebras of painted tree polytopes

Combinatorial Hopf algebras of trees exemplify the connections between operads and bialgebras. Painted trees were introduced recently as examples of how graded Hopf operads can bequeath Hopf structures upon compositions of coalgebras. We put these trees in context by exhibiting them as the minimal elements of face posets of certain convex polytopes. The full face posets themselves often possess the structure of graded Hopf algebras (with one-sided unit). We can enumerate faces using the fact that they are structure types of substitutions of combinatorial species. Species considered here include ordered and unordered binary trees and ordered lists (labeled corollas). Some of the polytopes that constitute our main results are well known in other contexts. First we see the classical permutohedra, and then certain generalized permutohedra: specifically the graph associahedra of suspensions of certain simple graphs. As an aside we show that the stellohedra also appear as liftings of generalized permutohedra: graph composihedra for complete graphs. Thus our results give examples of Hopf algebras of tubings and marked tubings of graphs. We also show an alternative associative algebra structure on the graph tubings of star graphs.

Using Catalan words and a q-shuffle algebra to describe a PBW basis for the positive part of [formula omitted

The positive part Uq+ of Uq(slˆ2) has a presentation with two generators A,B that satisfy the cubic q-Serre relations. In 1993 I. Damiani obtained a PBW basis for Uq+, consisting of some elements {Enδ+α0}n=0∞, {Enδ+α1}n=0∞, {Enδ}n=1∞ that are defined recursively. Our goal is to describe these elements in closed form. To reach our goal, start with the free associative algebra V on two generators x,y. The standard (linear) basis for V consists of the words in x,y. In 1995 M. Rosso introduced an associative algebra structure on V, called a q-shuffle algebra. For u,v∈{x,y} their q-shuffle product is u⋆v=uv+q〈u,v〉vu, where 〈u,v〉=2 (resp. 〈u,v〉=−2) if u=v (resp. u≠v). Rosso gave an injective algebra homomorphism ♮ from Uq+ into the q-shuffle algebra V, that sends A↦x and B↦y. We apply ♮ to the above PBW basis, and express the image in the standard basis for V. This image involves words of the following type. Define x‾=1 and y‾=−1. A word a1a2⋯an is Catalan whenever a‾1+a‾2+⋯+a‾i is nonnegative for 1≤i≤n−1 and zero for i=n. In this case n is even. For n≥0 defineCn=∑a1a2⋯a2n[1]q[1+a‾1]q×[1+a‾1+a‾2]q⋯[1+a‾1+a‾2+⋯+a‾2n]q,where the sum is over all the Catalan words a1a2⋯a2n in V that have length 2n. We show that ♮ sends Enδ+α0↦q−2n(q−q−1)2nxCn and Enδ+α1↦q−2n(q−q−1)2nCny for n≥0, and Enδ↦−q−2n(q−q−1)2n−1Cn for n≥1. It follows from this and earlier results of Damiani that {Cn}n=1∞ mutually commute in the q-shuffle algebra V.

On the composition and exterior products of double forms and p -pure manifolds

We translate into double forms formalism the basic Greub and Greub-Vanstone identities that were previously obtained in mixed exterior algebras. In particular, we introduce a second product in the space of double forms, namely the composition product, which provides this space with a second associative algebra structure. The composition product interacts with the exterior product of double forms; we show that the resulting relations provide simple alternative proofs to some classical linear algebra identities as well as to recent results in the exterior algebra of double forms. We define and study a refinement of the notion of pure curvature of Maillot, namely $p$-pure curvature, and we use one of the basic identities to prove that if a Riemannian $n$-manifold has $k$-pure curvature and $n\geq 4k$ then its Pontryagin class of degree $4k$ vanishes.

Certain numerical results in non-associative structures

The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer nge 2, the nth-commutativity degree of a finite algebraic structure S, denoted by P_n(S), is the probability that for chosen randomly two elements x and y of S, the relator x^ny=yx^n holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the nth-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for nth-commutativity degree of these loops, we will obtain best upper bounds for this probability.

Deformation Theory with Homotopy Algebra Structures on Tensor Products

In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras and algebras. When considering tensor products, such algebraic structures extend the Lie algebra or associative algebra structures that can be obtained by means of the Manin products of operads. These new homotopy algebra structures are proven to be compatible with the concepts of homotopy theory: \infty -morphisms and the Homotopy Transfer Theorem. We give a conceptual interpretation of their Maurer-Cartan elements. In the end, this allows us to construct the deformation complex for morphisms of algebras over an operad and to represent the deformation \infty -groupoid for differential graded Lie algebras.

Central extensions of the algebra of formal pseudo-differential symbols via Hochschild (co)homology and quadratic symplectic Lie algebras

We describe the space of central extensions of the associative algebra Ψn of formal pseudo-differential symbols in n≥1 independent variables using Hochschild (co)homology groups: we prove that the first Hochschild (co)homology group HH1(Ψn) is 2n-dimensional and we use this fact to calculate the first Lie (co)homology group HLie1(Ψn) of Ψn equipped with the Lie bracket induced by its associative algebra structure. As an application, we use our calculations to provide examples of infinite-dimensional quadratic symplectic Lie algebras.

Simple finite-dimensional double algebras

A double algebra is a linear space V equipped with linear map V⊗V→V⊗V. Additional conditions on this map lead to the notions of Lie and associative double algebras. We prove that simple finite-dimensional Lie double algebras do not exist over an arbitrary field, and all simple finite-dimensional associative double algebras over an algebraically closed field are trivial. Over an arbitrary field, every simple finite-dimensional associative double algebra is commutative. A double algebra structure on a finite-dimensional space V is naturally described by a linear operator R on the algebra EndV of linear transformations of V. Double Lie algebras correspond in this sense to skew-symmetric Rota–Baxter operators, double associative algebra structures – to (left) averaging operators.

On homology of associative shelves

Homology theories for associative algebraic structures are well established and have been studied for a long time. More recently, homology theories for self-distributive algebraic structures motivated by knot theory, such as quandles and their relatives, have been developed and investigated. In this paper, we study associative self-distributive algebraic structures and their one-term and two-term (rack) homology groups.

Application of soft sets to non-associative rings

Molodtsov developed the theory of soft sets which can be seen as an effective tool to deal with uncertainties. Since the introduction of this concept, the application of soft sets has been restricted to associative algebraic structures (groups, semigroups, associative rings, semi-rings etc). Acceptably, though the study of soft sets, where the base set of parameters is a commutative structure, has attracted the attention of many researchers for more than one decade. But on the other hand there are many sets which are naturally endowed by two compatible binary operations forming a non-associative ring and we may dig out examples which investigate a non-associative structure in the context of soft sets. Thus it seems natural to apply the concept of soft sets to non-commutative and non-associative structures. In present paper, we make a new approach to apply Molodtsov’s notion of soft sets to LA-ring (a class of non-associative ring). We extend the study of soft commutative rings from theoretical aspect.

Representations of U¯qsℓ(2|1) at even roots of unity

We construct all projective modules of the restricted quantum group U¯qsℓ(2|1) at an even, 2p th, root of unity. This 64p4-dimensional Hopf algebra is a common double bosonization of two rank-2 Nichols algebras 𝔅(X) with fermionic generator(s). We show that the category of U¯qsℓ(2|1)-modules is equivalent to the category of Yetter–Drinfeld 𝔅(X)-modules in Cρ=HHY\kern -1ptD for H = ℤ2p ⊗ ℤ2p, where the coaction is defined by a universal R-matrix ρ ∈ H ⊗ H. As an application of the projective module construction, we study the basic algebra of U¯qsℓ(2|1) and find the associative algebra structure and the dimension, 5p2 − p + 4, of its center.

Homotopy units in 𝐴-infinity algebras

We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra structures on a given object is up to homotopy a subset of connected components of the space of non-unital associative algebra structures.

QUASI-COMMUTATIVE SEMIGROUPS OF FINITE ORDER RELATED TO HAMILTONIAN GROUPS

If for every elements x and y of an associative algebraic structure (S, <TEX>${\cdot}$</TEX>) there exists a positive integer r such that <TEX>$ab=b^ra$</TEX>, then S is called quasi-commutative. Evidently, every abelian group or commutative semigroup is quasi-commutative. Also every finite Hamiltonian group that may be considered as a semigroup, is quasi-commutative however, there are quasi-commutative semigroups which are non-group and non commutative. In this paper, we provide three finitely presented non-commutative semigroups which are quasi-commutative. These are the first given concrete examples of finite semigroups of this type.

Frobenius map for the centers of Hecke algebras

We introduce a commutative associative graded algebra structure on the direct sum Z of the centers of the Hecke algebras associated to the symmetric groups in n letters for all n. As a natural deformation of the classical construction of Frobenius, we establish an algebra isomorphism from the algebra Z to the ring of symmetric functions. This isomorphism provides an identification between several distinguished bases for the centers (introduced by Geck-Rouquier, Jones, Lascoux) and explicit bases of symmetric functions.