Associative Algebra Structure Research Articles

Non-Commutative Poisson Algebra Structures on Lie Algebra [formula omitted] with Nullity M

Non-commutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on the Lie algebras sln(ℂ˜q) are determined.

Characterizations of semi-groups that are semi-lattices of left groups

Zadeh in 1965 introduce the concept of a fuzzy set which now have a wide range of applications in various fields of Engineering Sciences, Computer Science and Management Sciences. It is worth mentioning here that for applications of fuzzy sets mostly associative algebraic structures are used such as in 2003, Mordeson, Malik and Kuroki have discovered the applications of fuzzy semigroups in fuzzy coding, fuzzy finite-state machines and fuzzy languages. We have applied fuzzy set theory to associative algebraic structures and have discussed related properties. Specifically we have characterized semigroups which are semilattices of left groups by the properties of generalized fuzzy ideals. Key words: -fuzzy ideals, -fuzzy bi-ideals, -fuzzy quasi ideals.

On fully regular $$\mathcal{AG }$$ -groupoids

One of the best approaches to study one type of algebraic structure is to connect it with other type of algebraic structure which is better explored. In this paper we have accomplished this aim by connecting \({\mathcal{AG }}\)-groupoids with some useful associative and commutative algebraic structures. We have also introduced a fully regular class of an \( {\mathcal{AG }}\)-groupoid and shown that an \({\mathcal{AG }}\)-groupoid \(\mathcal S \) with left identity is fully regular if and only if \(\mathcal{L=L }^{i+1}\), for any left ideal \(\mathcal L \) of \(\mathcal S \), where \(i=1,\ldots ,n\).

Left regular $${\mathcal{AG}}$$ -groupoids in terms of fuzzy interior ideals

The left and right regular classes of a semigroup theory plays essential role in studying the structural properties of the associative algebraic structure. In this paper, we show that these two classes coincides in an \({\mathcal{AG}}\)-groupoid with left identity which is a non-associative algebraic structure. As an application of our results we get characterizations of a left regular \({\mathcal{AG}}\)-groupoid in terms of fuzzy interior ideals and study some important properties in different aspects. Further we introduce a very basic theory for an \({\mathcal{AG}}\)-groupoid in terms of classical and fuzzy (left, right, two-sided, interior) ideals and show that how similar is the theory of an \({\mathcal{AG}}\)-groupoid with the theory of fuzzy \({\mathcal{AG}}\)-groupoid.

A study of fuzzy Abel-Grassmann's groupoids

Fuzzy sets were first introduced by Zadeh in 1965 which have a wide range of applications in various fields of Engineering Sciences, Computer Science and Management Sciences. It is worth mentioning here that for applications of fuzzy sets mostly associative algebraic structures are used such as in 2003, Mordeson, Malik and Kuroki discovered the applications of fuzzy semigroups in fuzzy coding, fuzzy finite-state machines and fuzzy languages. Our aim in this paper is to apply fuzzy sets to non-associative algebraic structures and to develop some new related properties. Specifically we have characterized intra-regular Abel-Grassmann’s groupoids using generalized fuzzy bi-ideals. Key words: Abel-Grassman's groupoid (AG-groupoid), intra-regular, fuzzy sets, fuzzy AG-groupoids, -fuzzy ideals.

On a VOA Associated with the Simple Jordan Algebra of Type D

If a vertex operator algebra satisfies dim V 0 = 1 and V 1 = {0}, then V 2 has a commutative (not necessarily associative) algebra structure, called the Griess algebra. By using a vertex operator algebra associated with the simple Jordan algebra of type D, this article gives a counterexample to the following assertion: If R is a subalgebra of the Griess algebra, then the weight two space of the vertex operator subalgebra VOA(R) generated by R coincides with R.

ON THE ASSOCIATIVE ALGEBRA STRUCTURES CLOSEST TO ALGEBRA STRUCTURES

We shall define the associative algebra structure closest to a given algebra structure. When we face a new multiplication, being caused by noise and so on, it must be useful to compute with the closest associative multiplication to such a perturbed one. In this paper we shall give a procedure to find the closest associative structure and demonstrate our strategy for the 2-dimensional algebras over the field of real numbers. Then we trace the points of the sets of structure constants of the deformed algebras.

Quantum Group as Semi-infinite Cohomology

We obtain the quantum group SL q (2) as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges \({c+\bar{c}=26}\). Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of SL q (2) and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.

Endomorphism algebras and q-traces

For a braided vector space ( V , σ ) with braiding σ of Hecke type, we introduce three associative algebra structures on the space ⊕ p = 0 M End S σ p ( V ) of graded endomorphisms of the quantum symmetric algebra S σ ( V ) . We use the second product to construct a new trace. This trace is an algebra morphism with respect to the third product. In particular, when V is the fundamental representation of U q s l N + 1 and σ is the action of the R -matrix, this trace is a scalar multiple of the quantum trace of type A .

Algebraic structures in [formula omitted] associated to linear transformations

Given a linear transformation between finite-dimensional vector spaces T : W → V , we study the associative and Lie algebra structure that arise in Hom ( V , W ) , the space of all linear transformations V → W . As a consequence, we obtain a lower bound for the number of non-isomorphic Lie algebra structures that F N can admit.

Algebras with ternary law of composition and their realization by cubic matrices

We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping, we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions, we construct a ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.

Algebras with ternary law of composition and their realization by cubic matrices

We study partially and totally associative ternary algebras of first and second kind. Assuming the vector space underlying a ternary algebra to be a topological space and a triple product to be continuous mapping we consider the trivial vector bundle over a ternary algebra and show that a triple product induces a structure of binary algebra in each fiber of this vector bundle. We find the sufficient and necessary condition for a ternary multiplication to induce a structure of associative binary algebra in each fiber of this vector bundle. Given two modules over the algebras with involutions we construct a ternary algebra which is used as a building block for a Lie algebra. We construct ternary algebras of cubic matrices and find four different totally associative ternary multiplications of second kind of cubic matrices. It is proved that these are the only totally associative ternary multiplications of second kind in the case of cubic matrices. We describe a ternary analog of Lie algebra of cubic matrices of second order which is based on a notion of j-commutator and find all commutation relations of generators of this algebra.

Non-commutative Poisson Algebra Structures on the Lie Algebra $so_n\widetilde{({\Bbb C}_Q)}$

Non-commutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this paper, the non-commutative poisson algebra structures on [Formula: see text] are determined.

Poisson algebra structures on sp 2l( ) with nullity m

Noncommutative Poisson algebras are the algebras having both an associative algebra structure and a Lie algebra structure together with the Leibniz law. In this article, the noncommutative Poisson algebra structures on sp 2l( ▪) are determined.

Weak Bruhat order on the set of faces of the permutohedron and the associahedron

We define a partial order on the set of faces of the permutohedron, which extends the weak Bruhat order of the symmetric group. The restriction of this order gives a partially ordered set structure to the set of faces of the associahedron, which extends the Tamari order defined on the set of planar binary rooted trees. These orders are used to describe associative algebra structures on the vector spaces spanned by the set of faces of the permutohedra on one hand, and on the vector space spanned by the set of planar rooted trees on the other hand. Our results extend the description of the dendriform algebra structure on the spaces spanned by permutations and by planar binary trees in terms of the weak Bruhat order and the Tamari order, respectively, obtained by J.-L. Loday and the second author in [J.-L. Loday, M. Ronco, Order structure and the algebra of permutations and of planar binary trees, J. Algebraic Combin. 15 (2002) 253–270].

A generalization of the Kuga-Satake construction

The Kuga-Satake construction associates to a K3 type polarized weight 2 Hodge structure H an abelian variety A such that H is a quotient Hodge structure of H^2(A). The first step is to consider the Clifford algebra of H. It turns out that it is endowed with a weight 2 Hodge structure compatible with the algebra structure. We show more generally that a weight 2 polarized Hodge structure which carries a compatible (unitary, associative) algebra structure is a quotient of the H^2 of an abelian variety.

Combinatorial operad actions on cochains

A classical E-infinity operad is formed by the bar construction of the symmetric groups. Such an operad has been introduced by M. Barratt and P. Eccles in the context of simplicial sets in order to have an analogue of the Milnor FK-construction for infinite loop spaces. The purpose of this paper is to prove that the associative algebra structure on the normalized cochain complex of a simplicial set extends to the structure of an algebra over the Barratt–Eccles operad. We also prove that differential graded algebras over the Barratt–Eccles operad form a closed model category. Similar results hold for the normalized Hochschild cochain complex of an associative algebra. More precisely, the Hochschild cochain complex is acted on by a suboperad of the Barratt–Eccles operad which is equivalent to the classical little squares operad.

Some Results on Novikov–Poisson Algebras

Novikov algebras were introduced in connection with the Poisson brackets of hydrodynamic type and Hamiltonian operators in the formal variational calculus. A Novikov–Poisson algebra is a Novikov algebra with a compatible commutative associative algebraic structure, which was introduced to construct the tensor product of two Novikov algebras. In this paper, we commence a study of finite-dimensional Novikov–Poisson algebras. We show the commutative associative operation in a Novikov–Poisson algebra is a compatible global deformation of the associated Novikov algebra. We also discuss how to classify Novikov–Poisson algebras. And as an example, we give the classification of 2-dimensional Novikov–Poisson algebras.

Stratifying algebras with near-matrix algebras

Given a left module U and a right module V over an algebra D and a D– D-bilinear form β: U× V→ D, we may define an associative algebra structure on the tensor product V⊗ D U. This algebra is called a near-matrix algebra. In this paper, we shall investigate algebras filtered by near-matrix algebras in some nice way and give a unified treatment for quasi-hereditary algebras, cellular algebras, and stratified algebras.

ASSOCIATIVITY AND OPERATOR HAMILTONIAN QUANTIZATION OF GAUGE THEORIES

We show that the associative algebra structure can be incorporated in the BRST quantization formalism for gauge theories such that extension from the corresponding Lie algebra to the associative algebra is achieved using operator quantization of reducible gauge theories. The BRST differential that encodes the associativity of the algebra multiplication is constructed as a quadratic second-order differential operator on the bar resolution.