This is a sequel to [Li4] and [Li5] in a series to study vertex algebra-like structures arising from various algebras such as quantum affine algebras and Yangians. In this paper, we study two versions of the double Yangian DY~(sl2), denoted by DYq(sl2) and DY ∞ q (sl2) with q a nonzero complex number. For each nonzero complex number q, we construct a quantum vertex algebra Vq and prove that eve...

Let R be a commutative ring with identity , and M is unitary left R-module”, “A proper submodule E of an R-module called weakly quasi-prime if whenever r, s ∈ R, m M, 0 ≠ rsm implies that rm or sm E”. “We introduce the concept quasi 2-absorbing as generalization submodule”, where r,s,t ∈M 0≠ rstm rtm stm E. we study basic properties 2-absorbing. Furthermore, relationships other classes module a...

Let U denote the half quantum group for a fixed simple Lie algebra. We examine some properties and representation of U. We prove that the Hopf algebra U is not quasi-cocommutative, and hence the category of left U-module is not a braided monoidal category. In the weight module category W , we describe all the simple objects and the projective objects. We also describe all simple Yetter-Drinfel’...

We generalize various properties of Yetter-Drinfeld modules over Hopf algebras to quasi-Hopf algebras. The dual of a finite dimensional Yetter-Drinfeld module is again a Yetter-Drinfeld module. The algebra H 0 in the category of Yetter-Drinfeld modules that can be obtained by modifying the multiplication in a proper way is quantum commutative. We give a Structure Theorem for Hopf modules in the...

We find a bound for the Goldie dimension of hereditary modules in terms of the cardinality of the generator sets of its quasi-injective hull. Several consequences are deduced. In particular, it is shown that every right hereditary module with countably generated quasi-injective hull is noetherian. Or that every right hereditary ring with finitely generated injective hull is artinian, thus answe...

We introduce a notion of φ-coordinated (quasi) module for a nonlocal vertex algebra and we establish a conceptual construction of nonlocal vertex algebras and their φ-coordinated (quasi) modules. As an application, we associate quantum affine algebras with weak quantum vertex algebras in the sense of [Li3], and we also associate a certain quantum βγ-system with quantum vertex algebras and φ-coo...

The Hardy space on the unit ball in C provides examples of a quasi-free, finite rank Hilbert module which contains a pure submodule isometrically isomorphic to the module itself. For n = 1 the submodule has finite codimension. In this note we show that this phenomenon can only occur for modules over domains in C and for finitely-connected domains only for Hardy-like spaces, the bundle shifts. M...

Let X be a complex analytic manifold, D ⊂ X a locally quasi-homogeneous free divisor, E an integrable logarithmic connection with respect to D and L the local system of the horizontal sections of E on X − D. In this paper we give an algebraic description in terms of E of the regular holonomic DX-module whose de Rham complex is the intersection complex associated with L. As an application, we pe...

This is a paper in a series to study quantum vertex algebras and their relations with various quantum algebras. In this paper, we introduce a notion of T-type quantum vertex algebra and a notion ofG-equivariant φ-coordinated quasi module for a T -type quantum vertex algebra with an automorphism group G. We refine and extend several previous results and we obtain a commutator formula for G-equiv...

Let k be an algebraically closed field of characteristic 0, and let A = k[x, y]/(f) be a quasi-homogeneous plane curve. We show that for any graded torsion free A-module M without free summands, there exists a natural graded integrable connection, i.e. a graded A-linear homomorphism ∇ : Derk(A) → Endk(M) that satisfy the derivation property and preserves the Lie product. In particular, a torsio...