Throughout this paper, R will denote a commutative ring with identity and M is a unitary R- module and Z will denote the ring of integers. We introduce the graph Ω(M) of module M with the set of vertices contain all nontrivial non-essential submodules of M. We investigate the interplay between graph-theoretic properties of Ω(M) and algebraic properties of M. Also, we assign the values of natura...

Abstract Gendo-Frobenius algebras are a common generalisation of Frobenius and gendo-symmetric algebras. A comultiplication is constructed for gendo-Frobenius algebras, which specialises to the known comultiplications on In addition, shown be precisely those that have counit compatible with this comultiplication. Moreover, new characterisation given. This key constructing

In [8], V.G. Drinfeld gave a new set of generators and relations for the quantum affine algebra Uq(ĝ) (and also for the Yangian). He also gave an isomorphism between the two realizations, but there was no proof in that article. In [1], J. Beck found these new generators inside Uq(ĝ), and proved that they satisfy the relations given by Drinfeld. He also proved that the two realizations were isom...

1. The construction of T(A) is motivated by dualizing the fundamental group 7Ti(X) of a differentiable manifold X with a base point Xo. Let A be the -R-algebra of C functions on X equipped with the derivation d, which is the usual differentiation from A into the Amodule M = ÛA of C 1-forms on X. Recall that the shuffle algebra Sh(M) consists of the i£-module of the tensor algebra TR(M) and the ...

The anomalous bilinear commutation relations satisfied by the components of the Green's ansatz for paraparticles are shown to derive from the comultiplication of the paraboson or parafermion algebra. The same provides a generalization of the ansatz, wherein paraparticles of order p = r α=1 p α are constructed from r paraparticles of order p

We consider a special category of Hopf algebras, depending on parameters Σ which possess properties similar to the category of representations of simple Lie group with highest weight λ. We connect quantum groups to minimal objects in this categories—they correspond to irreducible representations in the category of representations with highest weight λ. Moreover, we want to correspond quantum gr...

For the current realization of the affine quantum groups, a simple comultiplication for the quantum current operators was given by Drinfeld. With this comultiplication, we prove that, for the integrable modules of Uq(ŝl(2)) of level k + 1, x(z)x(zq) · · · x(zq) are vertex operators satisfying certain q-difference equations, and we derive the quantum parafermions of Uq(ŝl(2)).

For the current realization of the affine quantum groups, a simple comultiplication for the quantum current operators was given by Drinfeld. With this comultiplication, we study the zeros and poles of the quantum current operators and present a condition of integrability on the quantum current of Uq ( ŝl(2) ) , which is a deformation of the corresponding condition for ŝl(2).