The Lawrence-Krammer representation of the braid groups recently came to prominence when it was shown to be faithful by myself and Krammer. It is an action of the braid group on a certain homology module H2(C̃) over the ring of Laurent polynomials in q and t . In this paper we describe some surfaces in C̃ representing elements of homology. We use these to give a new proof that H2(C̃) is a free mod...

We prove the following three closely related results: (1) Every finite simple group G has a profinite presentation with 2 generators and at most 18 relations. (2) If G is a finite simple group, F a field and M is an FG-module, then dim H.G;M/ .17:5/ dim M . (3) If G is a finite group, F a field and M is an irreducible faithful FG-module, then dim H.G;M/ .18:5/ dim M . Mathematics Subject Classi...

Let R be a Cohen–Macaulay local ring of dimension one with a canonical module KR. Let I be a faithful ideal of R. We explore the problem of when I ⊗R I ∨ is torsionfree, where I = HomR(I,KR). We prove that if R has multiplicity at most 6, then I is isomorphic to R or KR as an R-module, once I ⊗R I ∨ is torsionfree. This result is applied to monomial ideals of numerical semigroup rings. A higher...

Let R be an associative ring with non-zero identity. For a Serre subcategory C of the category R-mod of left R-modules, we consider the class AC of all modules that do not belong to C, but all of their proper submodules belong to C. Alongside of basic properties of such associated classes of modules, we will prove that every uniform module of AC has a local endomorphism ring. Moreover, if R is ...

This is the first paper in a series to study vertex algebra-like objects arising from infinite-dimensional quantum groups (quantum affine algebras and Yangians). In this paper we lay the foundation for this study. For any vector space W , we study what we call quasi compatible subsets of Hom(W,W ((x))) and we prove that any maximal quasi compatible subspace has a natural nonlocal (namely noncom...

We introduce and discuss the notion of naturally full func-tor. The definition is similar to the definition of separable functor: a naturally full functor is a functorial version of a full functor, while a separable functor is a functorial version of a faithful functor. We study general properties of naturally full functors. We also discuss when func-tors between module categories and between c...

We introduce and discuss the notion of naturally full func-tor. The definition is similar to the definition of separable functor: a naturally full functor is a functorial version of a full functor, while a separable functor is a functorial version of a faithful functor. We study general properties of naturally full functors. We also discuss when func-tors between module categories and between c...

The faithful quasi-dual H and strict quasi-dual H ′ of an infinite braided Hopf algebra H are introduced and it is proved that every strict quasi-dual H ′ is an HHopf module. The connection between the integrals and the maximal rational Hsubmodule H of H is found. That is, H ∼= ∫ l H ⊗H is proved. The existence and uniqueness of integrals for braided Hopf algebras in the Yetter-Drinfeld categor...

Let M be a 2-torsion free prime Γ-ring and X a nonzero faithful and prime ΓM -module. Then the existence of a nonzero Jordan left derivation d : M → X satisfying some appropriate conditions implies M is commutative. M is also commutative in the case that d : M → M is a derivation along with some suitable assumptions. AMS (MOS) Subject Classification Codes: 03E72, 54A40, 54B15

We define a family of braid group representations via the action R-matrix (of quasitriangular extension) restricted quantum $\mathfrak {sl}(2)$ on tensor power simple projective module. This is an extension Lawrence representation specialized at roots unity. Although center has finite order representations, this faithful for our extension.