Let C be a finite braided multitensor category. Then the end B=∫X∈CX⊗X⁎ is natural Hopf algebra in C. We show that Drinfeld center of isomorphic to category left B-comodules C, and decomposition B into direct sum indecomposable C-subcoalgebras leads B-ComodC C-module subcategories. As an application, we present explicit characterization structure irreducible Yetter-Drinfeld modules over semisim...

(1) If there is an indecomposable representation of dimension v, then v is a root. (2) If v is a real root, then there is a unique (up to an isomorphism) indecomposable representation of dimension v. (3) If v is primitive (meaning that GCD(vi) = 1) and there is an indecomposable representation of dimension v, then pv, the number of parameters for the isomorphism classes of indecomposable repres...

Let L be a free Lie algebra of finite rank r over a field . For each positive integer n, denote the degree n homogeneous component of L by L. The group of graded algebra automorphisms of L may be identified with GLr; ‘ in such a way that L1 becomes the natural module, and then the L are referred to as the Lie powers of this module. Understanding the GLr; ‘-module structure of the L may be tho...

T. Shudo and H. Miyamito [3] showed that C can be decomposed into a direct sum of its indecomposable subcoalgebras of C. Y.H. Xu [5] showed that the decomposition was unique. He also showed that M can uniquely be decomposed into a direct sum of the weak-closed indecomposable subcomodules of M(we call the decomposition the weak-closed indecomposable decomposition ) in [6]. In this paper, we give...

A (right R-) module N is said to be a Whitehead test module for projectivity (shortly: a p-test module) provided for each module M , ExtR(M,N) = 0 implies M is projective. Dually, i-test modules are defined. For example, Z is a p-test abelian group iff each Whitehead group is free. Our first main result says that if R is a right hereditary non-right perfect ring, then the existence of p-test mo...

Let (R,m) be a commutative Noetherian local ring. We establish some bounds for the sequence of Bass numbers and their dual for a finitely generated R-module. Introduction Throughout this paper, (R,m, k) is a non-trivial commutative Noetherian local ring with unique maximal ideal m and residue field k. Several authors have obtained results on the growth of the sequence of Betti numbers {βn(k)} (...

Abstract The automorphism group $\operatorname {Aut}(F_n)$ of the free $F_n$ acts on a space $A_d(n)$ Jacobi diagrams degree d n oriented arcs. We study -module structure by using two actions associated graded vector : an action general linear {GL}(n,\mathbb {Z})$ and Lie algebra $\mathrm {gr}(\operatorname {IA}(n))$ IA-automorphism {IA}(n)$ with its lower central series. extend to Andreadakis ...

In this paper, we study a close relationship between relative cluster tilting theory in extriangulated categories and τ-tilting module categories. Our main results show that rigid objects are bijection with τ-rigid pairs, also maximal support pairs under some assumptions. These generalize the work by Adachi-Iyama-Reiten, Yang-Zhu Fu-Geng-Liu. addition, introduce mutation of any basic almost obj...

Given a nite dimensional algebra over an algebraically closed eld one frequently disregards the projective objects in the category mod of nite dimensional-modules and focuses on the stable category mod. The objects of mod are those of mod but the Hom-groups are the ordinary Hom-groups modulo the subgroup of those morphisms which factor through a projective-module. In this paper we show that mod...

The space of summands (with respect to vector addition) of a convex polytope in n dimensions is studied. This space is shown to be isomorphic to a convex pointed cone in Euclidean space. The extreme rays of this cone correspond to similarity classes of indecomposable polytopes. The decomposition of a polytope is described and a bound is given for the number of indecomposable summands needed. A ...