To save content items to your account,
please confirm that you agree to abide by our usage policies.
If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account.
Find out more about saving content to .
To save content items to your Kindle, first ensure firstname.lastname@example.org
is added to your Approved Personal Document E-mail List under your Personal Document Settings
on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part
of your Kindle email address below.
Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations.
‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi.
‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
We give two generalizations of Kac's Theorem on representations of quivers. One is to representations of equipped graphs by relations, in the sense of Gelfand and Ponomarev. The other is to representations of quivers in which certain of the linear maps are required to have maximal rank.
It is easy to imagine that a subvariety of a vector bundle, whose intersection with every fibre is a vector subspace of constant dimension, must necessarily be a sub-bundle. We give two examples to show that this is not true, and several situations in which the implication does hold. For example it is true if the base is normal and the field has characteristic zero. A convenient test is whether or not the intersections with the fibres are reduced as schemes.
We decompose the Marsden–Weinstein reductions for the moment map associated to representations of a quiver. The decomposition involves symmetric products of deformations of Kleinian singularities, as well as other terms. As a corollary we deduce that the Marsden–Weinstein reductions are irreducible varieties.
We study the moment map associated to the cotangent bundle of the space of representations of a quiver, determining when it is flat, and giving a stratification of its Marsden–Weinstein reductions. In order to do this we determine the possible dimension vectors of simple representations of deformed preprojective algebras. In an appendix we use deformed preprojective algebras to give a simple proof of much of Kac's Theorem on representations of quivers in characteristic zero.
By an algebra Λ we mean an associative
k-algebra with identity, where k is an
algebraically closed field. All algebras are assumed to be finite dimensional
(except the path algebra kQ). An algebra is said to be biserial
if every indecomposable
projective left or right Λ-module P contains
uniserial submodules U and V such that
U+V=Rad(P) and U∩V
is either zero or simple. (Recall that a module is uniserial
if it has a unique composition series, and the radical Rad(M)
of a module M is the
intersection of its maximal submodules.) Biserial algebras arose as a natural
generalization of Nakayama's generalized uniserial algebras .
The condition first
appeared in the work of Tachikawa [6, Proposition 2.7],
and it was formalized by
Fuller . Examples include blocks of group algebras
with cyclic defect group; finite
dimensional quotients of the algebras (1)–(4) and (7)–(9)
in Ringel's list of tame local
algebras ; the special biserial algebras of
[5, 8] and the regularly biserial algebras
. An algebra Λ is basic if Λ/Rad(Λ)
a product of copies of k. This paper contains
a natural alternative characterization of basic biserial algebras, the
concept of a
bisected presentation. Using this characterization we can prove a number
about biserial algebras which were inaccessible before. In particular we
basic biserial algebras by means of quivers with relations.
Given a ring R (associative, with 1) one can define the endolength of an R–module M to be its length when it is regarded in the natural way as an EndR (M)–module, and thus one can consider the class of modules of finite endolength. The aim of this paper is to show that this is a useful concept. Briefly, the contents are as follows. In §§1–3 we cover some background machinery, in §§4–6 we discuss the modules of finite endolength for a general ring, and in §§7–9 we show how these modules control the behaviour of the finite length modules for noetherian and artin algebras. Although much of this paper has a survey nature, there are some new results proved here, the main ones being the characterization of the pure-injective modules which occur as the source of a left almost split map in §2, the character theory for modules of finite endolength in §5, and the characterization of the artin algebras with an indecomposable module of infinite length and finite endolength (a generic module) proved in §§8–9.
The author is supported by an SERC Advanced Research Fellowship. I would like to thank M. Prest and A. H. Schofield for some useful discussions.
THE FUNCTOR CATEGORY
If R is a ring, we denote by R–Mod the category of left R–modules, and by mod–R be the category of finitely presented (f.p.) right R–modules. We denote by D(R) the category of additive functors from mod–R to ℤ–Mod (the category of abelian groups). This category has a very rich structure, and is an invaluable tool for the study of R–modules.
Email your librarian or administrator to recommend adding this to your organisation's collection.