The geometry of uniserial representations of finite dimensional algebras. III: Finite uniserial type

Huisgen-Zimmermann, Birge

doi:10.1090/s0002-9947-96-01575-9

Cited by 17 publications

(15 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…D'este, Kaynarca, and Tütüncü [9] give a summary of progress in the Artin algebra case in their introduction, and present an example of an Artin algebra having two nonisomorphic uniserial modules of length two having the same composition factors. B. Huisgen-Zimmermann [18] provides the tools to answer the question in the case in which the ring S is an algebraically closed field. So, for the moment at least, the rest of the results of Section 3 remain unresolved for general Artinian P IRs.…”

Section: Is Bi-hopfian If and Only If E(m ) Is Bi-hopfianmentioning

confidence: 99%

Hopfian and co-Hopfian modules over Artinian rings

Leary¹

2021

Preprint

View full text Add to dashboard Cite

If R is a ring with 1, we call a unital left R-module M Hopfian (co-Hopfian) in the category of left R-modules if any epic (monic) R-module endomorphism of M is an automorphism. In the case R is a commutative Noetherian ring, we use a result of Matlis to characterize those injective R-modules that are co-Hopfian, and to characterize those that are Hopfian when R is also reduced. We show that if R is a commutative Artinian principal ideal ring, then an R module M is Hopfian (co-Hopfian) if and only if M is finitely generated if and only if its injective envelope E(M ) is Hopfian (co-Hopfian) if and only if E(M ) is finitely generated. We note that the "finite uniserial type" problem poses an obstacle to establishing this result for an arbitrary Artinian principal ideal ring.

show abstract

Section: Is Bi-hopfian If and Only If E(m ) Is Bi-hopfianmentioning

confidence: 99%

Hopfian and co-Hopfian modules over Artinian rings

Leary¹

2021

Preprint

View full text Add to dashboard Cite

show abstract

“…It is worth mentioning that, in the theory of finite dimensional representations of associative algebras, the class of uniserial ones is quite relevant, a foundational result here is due to T. Nakayama [28] (see also [1] or [2]) and it states that every finitely generated module over a serial ring is a direct sum of uniserial modules. For more information in the associative case we refer the reader mainly to [1,2,31], see also [5,23,29]. We point out that, for Lie algebras, when g is 1-dimensional, any representation is a direct sum of uniserial ones.…”

Section: Introductionmentioning

confidence: 99%

Tensor products and intertwining operators for uniserial representations of the Lie algebra $\mathfrak{sl}(2)\ltimes V(m)$

Cagliero¹,

Rivera²

2022

Preprint

View full text Add to dashboard Cite

Let gm = sl(2) ⋉ V (m), m ≥ 1, where V (m) is the irreducible sl(2)-module of dimension m + 1 viewed as an abelian Lie algebra. It is known that the isomorphism classes of uniserial gm-modules consist of a family, say of type Z, containing modules of arbitrary composition length, and some exceptional modules with composition length ≤ 4.Let V and W be two uniserial gm-modules of type Z. In this paper we obtain the sl(2)-module decomposition of soc(V ⊗ W ) by giving explicitly the highest weight vectors. It turns out that soc(V ⊗ W ) is multiplicity free. Roughly speaking, soc(V ⊗ W ) = soc(V ) ⊗ soc(W ) in half of the cases, and in these cases we obtain the full socle series of V ⊗ W by proving that soc t+1 (V ⊗ W ) = t i=0 soc i+1 (V ) ⊗ soc t+1−i (W ) for all t ≥ 0. As applications of these results, we obtain for which V and W , the space of gm-module homomorphisms Homg m (V, W ) is not zero, in which case is 1dimensional. Finally we prove, for m = 2, that if U is the tensor product of two uniserial gm-modules of type Z, then the factors are determined by U . We provide a procedure to identify the factors from U .

show abstract

“…A celebrated result of Hironaka [18] states that any scheme X over a field K of characteristic zero admits a desingularisation, meaning a map f : Y → X of schemes such that Y is smooth and f is an isomorphism over the non-singular locus of X. When K is algebraically closed and X is projective, it follows from the work of several authors [6,13,16,17,20,32,38,39] that X may be realised representation-theoretically as a quiver Grassmannian. Given a finite-dimensional basic algebra A (which we realise as an admissible quotient of the path algebra of a quiver Q), an A-module M, and a dimension vector d ∈ N Q 0 , the…”

Section: Introductionmentioning

confidence: 99%

On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

Pressland

Sauter

2021

Algebr Represent Theor

View full text Add to dashboard Cite

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module, whose tilted algebra B is related to A by a recollement. Let M be a gen-finite A-module, meaning there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.

show abstract

The geometry of uniserial representations of finite dimensional algebras. III: Finite uniserial type

Cited by 17 publications

References 8 publications

Hopfian and co-Hopfian modules over Artinian rings

Hopfian and co-Hopfian modules over Artinian rings

Tensor products and intertwining operators for uniserial representations of the Lie algebra $\mathfrak{sl}(2)\ltimes V(m)$

On Quiver Grassmannians and Orbit Closures for Gen-Finite Modules

Contact Info

Product

Resources

About