Algebras and Representation Theory

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ISSN / EISSN : 1386-923X / 1572-9079
Published by: Springer Nature (10.1007)
Total articles ≅ 1,280
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Published: 20 September 2021
Algebras and Representation Theory pp 1-34; https://doi.org/10.1007/s10468-021-10092-4

Abstract:
We study the category \(\text {Rep}(Q,\mathbb {F}_{1})\) of representations of a quiver Q over “the field with one element”, denoted by \(\mathbb {F}_{1}\), and the Hall algebra of \(\text {Rep}(Q,\mathbb {F}_{1})\). Representations of Q over \(\mathbb {F}_{1}\) often reflect combinatorics of those over \(\mathbb {F}_{q}\), but show some subtleties - for example, we prove that a connected quiver Q is of finite representation type over \(\mathbb {F}_{1}\) if and only if Q is a tree. Then, to each representation \(\mathbb {V}\) of Q over \(\mathbb {F}_{1}\) we associate a coefficient quiver \({\Gamma }_{\mathbb {V}}\) possessing the same information as \(\mathbb {V}\). This allows us to translate representations over \(\mathbb {F}_{1}\) purely in terms of combinatorics of associated coefficient quivers. We also explore the growth of indecomposable representations of Q over \(\mathbb {F}_{1}\) - there are also similarities to representations over a field, but with some subtle differences. Finally, we link the Hall algebra of the category of nilpotent representations of an n-loop quiver over \(\mathbb {F}_{1}\) with the Hopf algebra of skew shapes introduced by Szczesny.
Published: 15 September 2021
Algebras and Representation Theory pp 1-52; https://doi.org/10.1007/s10468-021-10077-3

Abstract:
The article is about the representation theory of an inner form G of a general linear group over a non-Archimedean local field. We introduce semisimple characters for G whose intertwining classes describe conjecturally via the Local Langlands correspondence the behaviour on wild inertia. These characters also play a potential role to understand the classification of irreducible smooth representations of inner forms of classical groups. We prove the intertwining formula for semisimple characters and an intertwining implies conjugacy like theorem. Further we show that endo-parameters for G, i.e. invariants consisting of simple endo-classes and a numerical part, classify the intertwining classes of semisimple characters for G. They should be the counter part for restrictions of Langlands-parameters to wild inertia under the Local Langlands correspondence.
Published: 13 September 2021
Algebras and Representation Theory pp 1-51; https://doi.org/10.1007/s10468-021-10097-z

Abstract:
Extending the main result of Lorscheid and Weist (2015), in the first part of this paper we show that every quiver Grassmannian of an indecomposable representation of a quiver of type $\tilde D_{n}$ D ~ n has a decomposition into affine spaces. In the case of real root representations of small defect, the non-empty cells are in one-to-one correspondence to certain, so called non-contradictory, subsets of the vertex set of a fixed tree-shaped coefficient quiver. In the second part, we use this characterization to determine the generating functions of the Euler characteristics of the quiver Grassmannians (resp. F-polynomials). Along these lines, we obtain explicit formulae for all cluster variables of cluster algebras coming from quivers of type $\tilde D_{n}$ D ~ n .
Published: 6 September 2021
Algebras and Representation Theory pp 1-11; https://doi.org/10.1007/s10468-021-10090-6

Abstract:
If A is a finite-dimensional symmetric algebra, then it is well-known that the only silting complexes in Kb(projA) are the tilting complexes. In this note we investigate to what extent the same can be said for weakly symmetric algebras. On one hand, we show that this holds for all tilting-discrete weakly symmetric algebras. In particular, a tilting-discrete weakly symmetric algebra is also silting-discrete. On the other hand, we also construct an example of a weakly symmetric algebra with silting complexes that are not tilting.
Algebras and Representation Theory pp 1-11; https://doi.org/10.1007/s10468-021-10095-1

Abstract:
In this short note, we give a description of semisimple orbits in the restricted Cartan type Lie algebras W, S, H.
Algebras and Representation Theory pp 1-26; https://doi.org/10.1007/s10468-021-10091-5

Abstract:
Exceptional sequences are important sequences of quiver representations in the study of representation theory of algebras. They are also closely related to the theory of cluster algebras and the combinatorics of Coxeter groups. We combinatorially classify exceptional sequences of a family of type D Dynkin quivers, and we show how our model for exceptional sequences connects to the combinatorics of type D noncrossing partitions.
Algebras and Representation Theory pp 1-22; https://doi.org/10.1007/s10468-021-10088-0

Abstract:
It is well known that the real spectrum of any commutative unital ring, and the ℓ-spectrum of any Abelian lattice-ordered group with order-unit, are all completely normal spectral spaces. We complete the existing list of containments and non-containments between the associated spectral spaces and their spectral subspaces, by proving the following results: (1) Every real spectrum can be embedded, as a spectral subspace, into some ℓ-spectrum. (2) Not every real spectrum is an ℓ-spectrum. (3) A spectral subspace of a real spectrum may not be a real spectrum. (4) Not every ℓ-spectrum can be embedded, as a spectral subspace, into a real spectrum. The commutative unital rings and Abelian lattice-ordered groups constructed in Eqs. 2, 3, and 4 all have cardinality ℵ1. Moreover, Eq. 3 solves a problem stated in 2012 by Mellor and Tressl.
Algebras and Representation Theory pp 1-22; https://doi.org/10.1007/s10468-021-10083-5

Abstract:
Let Λ be a basic finite dimensional algebra over an algebraically closed field \(\Bbbk \), and let \(\widehat {\Lambda }\) be the repetitive algebra of Λ. In this article, we prove that if \(\widehat {V}\) is a left \(\widehat {\Lambda }\)-module with finite dimension over \(\Bbbk \), then \(\widehat {V}\) has a well-defined versal deformation ring \(R(\widehat {\Lambda },\widehat {V})\), which is a local complete Noetherian commutative \(\Bbbk \)-algebra whose residue field is also isomorphic to \(\Bbbk \). We also prove that \(R(\widehat {\Lambda }, \widehat {V})\) is universal provided that \(\underline {\text {End}}_{\widehat {\Lambda }}(\widehat {V})=\Bbbk \) and that in this situation, \(R(\widehat {\Lambda }, \widehat {V})\) is stable after taking syzygies. We apply the obtained results to finite dimensional modules over the repetitive algebra of the 2-Kronecker algebra, which provides an alternative approach to the deformation theory of objects in the bounded derived category of coherent sheaves over \(\mathbb {P}^{1}_{\Bbbk }\).
Algebras and Representation Theory pp 1-9; https://doi.org/10.1007/s10468-021-10089-z

Abstract:
Let G be a semisimple algebraic group defined over an algebraically closed field. We provide some criteria for normality and rational singularities of G-saturation under certain circumstances. Our results are applied to determine when the commuting variety over simple Lie algebra of low rank is normal and Cohen-Macaulay. We also present some interesting connections between injective modules and normality (resp. rational singularities) of their G-saturations. Finally, we generalize a machinery used to study singularities of nilpotent orbit closures.
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