arXiv subjects

Karin Erdmann

Publications and source records attributed to Karin Erdmann.

At least 37 records · Page 2Linked to original sources

arXiv2021

Torsion pairs and Ringel duality for Schur algebras

Let $A$ be a finite-dimensional algebra over a field of characteristic $p>0$. We use a functorial approach involving torsion pairs to construct embeddings of endomorphism algebras of basic projective $A$--modules $P$ into those of the torsion submodules of $P$. As an application, we show that blocks of both the classical and quantum Schur algebras $S(2,r)$ and $S_q(2,r)$ are Morita equivalent as quasi-hereditary algebras to their Ringel duals if they contain $2p^k$ simple modules for some $k$.

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arXiv2021

Some cluster tilting modules for weighted surface algebras

Non-singular weighted surface algebras satisfy the necessary condition found in [6] for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiver is bipartite, has a module satisfying the necessary ext vanishing condition. We show that it is 3-cluster tilting precisely for the non-singular triangular or spherical algebras but not for any other weighted surface algebra with bipartite Gabriel quiver.

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arXiv2020

Cluster tilting modules for mesh algebras

We study cluster tilting modules in mesh algebras of Dynkin type, providing a new proof for their existence. In all but one case, we show that these are precisely the maximal rigid modules, and that they are equivariant for a certain automorphism. We further study their mutation, providing an example of mutation in an abelian category which is not stably 2-Calabi-Yau, and explicitly describe the combinatorics.

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arXiv2020

Weighted surface algebras: general version. Corrigendum

In [arXiv:1902.04063] we generalize the original definition of weighted surface algebras in [arXiv:1703.02346] by allowing the possibility that arrows might not be part of the Gabriel quiver, which gives a much larger class of algebras. This means that the zero relations need modification, to make sure that the algebras are symmetric, and of the appropriate dimension. We found recently that we had missed one necessary modification for the zero relations. Here we give the correct definition, and revise the parts of [arXiv:1902.04063] which are affected by this modification.

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arXiv2020

On l-parabolic Hecke algebras of symmetric groups

Let $H=H_q(n)$ be the Hecke algebra of the symmetric group of degree n, over a field of arbitrary characteristic, and where q is a primitive l-th root of unity in $K$. Let $H_ρ$ be an l-parabolic subalgebra of $H$. We give an elementary explicit construction for the basic algebra of a non-simple block of $H_ρ$. We also discuss homological properties of $H_ρ$-modules, in particular existence of varieties for modules, and some consequences.

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arXiv2019

Representation type of Borel-Schur algebras

In our previous work arXiv:1306.1770, we found all Borel-Schur algebra of finite representation type. In the present article, we determine which Borel-Schur algebras of infinite representation type are tame, and which are wild.

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arXiv2019

Modules of constant Jordan type over quantum complete intersections

We initiate the study of modules of constant Jordan type for quantum complete intersections, and prove a range of basic properties. We then show that for these algebras, constant Jordan type is an invariant of Auslander-Reiten components. Finally, we classify modules with stable constant Jordan type [1] or [n-1] in the 2-generator case.

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arXiv2019

Higher Spherical Algebras

We introduce and study higher spherical algebras, an exotic family of finite-dimensional algebras over an algebraically closed field. We prove that every such an algebra is derived equivalent to a higher tetrahedral algebra studied in [7], an hence that it is a tame symmetric periodic algebra of period four.

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arXiv2019

Cohomology of some local selfinjective algebras

We show that the cohomology for some 2-generated selfinjective algebras is finitely generated. This applies in particular to the algebras $A5(β)$ for $β\neq 0$ in the classification of connected Hopf algebras of dimension $p^3$ over characteristic $p$ by Nguyen-Wang-Wang.

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arXiv2019

Weighted surface algebras: general version

We introduce general weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular disc, triangle, tetrahedral and spherical algebras, are symmetric tame periodic algebras of period 4.

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arXiv2018

Algebras of generalized dihedral type

We provide a complete classification of all algebras of generalised dihedral type, which are natural generalizations of algebras which occurred in the study of blocks with dihedral defect groups. This gives a description by quivers and relations coming from surface triangulations.

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arXiv2018

From Brauer graph algebras to biserial weighted surface algebras

We prove that the class of Brauer graph algebras coincides with the class of indecomposable idempotent algebras of biserial weighted surface algebras. These algebras are associated to triangulated surfaces with arbitrarily oriented triangles, investigated in [17] and [18]. Moreover we prove that Brauer graph algebras are idempotent algebras of periodic weighted surface algebras, investigated in [17] and [19].

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arXiv2018

Stable Green ring of the Drinfeld doubles of the generalised Taft algebras (corrections and new results)

We return to the fusion rules for the Drinfeld double of the duals of the generalised Taft algebras that we studied in [9]. We first correct some proofs and statements in [9] that were incorrect, using stable homomorphisms. We then complete this with new results on fusion rules for the modules we had not studied in [9] and a classification of endotrivial and algebraic modules.

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arXiv2017

Higher Tetrahedral Algebras

We introduce and study the higher tetrahedral algebras, an exotic family of finite-dimensional tame symmetric algebras over an algebraically closed field. The Gabriel quiver of such an algebra is the triangulation quiver associated to the coherent orientation of the tetrahedron. Surprisingly, these algebras occurred in the classification of all algebras of generalised quaternion type, but are not weighted surface algebras. We prove that a higher tetrahedral algebra is periodic if and only if it is non-singular.

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arXiv2017

Hochschild cohomology for periodic algebras of polynomial growth

We describe the dimensions of low Hochschild cohomology spaces of exceptional periodic representation-infinite algebras of polynomial growth. As an application we obtain that an indecomposable non-standard periodic representation-infinite algebra of polynomial growth is not derived equivalent to a standard self-injective algebra.

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arXiv2017

The cohomology ring of some Hopf algebras

Let p be a prime, and k be a field of characteristic p. We investigate the algebra structure and the structure of the cohomology ring for the connected Hopf algebras of dimension p^3, which appear in the classification obtained in [V.C. Nguyen, L.-H. Wang and X.-T. Wang, Classification of connected Hopf algebras of dimension p^3, J. Algebra 424 (2015), 473-505]. The list consists of 23 algebras together with two infinite families. We identify the Morita type of the algebra, and in almost all cases this is sufficient to clarify the structure of the cohomology ring.

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arXiv2017

Weighted Surface Algebras

A finite-dimensional algebra $A$ over an algebraically closed field $K$ is called periodic if it is periodic under the action of the syzygy operator in the category of $A-A-$ bimodules. The periodic algebras are self-injective and occur naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period $4$. Moreover, we describe the socle deformations of the weighted surface algebras and prove that all these algebras are symmetric tame periodic algebras of period $4$. The main results of the paper form an important step towards a classification of all periodic symmetric tame algebras of non-polynomial growth, and lead to a complete description of all algebras of generalized quaternion type. Further, the orbit closures of the weighted surface algebras (and their socle deformations) in the affine varieties of associative $K$-algebra structures contain wide classes of tame symmetric algebras related to algebras of dihedral and semidihedral types, which occur in the study of blocks of group algebras with dihedral and semidihedral defect groups.

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arXiv2017

Algebras of generalized quaternion type

We introduce and study the algebras of generalized quaternion type, which are natural generalizations of algebras which occurred in the study of blocks of group algebras with generalized quaternion defect groups. We prove that all these algebras, with 2-regular Gabriel quivers, are periodic algebras of period 4 and very specific deformations of the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles. The main result o f the paper forms an important step towards the Morita equivalence classification of all periodic symmetric tame algebras of non-polynomial growth. Applying the main result, we establish existence of wild periodic algebras of period 4, with arbitrarily large number (at least 4) of pairwise non-isomorphic simple modules. These wild periodic algebras arise as stable endomorphism rings of cluster tilting Cohen-Macaulay modules over one-dimensional hypersurface singularities.

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