Tame Blocks Research Articles

Selfextensions of modules over group algebras

Let KG be a group algebra with G a finite group and K a field and M an indecomposable KG-module. We pose the question, whether ExtKG1(M,M)≠0 implies that ExtKGi(M,M)≠0 for all i≥1. We give a positive answer in several important special cases such as for periodic groups and give a positive answer also for all Nakayama algebras, which allows us to improve a classical result of Gustafson. We then specialise the question to the case where the module M is simple, where we obtain a positive answer also for all tame blocks of group algebras. For simple modules M, the appendix provides a Magma program that gives strong evidence for a positive answer to this question for groups of small order.

REAL BLOCKS WITH DIHEDRAL DEFECT GROUPS REVISITED

AbstractThe Frobenius–Schur indicators of characters in a real $2$ -block with dihedral defect groups have been determined by Murray [‘Real subpairs and Frobenius–Schur indicators of characters in 2-blocks’, J. Algebra322 (2009), 489–513]. We show that two infinite families described in his work do not exist and we construct examples for the remaining families. We further present some partial results on Frobenius–Schur indicators of characters in other tame blocks.

The first Hochschild cohomology as a Lie algebra

In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely, several families of self-injective tame algebras including all tame blocks of finite groups and some wild algebras including most quantum complete intersections.

Morita equivalence classes of tame blocks of finite groups

We show that several Morita equivalence classes of tame algebras do not occur as blocks of finite groups. This refines classifications by Erdmann of classes of blocks with dihedral, semidihedral, and generalised quaternion defect groups. In particular we now have a complete classification of the Morita equivalence classes of blocks of finite groups with dihedral defect groups.

A reduction theorem for tau -rigid modules

We prove a theorem which gives a bijection between the support tau -tilting modules over a given finite-dimensional algebra A and the support tau -tilting modules over A / I, where I is the ideal generated by the intersection of the center of A and the radical of A. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are tau -tilting-finite wild blocks with more than one simple module. We then go on to classify all support tau -tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most 32 different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field k, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all tau -rigid modules over (not necessarily symmetric) string algebras.

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 occurred 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 also symmetric tame periodic algebras of period 4. The main results of this 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 with 2-regular Gabriel quivers [36].

Universal deformation rings and tame blocks

Let k be an algebraically closed field of positive characteristic, and let G be a finite group. Suppose B is a block of kG of infinite tame representation type. We find all finitely generated kG-modules V that belong to B and whose endomorphism ring is isomorphic to k and determine the universal deformation ring R(G,V) for each of these modules.

Classifying tame blocks and related algebras up to stable equivalences of Morita type

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of Erdmann’s algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander–Reiten conjecture for stable equivalences of Morita type between two algebras, one of which is of dihedral, semi-dihedral or quaternion type.

The gluing problem for some block fusion systems

We answer the gluing problem of blocks of finite groups (Linckelmann (2004) [7, 4.2] ) for tame blocks and the principal p -block of PSL 3 ( p ) for p odd. In particular, we show that the gluing problem for the principal p -block of PSL 3 ( p ) does not have a unique solution when p ≡ 1 mod 3 .

BLOCKS WITH A QUATERNION DEFECT GROUP OVER A 2-ADIC RING: THE CASEÃ4

Abstract.Except for blocks with a cyclic or Klein four defect group, it is not known in general whether the Morita equivalence class of a block algebra over a field of prime characteristic determines that of the corresponding block algebra over ap-adic ring. We prove this to be the case when the defect group is quaternion of order 8 and the block algebra over an algebraically closed fieldkof characteristic 2 is Morita equivalent tokÃ4. The main ingredients are Erdmann's classification of tame blocks [6] and work of Cabanes and Picaronny [4, 5] on perfect isometries between tame blocks.

Tameness and complexity of finite group schemes

Using a representation-theoretic interpretation of support varieties due to Friedlander- Pevtsova (18), we show that the complexity of tame blocks of finite group schemes is bounded by 2. In this context, our result salvages a theorem by Rickard (23), whose proof is flawed.

Polyhedral groups, McKay quivers, and the finite algebraic groups with tame principal blocks

Given an algebraically closed field k of characteristic p≥3, we classify the finite algebraic k-groups whose algebras of measures afford a principal block of tame representation type. The structure of such a group $\mathcal{G}$ is largely determined by a linearly reductive subgroup scheme $\hat{\mathcal{G}}$ of SL(2), with the McKay quiver of $\hat{\mathcal{G}}$ relative to its standard module being the Gabriel quiver of the principal block $\mathcal{B}_0(\mathcal{G})$ . The graphs underlying these quivers are extended Dynkin diagrams of type $\tilde{A}, \tilde{D}$ or $\tilde{E}$ , and the tame blocks are Morita equivalent to generalizations of the trivial extensions of the radical square zero tame hereditary algebras of the corresponding type.

Block representation type of Frobenius kernels of smooth groups

Let be a smooth algebraic group, defined over an algebraically closed field k of characteristic p ≧ 3. We determine the structure of the representation-finite and tame blocks of the algebras of distributions associated to the Frobenius kernels of .

Harish-Chandra modules for Yangians

We study Harish-Chandra representations of the Yangian Y ( g l 2 ) \mathrm {Y}(\mathfrak {gl}_2) with respect to a natural maximal commutative subalgebra. We prove an analogue of the Kostant theorem showing that the restricted Yangian Y p ( g l 2 ) \mathrm {Y}_p(\mathfrak {gl}_2) is a free module over the corresponding subalgebra Γ \Gamma and show that every character of Γ \Gamma defines a finite number of irreducible Harish-Chandra modules over Y p ( g l 2 ) \mathrm {Y}_p(\mathfrak {gl}_2) . We study the categories of generic Harish-Chandra modules, describe their simple modules and indecomposable modules in tame blocks.

Eigenvalues and elementary divisors of Cartan matrices of cyclic blocks with [formula omitted] and tame blocks

We give a conjecture on relation between eigenvalues and elementary divisors of the Cartan matrix of a p-block B of a finite group. Then we prove the conjecture is true for cyclic blocks with l ( B ) ⩽ 5 and for tame blocks.

Hochschild cohomology of tame blocks

We study the additive structure of Hochschild cohomology for blocks of modular group algebras of tame representation type. We give explicit formulae for the dimensions of the cohomology groups and compute the Poincaré series of the Hochschild cohomology ring for all tame blocks having one or three simple modules.

Derived Equivalent Tame Blocks

In this article we provide derived equivalences between several blocks of tame representation type. Based on Erdmann's work on Morita equivalence classes of tame blocks we obtain a classification of tame blocks up to derived equivalence (as in Erdmann's work, in the case of two simple modules our classification is only complete up to scalars occurring in certain relations).

Dade's conjecture for tame blocks

Dade's conjecture for tame blocks

On the number of simple modules of certain tame blocks and algebras

On the number of simple modules of certain tame blocks and algebras

Tame biserial algebras

A similar result is suspected if we drop the property “representationfinite” in classes 1 and 2, and we will start an investigation of biserial algebras of infinite representation type. The key to the result for representation-finite algebras was to observe that each representation-finite biserial algebra is special [13] (see also the basic notations below), which means that it can be presented by a quiver with “nice” relations. Special algebras play an important role in the modular representation theory of finite groups. Namely, each representation-finite block of a group algebra and some of the tame blocks (they occur only in characteristic 2) are special [7, 10,4]. Moreover in the complex representation theory of the Lorentz group the so-called Harish-Chandra modules are defined over (tame) special algebras [6]. In this paper we show that /l(A)62 for any special algebra A and that special algebras of infinite representation type are tame. Our paper consists of four parts. First we prove that each special algebra is a factor of a special symmetric algebra (Theorem 1.5). For the latter the methods of Gelfand and Ponomarev used in the classification of the indecomposable Harish-Chandra modules of the Lorentz group apply and furnish a complete list of indecomposable finitely generated modules over special algebras (Proposition 2.3) presented in Section 2. From this list we 480 0021~8693/85 $3.00