On wildness of idempotent generated algebras associated with extended Dynkin diagrams

Let Λ denote an extended Dynkin diagram with vertex set Λ0 = {0, 1,... ,n}. For a vertex i, denote by S(i) the set of vertices j such that there is an edge joining i and j; one assumes the diagram has a unique vertex p, say p = 0, with |S(p)| = 3. Further, denote by Λ \ 0 the full subgraph of Λ with vertex set Λ0 \ {0}. Let ∆ = (δi |i ∈ Λ0) ∈ Z |Λ0| be an imaginary root of Λ, and let k be a field of arbitrary characteristic (with unit element 1). We prove that if Λ is an extended Dynkin diagram of type D₄, E₆ or E₇, then the k-algebra Qk(Λ, ∆) with generators ei , i ∈ Λ0 \ {0}, and relations e 2 i = ei , eiej = 0 if i and j 6= i belong to the same connected component of Λ \ 0, and Pn i=1 δi ei = δ01 has wild representation time

On dispersing representations of quivers and their connection with representations of bundles of semichains

In the paper we discuss the notion of “dispersing representation of a quiver” and give, in a natural special case, a criterion for the problem of classifying such representations to be tame. In proving the criterion we essentially use representations of bundles of semichains, introduced about fifteen years ago by the author

On representation type of a pair of posets with involution

In this paper we consider the problem on classifying the representations of a pair of posets with involution. We prove that if one of these is a chain of length at least 4 with trivial involution and the other is with nontrivial one, then the pair is tame ⇔ it is of finite type ⇔ the poset with nontrivial involution is a ∗-semichain (∗ being the involution). The case that each of the posets with involution is not a chain with trivial one was considered by the author earlier. In proving our result we do not use the known technically difficult results on representation type of posets with involution

On types of local deformations of quadratic forms

We consider some aspects of the theory of quadratic forms concerning their local deformations

Minimax isomorphism algorithm and primitive posets

The notion of minimax equivalence of posets, and a close notion of minimax isomorphism, introduced by the author are widely used in the study of quadratic Tits forms (in particular, for the description of P-critical and P-supercritical posets). In this paper, for an important special case, we modify an algorithm of classifying all posets minimax isomorphic to a given one (described earlier by the author together with M.V.Stepochkina) by introducing the concept of weak isomorphism

On tame semigroups generated by idempotents with partial null multiplication

Let I be a finite set without 0 and J a subset in I×I without diagonal elements (i,i). We define S(I,J) to be the semigroup with generators ei, where i∈I∪0, and the following relations: e0=0; e2i=ei for any i∈I; eiej=0 for any (i,j)∈J. In this paper we study finite-dimensional representations of such semigroups over a field k. In particular, we describe all finite semigroups S(I,J) of tame representation type

The representation type of elementary abelian p-groups with respect to the modules of constant Jordan type

We describe the representation type of elementary abelian p-groups with respect to the modules of constant Jordan type and offer two conjectures (for such modules) in the general case, one of which suggests that any non-wild group is of finite representation type in each dimension

The classification of serial posets with the non-negative quadratic Tits form being principal

Using (introduced by the first author) the method of (min, max)-equivalence, we classify all serial principal posets, i.e. the posets S satisfying the following conditions: (1) the quadratic Tits form qS(z) : Zˢ⁺¹ → Z of S is non-negative; (2) KerqS(z) := {t | qS(t) = 0} is an infinite cyclic group (equivalently, the corank of the symmetric matrix of qS(z) is equal to 1); (3) for any m ∈ N, there is a poset S(m) ⊃ S such that S(m) satisfies (1), (2) and |S(m) \ S| =

On characteristic properties of semigroups

Let K be a class of semigroups and P be a set of general properties of semigroups. We call a subset Q of P cha\-racteristic for a semigroup S∈ K if, up to isomorphism and anti-isomorphism, S is the only semigroup in K, which satisfies all the properties from Q. The set of properties P is called char-complete for K if for any S∈ K the set of all properties P∈ P, which hold for the semigroup S, is characteristic for S. We indicate a 7-element set of properties of semigroups which is a minimal char-complete setfor the class of semigroups of order 3

Minimax sums of posets and the quadratic Tits form

In this paper we study the structure of infinite posets with positive Tits form. In particular, there arise posets of specific form which we call minimax sums of posets