We describe the endomorphism rings of maximal rigid objects in the cluster categories of tubes. Moreover, we show that they are gentle and have Gorenstein dimension 1. We analyse their representation theory and prove that they are of finite type. Finally, we study the relationship between the module category and the cluster tube via the Hom-functor.
Let H be a connected wild hereditary path algebra. We prove that if Z is a quasi-simple regular brick, and [r]Z indecomposable regular of quasi-length r and with quasi-top Z, then .
We introduce equivariant formal deformation theory of associative algebra morphisms. We also present an equivariant deformation cohomology of associative algebra morphisms and using this we study the equivariant formal deformation theory of associative algebra morphisms. We discuss some examples of equivariant deformations and use the Maurer-Cartan equation to characterize equivariant deformations.
2000 Mathematics Subject Classification: 16N80, 16S70, 16D25, 13G05.We construct some new examples showing that Heyman and Roos construction of the essential closure in the class of associative rings can terminate at any finite or the first infinite ordinal.
To a commutative ring K, and a family of K-algebras indexed by the vertex set of a graph, we associate a K-algebra obtained by a mixture of coproduct and tensor product constructions. For this, and related constructions, we give exact sequences and deduce homological properties.
In this paper we introduce the notion of existentially closed Leibniz algebras. Then we use HNN-extensions of Leibniz algebras in order to prove an embedding theorem.
Partially supported by grant RFFI 98-01-01020.Let Uc be the variety of associative algebras generated by the algebra of all upper triangular matrices, the field being arbitrary. We prove that the upper exponent of any subvariety V ⊂ Uc coincides with the lower exponent and is an integer.