The terms in the Ringel-Hall product of preinjective Kronecker modules
- PDF / 232,516 Bytes
- 18 Pages / 595 x 842 pts (A4) Page_size
- 58 Downloads / 169 Views
THE TERMS IN THE RINGEL–HALL PRODUCT OF PREINJECTIVE KRONECKER MODULES ´nto ´ 1 and Istva ´n Szo ¨ llo ˝ si2 Csaba Sza 1
Faculty of Mathematics and Computer Science, Babe¸s–Bolyai University str. M. Kog˘ alniceanu, nr. 1, 400084, Cluj-Napoca, Romania
2
Faculty of Mathematics and Computer Science, Babe¸s–Bolyai University str. M. Kog˘ alniceanu, nr. 1, 400084, Cluj-Napoca, Romania E-mail: [email protected] (Received September 6, 2010; Accepted February 22, 2011)
Abstract We give a description of the terms in the Ringel–Hall product of preinjective Kronecker modules. We characterize in this way all the short exact sequences of preinjective modules. As an application we also give an explicit solution to the column completion challenge for pencils with only minimal indices for columns (corresponding to preinjective modules) and to the row completion challenge for pencils with only minimal indicies for rows (corresponding to preprojective modules).
1. Kronecker modules Let κ be a field. The Kronecker algebra κK, i.e., the path algebra over the Kronecker quiver α K: 1 ←− ←− 2, β
is a special but important tame hereditary algebra, because in some sense it models the behavior of all tame hereditary algebras. Denote by mod-κK the category of finite-dimensional right modules over the Kronecker algebra. These modules, called Kronecker modules, correspond to matrix pencils in linear algebra, so the Kronecker algebra relates representation theory with numerical linear algebra and matrix theory. The category mod-κK can and will be identified with the category rep-κK of the finite-dimensional κ-representations of the Kronecker quiver. Recall Mathematics subject classification number : 16G20. Key words and phrases: Kronecker algebra, matrix pencil, preinjective Kronecker module, Ringel–Hall algebra. This work was supported by Grant PN-II-ID-PCE-2008-2 project ID 2271. 0031-5303/2011/$20.00 c Akad´emiai Kiad´o, Budapest
Akad´ emiai Kiad´ o, Budapest Springer, Dordrecht
228
´ ´ and I. SZOLL ¨ ˝ CS. SZANT O OSI
that such a representation is defined as a quadruple (M1 , M2 ; f, g) where M1 , M2 are finite-dimensional κ-vector spaces (corresponding to the vertices) and f, g: M2 → M1 are κ-linear maps (corresponding to the arrows). The dimension vector of a module (viewed as a representation) M = (M1 , M2 ; f, g) ∈ mod-κK is dim(M ) = (dimκ M1 , dimκ M2 ). Up to isomorphism we will have two simple objects in mod-κK corresponding to the two vertices. We shall denote them by S1 and S2 . For a module M ∈ mod-κK, [M ] will denote the isomorphism class of M . Let tM := M ⊕ · · ·⊕ M (t-times). For two modules M, M ∈ mod-κK we will denote by M → M the fact that M can be embedded in M (i.e., M is isomorphic with a submodule of M ) and by M M the fact that M projects on M (i.e., M is isomorphic with a factor module of M ). The indecomposables in mod-κK are divided into three families: the preprojectives, the regulars and the preinjectives (see [1], [2], [6]). We will be mainly interested in the preinjective and preprojectiv
Data Loading...
