On Coxeter type study of non-negative posets using matrix morsifications and isotropy groups of Dynkin and Euclidean diagrams

The incidence coalgebras K I of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form q • : Z (I) → Z, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category K I-comod of finite-dimensional left K I-modules is equivalent to the tameness of the category K I-Comod fc of finitely copresented left K I-modules. Hence, the tame-wild dichotomy for the coalgebras K I is deduced. Moreover, we prove that for an interval finite A * m -free poset I the incidence coalgebra K I is of tame comodule type if and only if the quadratic form q • is weakly non-negative. Finally, we give a complete list of all infinite connected interval finite A * m -free posets I such that K I is of tame comodule type. In this case we prove that, for any pair of finite-dimensional left K I-comodules M and N , bK I (dim M, dim N ) = ∞ j=0 (−1) j dimK Ext j K I (M, N ), where bK I : Z (I) × Z (I) → Z is the Euler Z-bilinear form of I and dim M , dim N are the dimension vectors of M and N .

show abstract

“…A straightforward calculation shows that the Coxeter polynomial and the reduced Coxeter-Euler polynomial of I are (see [13])…”

Section: Preliminaries On Incidence Coalgebras and Their Comodulesmentioning

confidence: 99%

“…the Coxeter number c I equals 6, whereas the reduced Coxeter-Euler number c • I is infinite (see [13,47,[50][51][52]). The Euler form q I : Z 8 → Z is given by…”

Section: Preliminaries On Incidence Coalgebras and Their Comodulesmentioning

confidence: 99%