Incidence coalgebras of interval finite posets of tame comodule type

Zbigniew Leszczyński, and Daniel Simson. "Incidence coalgebras of interval finite posets of tame comodule type." Colloquium Mathematicae 141.2 (2015): 261-295. <http://eudml.org/doc/284146>.

@article{ZbigniewLeszczyński2015,
abstract = {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 ̃ *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 ̃ *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, $b̅_\{K^\{□\} I\} (dim M,dim N) = ∑_\{j=0\}^\{∞\} (-1)^\{j\} dim_\{K\} Ext_\{K^\{□\} I\}^\{j\}(M,N)$, where $b̅_\{K^\{□\} I\}: ℤ^\{(I)\} × ℤ^\{(I)\} → ℤ$ is the Euler ℤ-bilinear form of I and dim M, dim N are the dimension vectors of M and N.},
author = {Zbigniew Leszczyński, Daniel Simson},
journal = {Colloquium Mathematicae},
keywords = {incidence coalgebras; finitely copresented comodules; nilpotent representations; Grothendieck groups; Euler characteristic; Euler coalgebras; integral quadratic forms; Coxeter polynomials; interval finite posets},
language = {eng},
number = {2},
pages = {261-295},
title = {Incidence coalgebras of interval finite posets of tame comodule type},
url = {http://eudml.org/doc/284146},
volume = {141},
year = {2015},
}

TY - JOUR
AU - Zbigniew Leszczyński
AU - Daniel Simson
TI - Incidence coalgebras of interval finite posets of tame comodule type
JO - Colloquium Mathematicae
PY - 2015
VL - 141
IS - 2
SP - 261
EP - 295
AB - 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 ̃ *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 ̃ *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, $b̅_{K^{□} I} (dim M,dim N) = ∑_{j=0}^{∞} (-1)^{j} dim_{K} Ext_{K^{□} I}^{j}(M,N)$, where $b̅_{K^{□} I}: ℤ^{(I)} × ℤ^{(I)} → ℤ$ is the Euler ℤ-bilinear form of I and dim M, dim N are the dimension vectors of M and N.
LA - eng
KW - incidence coalgebras; finitely copresented comodules; nilpotent representations; Grothendieck groups; Euler characteristic; Euler coalgebras; integral quadratic forms; Coxeter polynomials; interval finite posets
UR - http://eudml.org/doc/284146
ER -