Bound quivers of three-separate stratified posets, their Galois coverings and socle projective representations

Stanisław Kasjan

Fundamenta Mathematicae (1993)

  • Volume: 143, Issue: 3, page 259-279
  • ISSN: 0016-2736

Abstract

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A class of stratified posets I * ϱ is investigated and their incidence algebras K I * ϱ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on I * ϱ we associate with I * ϱ a bound quiver (Q, Ω) in such a way that K I * ϱ ≃ K ( Q , Ω ) . We show that the fundamental group of (Q, Ω) is the free group with two free generators if I * ϱ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition I ϱ is three-partite a fundamental domain I * + × of this covering is constructed and a functorial connection between m o d s p ( K I ϱ * + × ) and m o d s p ( K I * ϱ ) is given.

How to cite

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Kasjan, Stanisław. "Bound quivers of three-separate stratified posets, their Galois coverings and socle projective representations." Fundamenta Mathematicae 143.3 (1993): 259-279. <http://eudml.org/doc/212008>.

@article{Kasjan1993,
abstract = {A class of stratified posets $I*_ϱ$ is investigated and their incidence algebras $KI*_ϱ$ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on $I*_ϱ$ we associate with $I*_ϱ$ a bound quiver (Q, Ω) in such a way that $KI*_ϱ ≃ K(Q, Ω)$. We show that the fundamental group of (Q, Ω) is the free group with two free generators if $I*_ϱ$ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition $I_ϱ$ is three-partite a fundamental domain $I^\{*+×\}$ of this covering is constructed and a functorial connection between $mod_\{sp\} (KI^\{*+×\}_ϱ)$ and $mod_\{sp\}(KI*_ϱ)$ is given.},
author = {Kasjan, Stanisław},
journal = {Fundamenta Mathematicae},
keywords = {socle-projective representations; incidence algebras; stratified posets; socle-projective modules; bound quiver; universal covering; three-peak algebra; representation type; Auslander-Reiten quiver},
language = {eng},
number = {3},
pages = {259-279},
title = {Bound quivers of three-separate stratified posets, their Galois coverings and socle projective representations},
url = {http://eudml.org/doc/212008},
volume = {143},
year = {1993},
}

TY - JOUR
AU - Kasjan, Stanisław
TI - Bound quivers of three-separate stratified posets, their Galois coverings and socle projective representations
JO - Fundamenta Mathematicae
PY - 1993
VL - 143
IS - 3
SP - 259
EP - 279
AB - A class of stratified posets $I*_ϱ$ is investigated and their incidence algebras $KI*_ϱ$ are studied in connection with a class of non-shurian vector space categories. Under some assumptions on $I*_ϱ$ we associate with $I*_ϱ$ a bound quiver (Q, Ω) in such a way that $KI*_ϱ ≃ K(Q, Ω)$. We show that the fundamental group of (Q, Ω) is the free group with two free generators if $I*_ϱ$ is rib-convex. In this case the universal Galois covering of (Q, Ω) is described. If in addition $I_ϱ$ is three-partite a fundamental domain $I^{*+×}$ of this covering is constructed and a functorial connection between $mod_{sp} (KI^{*+×}_ϱ)$ and $mod_{sp}(KI*_ϱ)$ is given.
LA - eng
KW - socle-projective representations; incidence algebras; stratified posets; socle-projective modules; bound quiver; universal covering; three-peak algebra; representation type; Auslander-Reiten quiver
UR - http://eudml.org/doc/212008
ER -

References

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  5. [S1] D. Simson, On the representation type of stratified posets, C. R. Acad. Sci. Paris 311 (1990), 5-10. Zbl0735.16008
  6. [S2] D. Simson, Representations of bounded stratified posets, coverings and socle projective modules, in: Topics in Algebra, Banach Center Publ. 26, Part 1, PWN, Warszawa, 1990, 499-533. 
  7. [S3] D. Simson, A splitting theorem for multipeak path algebras, Fund. Math. 138 (1991), 113-137. Zbl0780.16010
  8. [S4] D. Simson, Right peak algebras of two-separate stratified posets, their Galois covering and socle projective modules, Comm. Algebra 20 (1992), 3541-3591. Zbl0791.16011
  9. [S5] D. Simson, Linear Representations of Partially Ordered Sets and Vector Space Categories, Algebra, Logic Appl. 4, Gordon & Breach, 1992. 
  10. [Sp] E. H. Spanier, Algebraic Topology, McGraw-Hill, New York, 1966. 
  11. [W] Th. Weichert, Darstellungstheorie von Algebren mit projektivem Sockel, Doctoral Thesis, Universität Stuttgart, 1989. Zbl0677.16017

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