Representation-Directed Incidence Coalgebras of Intervally Finite Posets and the Tame-Wild Dichotomy

Abstract

Incidence coalgebras C = K ^□ I of intervally finite posets I that are representation-directed are characterized in the article, and the posets I with this property are described. In particular, it is shown that the coalgebra C = K ^□ I is representation-directed if and only if the Euler quadratic form q _C : ℤ^(I) → ℤ of C is weakly positive. Every such a coalgebra C is tame of discrete comodule type and gl. dimC ≤ 2. As a consequence, we get a characterization of the incidence coalgebras C = K ^□ I that are left pure semisimple in the sense that every left C-comodule is a direct sum of finite dimensional subcomodules. It is shown that every such coalgebra C = K ^□ I is representation-directed and gl. dimC ≤ 2. Finally, the tame-wild dichotomy theorem is proved, for the coalgebras K ^□ I that are right semiperfect.

Key Words:

• Coalgebra
• Colocalization
• Comodule type
• Euler form
• Poset
• Tame-wild dichotomy

2000 Mathematics Subject Classification:

• Primary 16G20, 16W30
• Secondary 15A63, 18E35

ACKNOWLEDGMENT

Supported by Polish KBN Grant 1 P03A 014 28. Dedicated to Kunio Yamagata on the occasion of his 60th birthday.

Notes

Communicated by M. Ferrero.