Incidence coalgebras of interval finite posets of tame comodule type

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 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 .

Inverses of Cartan matrices of Lie algebras and Lie superalgebras

“…The Cartan matrices of types A, B, C and D can be generalized to infinite case naturally (see pp.112-114 in [8] and references [10,17]). We give the inverse formulas for these matrices and the corresponding supercases.…”