Portal de Periódicos da CAPES

Reflexivity and coalgebras of finite type

1974; Elsevier BV; Volume: 28; Issue: 2 Linguagem: Inglês

10.1016/0021-8693(74)90035-0

1090-266X

Robert G Heyneman, David E. Radford,

Algebraic structures and combinatorial models

Tn this papes we study various finiteness conditions on a coalgebra C and the dual algebra C* of all linear functionals on C. C is a ru$~k~ coalgebra if every finite dirne~jon~ C*-module is rational; e~u~vale~tI~, ever>“continuous” linear functional on C* (i.e., every functional which vanishes on an ideal of C* having finite codimension) is induced by an element of C. IJsing topological considerations introduced in Section 1 together tx-ith a mx2commutative counterpart of the Hilbert “basis” theorem we show in Section 3 that a coalgebra C is reflexive if the algebra C* contains a dense subalgebra which is finitely generated. In particular w-e show that the cofree coaIgebra I(r’)% the shuffle coalgebra Sh(F) and the ~‘Bir~hoff-~~itt~~ coalgebra of divided powers B(V), are all reAexive coalgebras II-hen F is a finite dimensional vector space. Since any connected coalgebra C may be imbedded in the shufXe coalgebra Sh( F) (where si is the space of primitive elements of C), it then follows that a connected coalgebra C is reflexive if and on& if the space of primitive elements of C is finite dimensional. This condition simpiy means that C is of “‘finite type” in the foIlowing sense. Let C, be the sum of the simple subcoalgebras of a coalgebra C and define indu~t~~e~~an increasing coalgebm filtration bq’