Generalized dual coalgebras of algebras, with applications to cofree coalgebras (Q761527)

The authors give a new description of the cofree coalgebra on a vector space V over a field k [see \textit{M. Sweedler}, Hopf algebras (1969; Zbl 0194.32901)]. Let A be a graded k-algebra, TV the (graded) tensor algebra on V. A graded linear map f of degree zero from A to TV is called representative if for some graded linear maps \(g_ 1,...,g_ n\), \(h_ 1,...,h_ n\) of degree zero from A to TV, \(f(ab)=\sum^{n}_{i=1}(g_ ia)(h_ ib)\) for all a,b in A. This generalizes the notion of representative function on a group G, e.g., \(A=the\) group algebra kG, \(V=k\), and restrict to G [see \textit{G. Hochschild}, Introduction to affine algebraic groups (1971; Zbl 0221.20055)]. Let \(A^ 0_ v\) be the k- vector space of representative maps from A to TV. Using the above notation, the authors show that \(\Delta f=\sum^{n}_{i=1}g_ i\otimes h_ i\) is uniquely determined by f, and that \(g_ i\), \(h_ i\) are in \(A^ 0_ v\), so that \((A^ 0_ v,\Delta)\) is a coalgebra with counit which is evaluation at the unit element of A. For \(V=k\), this reduces to the usual notion of \(A^ 0\) [cf. M. Sweedler]. Now let A be the polynomial algebra k[x] with the usual grading, and \(\pi\) the map from \(k[x]^ 0_ v\) to V which is evaluation at x. The authors show that \((k[x]^ 0_ v,\pi)\) is the cofree coalgebra on V. By using symmetric functions, the authors also give an analogous construction of the cofree cocommutative coalgebra on V.