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Generalized dual coalgebras of algebras, with applications to cofree coalgebras. (English) Zbl 0556.16005
The authors give a new description of the cofree coalgebra on a vector space V over a field k [see 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)= i=1 n (g i a)(h i b) 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 G. Hochschild, Introduction to affine algebraic groups (1971; Zbl 0221.20055)]. Let A v 0 be the k- vector space of representative maps from A to TV. Using the above notation, the authors show that Δf= i=1 n g i h i is uniquely determined by f, and that g i , h i are in A v 0 , so that (A v 0 ,Δ) 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 π the map from k[x] v 0 to V which is evaluation at x. The authors show that (k[x] v 0 ,π) is the cofree coalgebra on V. By using symmetric functions, the authors also give an analogous construction of the cofree cocommutative coalgebra on V.
Reviewer: E.J.Taft

MSC:
16W30Hopf algebras (assoc. rings and algebras) (MSC2000)
16W50Graded associative rings and modules
15A69Multilinear algebra, tensor products