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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, $f\left(ab\right)={\sum }_{i=1}^{n}\left({g}_{i}a\right)\left({h}_{i}b\right)$ 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 ${\Delta }f={\sum }_{i=1}^{n}{g}_{i}\otimes {h}_{i}$ is uniquely determined by f, and that ${g}_{i}$, ${h}_{i}$ are in ${A}_{v}^{0}$, so that $\left({A}_{v}^{0},{\Delta }\right)$ 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 $k{\left[x\right]}_{v}^{0}$ to V which is evaluation at x. The authors show that $\left(k{\left[x\right]}_{v}^{0},\pi \right)$ 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:
 16W30 Hopf algebras (assoc. rings and algebras) (MSC2000) 16W50 Graded associative rings and modules 15A69 Multilinear algebra, tensor products