## 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, $$f(ab)=\sum^{n}_{i=1}(g_ ia)(h_ ib)$$ 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 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.
Reviewer: E.J.Taft

### MSC:

 16W30 Hopf algebras (associative rings and algebras) (MSC2000) 16W50 Graded rings and modules (associative rings and algebras) 15A69 Multilinear algebra, tensor calculus

### Citations:

Zbl 0194.32901; Zbl 0221.20055
Full Text:

### References:

 [1] Allen, H.P., Invariant radical splitting, a Hopf approach, J. pure appl. algebra, 3, 1-19, (1973) · Zbl 0271.17004 [2] Hochschild, G., Introduction to affine algebraic groups, (1971), Holden-Day San Francisco · Zbl 0221.20055 [3] Sweedler, M.E., Hopf algebras, (1969), W.A. Benjamin New York · Zbl 0194.32901
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.