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
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