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Abel transformation and harmonic analysis. I. (Russian) Zbl 0585.43004
Essentially, the subject is abstract Hopf algebras (cocommutative bigebras with antipode) in a symmetric monoidal closed k-category \({\mathcal R}\) for a field k; but the presentation is complicated by minimal, hence variable, hypotheses. One of the two main theorems [cf. M. E. Sweedler, Publ. Math., Inst. Hautes √Čtud. Sci. 44 (1974), 79-189 (1975; Zbl 0314.16008)] gives a functor \({\hat \alpha}\) from certain diagrams of \({\mathcal R}\)-Hopf algebras to \({\mathcal R}\)-algebras taking \(\Delta =G\leftarrow S\to K\) to part of a pseudo-pushout \(\Pi\) with suitable multiplication. \(\Pi\) is constructed like a pushout but with \(\otimes\) instead of coproduct. \(\Pi\) itself becomes an \({\hat \alpha}(\Delta)\)-module. If S has a total integral \(S\to k\), there is also an embedding of \({\hat \alpha}(\Delta)\) in \(G\otimes K.\)
The second theorem involves an S-admissible sub-bigebra P of G (i.e. G as coalgebra is the pseudo-pushout after pulling back to \(p=P\times_{G}S)\) and an ideal j of P which is a P-bimodule direct summand, \(P=j\oplus d\). Then from \(d=P/j\) one has \(d\leftarrow p\to K=\Delta '\), say. The theorem gives an algebra morphism \({\hat \alpha}(\Delta)\to {\hat \alpha}(\Delta')\) and a coalgebra morphism of the pseudo-pushouts.
Reviewer: J.R.Isbell

43A95 Categorical methods for abstract harmonic analysis
18D15 Closed categories (closed monoidal and Cartesian closed categories, etc.)
16W30 Hopf algebras (associative rings and algebras) (MSC2000)
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