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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 Étud. 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

##### MSC:
 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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