Smash (co)products from adjunctions. (English) Zbl 0905.16017

Dăscălescu, S.; Raianu, Ş.; Van Oystaeyen, F.

Caenepeel, Stefaan (ed.) et al., Rings, Hopf algebras, and Brauer groups. Proceedings of the fourth week on algebra and algebraic geometry, SAGA-4, Antwerp and Brussels, Belgium, September 12–17, 1996. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 197, 103-110 (1998).

The authors study the comultiplication on the smash coproduct of \(C\) and \(H\), where \(H\) is a finite dimensional Hopf algebra over a field \(k\) and \(C\) is a left \(H\)-comodule coalgebra. They show that this comultiplication is induced by an adjunction. They start with \(H\otimes C\) as an object in the category \(M(H^*)^C_{H^*}\) of generalized Hopf modules in the sense of Doi. Tensoring on the right by \(H\otimes C\) is a functor from \(M_k\), the category of \(k\)-vector spaces, to \(M(H^*)^C_{H^*}\), and is the right adjoint of the forgetful functor \(F\) from \(M(H^*)^C_{H^*}\) to \(M_k\). This adjunction induces a coalgebra structure on \(F(H\otimes C)\) which is isomorphic as \(k\)-vector space to \(C\otimes H\), and the comultiplication obtained on \(C\otimes H\) is the comultiplication of the smash coproduct.
A similar discussion for the dual case of smash product algebras is also possible. This is briefly indicated at the end of the article.
For the entire collection see [Zbl 0884.00036].

Reviewer: E.J.Taft (New Brunswick)

MSC:

16W30	Hopf algebras (associative rings and algebras) (MSC2000)
16S40	Smash products of general Hopf actions
16D90	Module categories in associative algebras

Keywords:

comodule coalgebras; comultiplications; smash coproducts; finite dimensional Hopf algebras; adjunctions; generalized Hopf modules; smash product algebras