Freeness of infinite dimensional Hopf algebras. (English) Zbl 0804.16042
Let \(H\) be a Hopf algebra over a field \(k\) and \(B\) a Hopf subalgebra of \(H\). It is known [see W. D. Nichols and M. B. Zoeller, Am. J. Math. 111, 381-385 (1989; Zbl 0672.16006)] that, if \(\dim_ k H\) is finite, then \(H\) is a free \(B\)-module. It is shown in the paper that \(H\) is also a free \(B\)-module in case \(\dim_ k H\) is infinite and \(B\) is a finite dimensional semisimple Hopf subalgebra of \(H\). We note that there are examples of infinite dimensional Hopf algebras which are not free over certain Hopf subalgebras [see U. Oberst and H. J. Schneider, J. Algebra 31, 10-44 (1974; Zbl 0304.14028)].
Reviewer: A.Skowroński (Toruń)
MSC:
| 16W30 | Hopf algebras (associative rings and algebras) (MSC2000) |
| 16D40 | Free, projective, and flat modules and ideals in associative algebras |
| 16S10 | Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) |
| 16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
Keywords:
Hopf algebras; Hopf subalgebras; free \(B\)-modules; finite dimensional semisimple Hopf subalgebras; infinite dimensional Hopf algebrasReferences:
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