Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T03:02:59.562Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Hopf algebra dual of an enveloping algebra

Published online by Cambridge University Press:  24 October 2008

Stephen Donkin
Stephen Donkin
Affiliation:
King's College, Cambridge
Get access Rights & Permissions [Opens in a new window]

Extract

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 91 , Issue 2 , March 1982 , pp. 215 - 224
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Donkin, S.On the Noetherian property in endomorphism rings of certain comodules. J. Algebra 70 (1981), 394–419.CrossRefGoogle Scholar
(2)Donkin, S. Locally finite representations of poly cyclic-by-finite groups. Proc. London Math. Soc. (to appear).Google Scholar
(3)Donkin, S.Polycyclic groups, Lie algebras and algebraic groups. J. Reine Angew. Math. (Crelle) 326 (1981), 104–123.Google Scholar
(4)Green, J. A.Locally finite representations. J. Algebra 41 (1976), 137–171.CrossRefGoogle Scholar
(5)Jacobson, N.Lie algebras. New York-London, Wiley Interscience 1962.Google Scholar
(6)McConnell, J. C.Localization in enveloping rings. J. London Math. Soc. 43 (1968), 421–428.CrossRefGoogle Scholar
(7)Müller, B. J.Localisation in noncommutative Noetherian rings. Canadian J. Math. 28 (1976), 600–610.Google Scholar
(8)Passman, D. S.The algebraic structure of group rings. Interscience, New York, 1977.Google Scholar
(9)Pickel, P. F.Rational cohomology of nilpotent groups and Lie algebras. Communications in algebra, 6 (1978), 409–419.Google Scholar
(10)Sweedler, M. E.Hopf algebras. Benjamin, New York, 1969.Google Scholar
(11)Takeuchi, M.A correspondence between Hopf ideals and sub-Hopf algebras. Manuscripta Math. 7 (1972), 251–270.CrossRefGoogle Scholar