Burnside's theorem for Hopf algebras
Authors:
D. S. Passman and Declan Quinn
Journal:
Proc. Amer. Math. Soc. 123 (1995), 327333
MSC:
Primary 16W30; Secondary 16S30, 16S34
MathSciNet review:
1215204
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: A classical theorem of Burnside asserts that if is a faithful complex character for the finite group G, then every irreducible character of G is a constituent of some power of . Fifty years after this appeared, Steinberg generalized it to a result on semigroup algebras with K an arbitrary field and with G a semigroup, finite or infinite. Five years later, Rieffel showed that the theorem really concerns bialgebras and Hopf algebras. In this note, we simplify and amplify the latter work.
 [B]
W.
Burnside, Theory of groups of finite order, Dover
Publications, Inc., New York, 1955. 2d ed. MR 0069818
(16,1086c)
 [H]
HarishChandra,
On representations of Lie algebras, Ann. of Math. (2)
50 (1949), 900–915. MR 0030945
(11,77b)
 [J]
Nathan
Jacobson, Lie algebras, Interscience Tracts in Pure and
Applied Mathematics, No. 10, Interscience Publishers (a division of John
Wiley & Sons), New YorkLondon, 1962. MR 0143793
(26 #1345)
 [LS]
Richard
Gustavus Larson and Moss
Eisenberg Sweedler, An associative orthogonal bilinear form for
Hopf algebras, Amer. J. Math. 91 (1969), 75–94.
MR
0240169 (39 #1523)
 [M]
Walter
Michaelis, Properness of Lie algebras and
enveloping algebras. I, Proc. Amer. Math.
Soc. 101 (1987), no. 1, 17–23. MR 897064
(89f:17015a), http://dx.doi.org/10.1090/S00029939198708970641
 [N]
Warren
D. Nichols, Quotients of Hopf algebras, Comm. Algebra
6 (1978), no. 17, 1789–1800. MR 508081
(80a:16017), http://dx.doi.org/10.1080/00927877808822321
 [Ra]
David
E. Radford, On the quasitriangular structures of a semisimple Hopf
algebra, J. Algebra 141 (1991), no. 2,
354–358. MR 1125700
(92j:16026), http://dx.doi.org/10.1016/00218693(91)902362
 [Ri]
M.
A. Rieffel, Burnside’s theorem for representations of Hopf
algebras, J. Algebra 6 (1967), 123–130. MR 0210794
(35 #1680)
 [St]
Robert
Steinberg, Complete sets of representations of
algebras, Proc. Amer. Math. Soc. 13 (1962), 746–747. MR 0141710
(25 #5107), http://dx.doi.org/10.1090/S0002993919620141710X
 [Sw]
Moss
E. Sweedler, Hopf algebras, Mathematics Lecture Note Series,
W. A. Benjamin, Inc., New York, 1969. MR 0252485
(40 #5705)
 [B]
 W. Burnside, Theory of groups of finite order (reprint of 2nd edition, 1911), Dover, New York, 1955. MR 0069818 (16:1086c)
 [H]
 HarishChandra, On representations of Lie algebras, Ann. of Math. (2) 50 (1949), 900915. MR 0030945 (11:77b)
 [J]
 N. Jacobson, Lie algebras, Interscience, New York, 1962. MR 0143793 (26:1345)
 [LS]
 R. G. Larson and M. E. Sweedler, An associative orthogonal bilinear form for Hopf algebras, Amer. J. Math. 91 (1969), 7594. MR 0240169 (39:1523)
 [M]
 W. Michaelis, Properness of Lie algebras and enveloping algebras I, Proc. Amer. Math. Soc. 101 (1987), 1723. MR 897064 (89f:17015a)
 [N]
 W. D. Nichols, Quotients of Hopf algebras, Comm. Algebra 6 (1978), 17891800. MR 508081 (80a:16017)
 [Ra]
 D. E. Radford, On the quasitriangular structure of a semisimple Hopf algebra, J. Algebra 141 (1991), 354358. MR 1125700 (92j:16026)
 [Ri]
 M. A. Rieffel, Burnside's theorem for representations of Hopf algebras, J. Algebra 6 (1967), 123130. MR 0210794 (35:1680)
 [St]
 R. Steinberg, Complete sets of representations of algebras, Proc. Amer. Math. Soc. 13 (1962), 746747. MR 0141710 (25:5107)
 [Sw]
 M. E. Sweedler, Hopf Algebras, Benjamin, New York, 1969. MR 0252485 (40:5705)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC:
16W30,
16S30,
16S34
Retrieve articles in all journals
with MSC:
16W30,
16S30,
16S34
Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199512152046
PII:
S 00029939(1995)12152046
Article copyright:
© Copyright 1995
American Mathematical Society
