An affine PI Hopf algebra not finite over a normal commutative Hopf subalgebra
Authors:
Shlomo Gelaki and Edward S. Letzter
Journal:
Proc. Amer. Math. Soc. 131 (2003), 26732679
MSC (2000):
Primary 16W30; Secondary 16R20, 16W55
Published electronically:
February 20, 2003
MathSciNet review:
1974322
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: In formulating a generalized framework to study certain noncommutative algebras naturally arising in representation theory, K. A. Brown asked if every finitely generated Hopf algebra satisfying a polynomial identity was finite over a normal commutative Hopf subalgebra. In this note we show that Radford's biproduct, applied to the enveloping algebra of the Lie superalgebra , provides a noetherian prime counterexample.
 1.
N. Andruskiewitsch, P. Etingof, and S. Gelaki, Triangular Hopf algebras with the Chevalley property, Michigan Journal of Mathematics 49 (2001), 277298. MR 2002h:16057
 2.
E. J. Behr, Enveloping algebras of Lie superalgebras, Pacific J. Math 130 (1987), 925. MR 89b:17023
 3.
K. A. Brown, Representation theory of noetherian Hopf algebras satisfying a polynomial identity, Trends in the Representation Theory of Finite Dimensional Algebras, Contemporary Mathematics, vol. 229, American Mathematical Society, Providence, 1998, pp. 4979. MR 99m:16056
 4.
K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups (to appear).
 5.
D. Fischman, Schur's double centralizer theorem: a Hopf algebra approach, J. Algebra 157 (1993), 331340. MR 94e:16043
 6.
K. R. Goodearl and E. S. Letzter, Prime ideals in skew and qskew polynomial rings, Mem. Amer. Math. Soc. 521 (1994). MR 94j:16051
 7.
B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometrical Methods in Mathematical Physics, Proc. Symp. Bonn 1975, Lect. Notes Math., vol. 570, Springer, New York, 1977, pp. 177306. MR 58:28326
 8.
G. Krause and T. H. Lenagan, Growth of Algebras and GelfandKirillov Dimension, revised edition, American Mathematical Society, Providence, 2000. MR 2000j:16035
 9.
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Grad. Stud. Math., vol. 30, Amer. Math. Soc., Providence, 2000. MR 2001i:16039
 10.
S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995. MR 97g:17016
 11.
S. Montgomery, Hopf Algebras and Their Actions on Rings, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 82, American Mathematical Society, providence, 1993. MR 94i:16019
 12.
D. Radford, The structure of Hopf algebras with a projection, J. Algebra 92 (1985), 322347. MR 86k:16004
 13.
M. Scheunert, The Theory of Lie Superalgebras, Lect. Notes Math., vol. 716, Springer, New York, 1979, pp. 177306. MR 80i:17005
 14.
L. W. Small and R. B. Warfield, Jr., Prime affine algebras of Gel'fandKirillov dimension one, J. Algebra 91 (1984), 386389. MR 86h:16006
 1.
 N. Andruskiewitsch, P. Etingof, and S. Gelaki, Triangular Hopf algebras with the Chevalley property, Michigan Journal of Mathematics 49 (2001), 277298. MR 2002h:16057
 2.
 E. J. Behr, Enveloping algebras of Lie superalgebras, Pacific J. Math 130 (1987), 925. MR 89b:17023
 3.
 K. A. Brown, Representation theory of noetherian Hopf algebras satisfying a polynomial identity, Trends in the Representation Theory of Finite Dimensional Algebras, Contemporary Mathematics, vol. 229, American Mathematical Society, Providence, 1998, pp. 4979. MR 99m:16056
 4.
 K. A. Brown and K. R. Goodearl, Lectures on Algebraic Quantum Groups (to appear).
 5.
 D. Fischman, Schur's double centralizer theorem: a Hopf algebra approach, J. Algebra 157 (1993), 331340. MR 94e:16043
 6.
 K. R. Goodearl and E. S. Letzter, Prime ideals in skew and qskew polynomial rings, Mem. Amer. Math. Soc. 521 (1994). MR 94j:16051
 7.
 B. Kostant, Graded manifolds, graded Lie theory, and prequantization, Differential Geometrical Methods in Mathematical Physics, Proc. Symp. Bonn 1975, Lect. Notes Math., vol. 570, Springer, New York, 1977, pp. 177306. MR 58:28326
 8.
 G. Krause and T. H. Lenagan, Growth of Algebras and GelfandKirillov Dimension, revised edition, American Mathematical Society, Providence, 2000. MR 2000j:16035
 9.
 J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Grad. Stud. Math., vol. 30, Amer. Math. Soc., Providence, 2000. MR 2001i:16039
 10.
 S. Majid, Foundations of Quantum Group Theory, Cambridge University Press, Cambridge, 1995. MR 97g:17016
 11.
 S. Montgomery, Hopf Algebras and Their Actions on Rings, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, vol. 82, American Mathematical Society, providence, 1993. MR 94i:16019
 12.
 D. Radford, The structure of Hopf algebras with a projection, J. Algebra 92 (1985), 322347. MR 86k:16004
 13.
 M. Scheunert, The Theory of Lie Superalgebras, Lect. Notes Math., vol. 716, Springer, New York, 1979, pp. 177306. MR 80i:17005
 14.
 L. W. Small and R. B. Warfield, Jr., Prime affine algebras of Gel'fandKirillov dimension one, J. Algebra 91 (1984), 386389. MR 86h:16006
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Additional Information
Shlomo Gelaki
Affiliation:
Department of Mathematics, TechnionIsrael Institute of Technology, Haifa 32000, Israel
Email:
gelaki@math.technion.ac.il
Edward S. Letzter
Affiliation:
Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email:
letzter@math.temple.edu
DOI:
http://dx.doi.org/10.1090/S0002993903068151
PII:
S 00029939(03)068151
Received by editor(s):
December 5, 2001
Received by editor(s) in revised form:
April 5, 2002
Published electronically:
February 20, 2003
Additional Notes:
The first author’s research was supported by the Technion V.P.R. Fund–Loewengart Research Fund, and by the Fund for the Promotion of Research at the Technion. The second author’s research was supported in part by NSF grant DMS9970413. This research was begun during the second author’s visit to the Technion in August 2001, and he is grateful to the Technion for its hospitality.
Communicated by:
Martin Lorenz
Article copyright:
© Copyright 2003
American Mathematical Society
