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), 2673-2679

MSC (2000):
Primary 16W30; Secondary 16R20, 16W55

DOI:
https://doi.org/10.1090/S0002-9939-03-06815-1

Published electronically:
February 20, 2003

MathSciNet review:
1974322

Full-text PDF

Abstract | References | Similar Articles | 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), 277-298. MR**2002h:16057****2.**E. J. Behr,*Enveloping algebras of Lie superalgebras*, Pacific J. Math**130**(1987), 9-25. 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. 49-79. 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), 331-340. MR**94e:16043****6.**K. R. Goodearl and E. S. Letzter,*Prime ideals in skew and q-skew 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. 177-306. MR**58:28326****8.**G. Krause and T. H. Lenagan,*Growth of Algebras and Gelfand-Kirillov 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), 322-347. MR**86k:16004****13.**M. Scheunert,*The Theory of Lie Superalgebras*, Lect. Notes Math., vol. 716, Springer, New York, 1979, pp. 177-306. MR**80i:17005****14.**L. W. Small and R. B. Warfield, Jr.,*Prime affine algebras of Gel'fand-Kirillov dimension one*, J. Algebra**91**(1984), 386-389. MR**86h:16006**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
16W30,
16R20,
16W55

Retrieve articles in all journals with MSC (2000): 16W30, 16R20, 16W55

Additional Information

**Shlomo Gelaki**

Affiliation:
Department of Mathematics, Technion-Israel 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:
https://doi.org/10.1090/S0002-9939-03-06815-1

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 DMS-9970413. 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