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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

Rigidity of graded regular algebras

Author(s): E. Kirkman; J. Kuzmanovich; J. J. Zhang
Journal: Trans. Amer. Math. Soc. 360 (2008), 6331-6369.
MSC (2000): Primary 16E10, 16W30, 20J05
Posted: June 26, 2008
MathSciNet review: 2434290
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Abstract | References | Similar articles | Additional information

Abstract: We prove a graded version of Alev-Polo's rigidity theorem: the homogenization of the universal enveloping algebra of a semisimple Lie algebra and the Rees ring of the Weyl algebras $ A_n(k)$ cannot be isomorphic to their fixed subring under any finite group action. We also show the same result for other classes of graded regular algebras including the Sklyanin algebras.

References:

[AD1]
J. Alev and F. Dumas, Sur les invariants des algèbres de Weyl et de leurs corps de fractions. Rings, Hopf algebras, and Brauer groups (Antwerp/Brussels, 1996), 1-10, Lecture Notes in Pure and Appl. Math., 197, Dekker, New York, 1998. MR 1615833 (99g:16042)

[AD2]
J. Alev and F. Dumas, Invariants du corps de Weyl sous l'action de groupes finis. Comm. Algebra 25 (1997), no. 5, 1655-1672. MR 1444026 (99c:16030)

[AP]
J. Alev and P. Polo, A rigidity theorem for finite group actions on enveloping algebras of semisimple Lie algebras, Adv. Math. 111 (1995), no. 2, 208-226. MR 1318528 (95m:16015)

[AS]
M. Artin and W. F. Schelter, Graded algebras of global dimension $ 3$, Adv. in Math. 66 (1987), no. 2, 171-216. MR 917738 (88k:16003)

[ATV1]
M. Artin, J. Tate and M. Van den Bergh, Some algebras associated to automorphisms of elliptic curves, ``The Grothendieck Festschrift,'' Vol. I, ed. P. Cartier et al., Birkhäuser Boston, 1990, 33-85. MR 1086882 (92e:14002)

[ATV2]
M. Artin, J. Tate and M. Van den Bergh, Modules over regular algebras of dimension $ 3$, Invent. Math. 106 (1991), no. 2, 335-388. MR 1128218 (93e:16055)

[AZ]
M. Artin and J. J. Zhang, Noncommutative projective schemes, Adv. Math. 109 (1994), 228-287. MR 1304753 (96a:14004)

[BR]
G. Benkart and T. Roby, Down-up algebras, J. Algebra 209 (1998), 305-344. Addendum, J. Algebra 213 (1999), no. 1, 378. MR 1652138 (2000e:06001a); MR1674692 (2000e:06001b)

[Be]
D.J. Benson, Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, 190. Cambridge University Press, Cambridge, 1993. MR 1249931 (94j:13003)

[CE]
A. Clark and J. Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425-434. MR 0367979 (51:4221)

[Co]
H.S.M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2) 35 (1934), no. 3, 588-621. MR 1503182

[Go]
N.L. Gordeev, Invariants of linear groups generated by matrices with two nonidentity eigenvalues, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 114 (1982), 120-130; English translation in J. Soviet Math. 27 (1984), no. 4. MR 669563 (83m:20067)

[HW]
G. H. Hardy and E. M Wright, ``An Introduction to the Theory of Numbers'', Fourth Edition, Oxford University Press, London, 1960. MR 0067125 (16:673c)

[JiZ]
N. Jing and J.J. Zhang, On the trace of graded automorphisms, J. Algebra 189 (1997), no. 2, 353-376. MR 1438180 (98f:16029)

[JoZ]
P. Jørgensen and J.J. Zhang, Gourmet's guide to Gorensteinness, Adv. Math. 151 (2000), no. 2, 313-345. MR 1758250 (2001d:16023)

[Jos]
A. Joseph, Coxeter structure and finite group action, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995), 185-219, Sémin. Congr., 2, Soc. Math. France, Paris, 1997. MR 1601143 (98k:17011)

[KW]
V. Kac and K. Watanabe, Finite linear groups whose ring of invariants is a complete intersection, Bull, Amer. Math. Soc. (N.S.) 6 (1982), no. 2, 221-223. MR 640951 (83h:14042)

[KKZ1]
E. Kirkman, J. Kuzmanovich and J.J. Zhang, A Shephard-Todd-Chevalley theorem for noncommutative regular algebras, in preparation.

[KKZ2]
E. Kirkman, J. Kuzmanovich and J.J. Zhang, Hopf algebra (co)-actions on Artin-Schelter regular algebras, in preparation.

[KMP]
E. Kirkman, I. Musson and D. Passman, Noetherian down-up algebras, Proc. Amer. Math. Soc. 127 (1999), 3161-3167. MR 1610796 (2000b:16042)

[KL]
G. Krause and T. Lenagan, ``Growth of Algebras and Gelfand-Kirillov Dimension, revised edition'', Graduate Studies in Mathematics, Vol. 22, AMS, Providence, 2000. MR 1721834 (2000j:16035)

[Le]
T. Levasseur, Some properties of noncommutative regular rings, Glasgow Math. J. 34 (1992), 277-300. MR 1181768 (93k:16045)

[MR]
J. C. McConnell and J. C . Robson, ``Noncommutative Noetherian Rings,'' Wiley, Chichester, 1987. MR 934572 (89j:16023)

[Mo]
R. Mollin, ``Algebraic Number Theory''. Chapman & Hall/CRC, Boca Raton, FL, 1999. MR 1682930 (2000e:11130)

[Mon1]
S. Montgomery, ``Fixed rings of finite automorphism groups of associative rings'', Lecture Notes in Mathematics, 818. Springer, Berlin, 1980 MR 590245 (81j:16041)

[Mon2]
S. Montgomery, ``Hopf algebras and their actions on rings'', CBMS Regional Conference Series in Mathematics, 82, Providence, RI, 1993. MR 1243637 (94i:16019)

[RRZ]
Z. Reichstein, D. Rogalski and J.J. Zhang, Projectively simple rings, Adv. in Math, 203 (2006), 365-407. MR 2227726 (2007i:16071)

[ShT]
G.C. Shephard and J.A. Todd, Finite unitary reflection groups, Canadian J. Math. 6, (1954). 274-304. MR 0059914 (15:600b)

[Sm1]
S.P. Smith, Can the Weyl algebra be a fixed ring?, Proc. Amer. Math. Soc. 107 (1989), no. 3, 587-589. MR 962247 (90b:16010)

[Sm2]
S.P. Smith, Some finite-dimensional algebras related to elliptic curves, Representation theory of algebras and related topics (Mexico City, 1994), 315-348, CMS Conf. Proc., 19, AMS, Providence, RI, 1996. MR 1388568 (97e:16053)

[SmS]
S. P. Smith and J. T. Stafford, Regularity of the four-dimensional Sklyanin algebra, Compositio Math. 83 (1992), no. 3, 259-289. MR 1175941 (93h:16037)

[StZ]
D.R. Stephenson and J.J. Zhang, Growth of graded Noetherian rings, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1593-1605. MR 1371143 (97g:16033)

[TV]
J. Tate and M. Van den Bergh, Homological properties of Sklyanin algebras, Invent. Math. 124 (1996), no. 1-3, 619-647. MR 1369430 (98c:16057)

[Zh]
J.J. Zhang, Connected graded Gorenstein algebras with enough normal elements, J. Algebra 189 (1997), no. 2, 390-405. MR 1438182 (97m:16018)

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Additional Information:

E. Kirkman
Affiliation: Department of Mathematics, Wake Forest University, P.O. Box 7388, Winston-Salem, North Carolina 27109
Email: kirkman@wfu.edu

J. Kuzmanovich
Affiliation: Department of Mathematics, Wake Forest University, P.O. Box 7388, Winston-Salem, North Carolina 27109
Email: kuz@wfu.edu

J. J. Zhang
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
Email: zhang@math.washington.edu

DOI: 10.1090/S0002-9947-08-04571-6
PII: S 0002-9947(08)04571-6
Keywords: Artin-Schelter regular algebra, group action, reflection, trace, Hilbert series, fixed subring, quantum polynomial rings
Received by editor(s): November 6, 2006
Posted: June 26, 2008
Copyright of article: Copyright 2008, American Mathematical Society




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