Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Block representation type of reduced enveloping algebras

Author(s): Iain Gordon; Alexander Premet
Journal: Trans. Amer. Math. Soc. 354 (2002), 1549-1581.
MSC (2000): Primary 20G05; Secondary 17B20
Posted: December 7, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: Let $K$ be an algebraically closed field of characteristic $p$, $G$ a connected, reductive $K$-group, $\mathfrak{g}=\text{Lie}(G)$, $\chi\in\mathfrak{g}^*$ and $U_\chi(\mathfrak{g})$ the reduced enveloping algebra of $\mathfrak{g}$ associated with $\chi$. Assume that $G^{(1)}$ is simply-connected, $p$ is good for $G$ and $\mathfrak{g}$ has a non-degenerate $G$-invariant bilinear form. All blocks of $U_\chi(\mathfrak{g})$ having finite and tame representation type are determined.

References:

1.
D.J. Benson.
Representations and Cohomology I.
Cambridge studies in advanced mathematics, number 30. Cambridge University Press, 1991. MR 92m:20005

2.
A. Borel.
Linear Algebraic Groups.
Number 126 in Graduate Texts in Mathematics. Springer-Verlag, second edition, 1991. MR 92d:20001

3.
K.A. Brown and I. Gordon.
The ramification of centres: Lie algebras in positive characteristic and quantised enveloping algebras.
To appear in Math. Zeit.

4.
W.W. Crawley-Boevey.
Tameness of biserial algebras.
Arch. Math. (Basel), 65:399-407, 1995. MR 96i:16021

5.
Yu. A. Drozd.
On representations of the Lie algebra $sl_2$.
Visn. Kiiv. Univ. Mat. i Mekh. 25:70-77, 1983 [Ukranian]. MR 86j:17010

6.
K. Erdmann.
Blocks of tame representation type and related algebras.
Number 1428 in Lecture Notes in Mathematics. Springer-Verlag, 1990. MR 91c:20016

7.
R. Farnsteiner.
Periodicity and representation type of modular Lie algebras.
J. reine angew. Math., 464:47-65, 1995. MR 97a:17019

8.
G. Fischer.
Dastellungtheorie des ersten Frobeniuskerns der $SL_2$.
PhD thesis, Universität Bielefeld, 1982.

9.
E.M. Friedlander and B.J. Parshall.
Geometry of $p$-unipotent Lie algebras.
J. Alg., 109:25-45, 1987. MR 89a:17017

10.
E.M. Friedlander and B.J. Parshall.
Rational actions associated to the adjoint representation.
Ann. scient. Éc. Norm. Sup., 20(4):215-226, 1987. MR 88k:14026

11.
E.M. Friedlander and B.J. Parshall.
Modular representation theory of Lie algebras.
Amer. J. Math., 110(6):1055-1093, 1988. MR 89j:17015

12.
E.M. Friedlander and B.J. Parshall.
Deformations of Lie algebra representations.
Amer. J. Math., 112(3):375-395, 1990. MR 91e:17012

13.
P. Gabriel.
Finite representation type is open.
In V. Dlab and P. Gabriel, editors, Representations of Algebras, number 488 in Springer Lecture Notes in Mathematics, pages 132-155, 1974. MR 51:12944

14.
C. Geiß.
On degenerations of tame and wild algebras.
Arch. Math., 64:11-16, 1995. MR 95k:16014

15.
I.M. Gelfand and V.A. Ponomarev.
Indecomposable representations of the Lorenz group.
Usp. Mat. Nauk, 232:3-60, 1968; English transl. in Russian Math Surveys 23 2:1-58. MR 37:5325

16.
I. Gordon.
Representations of semisimple Lie algebras in positive characteristic and quantum groups at roots of unity.
SFB preprint 00-007, Universität Bielefeld.

17.
R. Gordon and E.L. Green.
Graded Artin algebras.
J. Algebra, 76:111-137, 1982. MR 83m:16028

18.
J.E. Humphreys.
Reflection Groups and Coxeter Groups.
Number 29 in Cambridge studies in advanced mathematics. Cambridge University Press, 1990. MR 92h:20002

19.
J.E. Humphreys.
Conjugacy Classes in Semisimple Algebraic Groups, volume 43 of Math. Surveys Monographs.
Amer. Math. Soc., Providence, RI, 1995. MR 97i:200567

20.
L.E.P. Hupert.
Homological characteristics of pro-uniserial rings.
J. Algebra, 69:43-66, 1981. MR 83b:16018b

21.
J.C. Jantzen.
Kohomologie von $p$-Lie-Algebren und nilpotente Elemente.
Abh. Math. Sem. Univ. Hamburg, 56:191-219, 1986. MR 88e:17019

22.
J.C. Jantzen.
Representations of Algebraic Groups.
Academic Press, Boston, 1987. MR 89c:20001

23.
J.C. Jantzen.
Support varieties of Weyl modules.
Bull. London Math. Soc., 19:238-244, 1987. MR 88e:17008

24.
J.C. Jantzen.
Representations of Lie algebras in prime characteristic.
In Representation Theories and Algebraic Geometry, A. Broer, editor, Proceedings Montréal 1997 (NATO ASI series C 514), pages 185-235. Dordrecht etc, Kluwer, 1998. MR 99h:17026

25.
J. C. Jantzen.
Modular representations of reductive Lie algebras.
J. Pure Appl. Algebra 152: 133-185, 2000. MR 2001j:17016

26.
J.C. Jantzen.
Subregular nilpotent representations of $\mathfrak{sl}_n$ and $\mathfrak{so}_{2n+1}$.
Math. Proc. Camb. Phil. Soc., 126:223-257, 1999. MR 99k:17037

27.
H. Kraft.
Geometric methods in representation theory.
In Representations of Algebras, M. Auslander and E. Lluis, editors, number 944 in Springer Lecture Notes in Mathematics, pages 80-258, 1982. MR 84c:14007

28.
M. Meyer-ter-Vehn.
Köcher von reduzierten Einhüllenden.
Diplomarbeit, Universität Freiburg, 2000.
29.
A.A. Mil'ner.
The maximal degree of irreducible representations of a Lie algebra over a field of positive characteristic.
Funkt. Anal. i Prilozen., 14, 1980 [Russian]; English transl. in Funct. Anal. Appl., 14:136-137, 1980. MR 81h:17012

30.
I. Mirkovic and D. Rumynin.
Centers of reduced enveloping algebras.
Math. Zeit., 231:123-132, 1999. MR 2001i:17032

31.
D.K. Nakano, B.J. Parshall and D.C.Vella.
Support varieties for algebraic groups. Preprint, 2000.

32.
D. K. Nakano and R. D. Pollack.
Blocks of finite type in reduced enveloping algebras for classical Lie algebras.
J. London Math. Soc (2) 61: 374-394, 2000. MR 2001f:17040

33.
R.D. Pollack
Restricted Lie algebras of bounded type.
Bull. Amer. Math. Soc. 74(2): 326-331, 1968. MR 36:2661

34.
A. Premet.
The Green ring of the simple three-dimensional Lie $p$-algebra.
Soviet Math. (Iz. VUZ), 35(10):51-60, 1991. MR 93i:17007

35.
A. Premet.
An analogue of the Jacobson-Morozov theorem for Lie algebras of reductive groups of good characteristics.
Trans. Amer. Math. Soc., 347(8):2961-2988, 1995. MR 95k:17012

36.
A. Premet.
Irreducible representations of Lie algebras of reductive groups and the Kac-Weisfeiler conjecture.
Invent. Math., 121(1):79-117, 1995. MR 96g:17007

37.
A. Premet.
Support varieties of non-restricted modules over Lie algebras of reductive groups.
J. London Math. Soc., 55(2):236-250, 1997. MR 98a:17076

38.
A. Premet.
Complexity of Lie algebra representations and nilpotent elements of the stabilizers of linear forms.
Math. Zeit., 228(2):225-282, 1998. MR 99i:17023

39.
J. Rickard.
The representation type of self-injective algebras.
Bull. London Math. Soc., 22:540-546, 1990. MR 92f:16015

40.
C.M. Ringel.
The representation type of local algebras.
In V. Dlab and P. Gabriel, editors, Representations of Algebras, number 488 in Springer Lecture Notes in Mathematics, pages 282-305, 1974. MR 52:3241

41.
C.M. Ringel.
The indecomposable representations of the dihedral $2$-groups.
Math. Ann. 214:19-34, 1975. MR 51:680

42.
A.N. Rudakov.
Reducible $p$-representations of a simple three-dimensional Lie $p$-algebra.
Vestnik Moskov. Univ. Ser. I Mat. Mekh. 6: 45-49, 1982 [Russian]; English transl. in Moscow Univ. Math. Bull. 37(6):51-56, 1982. MR 84g:17016

43.
J. R. Schue.
Symmetry for the enveloping algebra of a restricted Lie algebra.
Proc. Amer. Math. Soc., 16:1123-1124, 1965. MR 32:2515

44.
T. A. Springer and R. Steinberg.
Conjugacy classes.
In Seminar on Algebraic Groups and Related Finite Groups, volume 131 of Lecture Notes in Mathematics, pages 167-276. Springer-Verlag, 1970. MR 42:3091

45.
T.A. Springer.
Some arithmetical results on semi-simple Lie algebras.
Publ. I.H.E.S., 30:115-141, 1966. MR 34:5993

46.
B. Ju. Veisfeiler and V.G. Kac.
The irreducible representations of Lie $p$-algebras.
Funkt. Anal. i Prilozen., 5(2):28-36, 1971 [Russian]; English transl. in Funct. Anal. Appl. 5:111-117, 1971. MR 44:2793

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20G05, 17B20

Retrieve articles in all Journals with MSC (2000): 20G05, 17B20

Additional Information:

Iain Gordon
Affiliation: Department of Mathematics, University of Glasgow, Glasgow G12 8QW, Scotland
Email: ig@maths.gla.ac.uk

Alexander Premet
Affiliation: Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, England
Email: sashap@ma.man.ac.uk

DOI: 10.1090/S0002-9947-01-02826-4
PII: S 0002-9947(01)02826-4
Received by editor(s): July 24, 2000
Received by editor(s) in revised form: January 2, 2001
Posted: December 7, 2001
Additional Notes: The authors would like to thank the London Mathematical Society for supporting a visit of the first author to Manchester through a travel grant scheme. Further financial support for the first author was provided by TMR grant ERB FMRX-CT97-0100 at the University of Bielefeld.
Copyright of article: Copyright 2001, American Mathematical Society

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google