The nucleus for restricted Lie algebras

Benson, David; Nakano, Daniel

doi:10.1090/S0002-9939-03-06939-9

The nucleus for restricted Lie algebras
HTML articles powered by AMS MathViewer

by David J. Benson and Daniel K. Nakano PDF

Proc. Amer. Math. Soc. 131 (2003), 3395-3405 Request permission

Abstract:

The nucleus was a concept first developed in the cohomology theory for finite groups. In this paper the authors investigate the nucleus for restricted Lie algebras. The nucleus is explicitly described for several important classes of Lie algebras.

References

Christopher P. Bendel and Daniel K. Nakano, Complexes and vanishing of cohomology for group schemes, J. Algebra 214 (1999), no. 2, 668–713. MR 1680605, DOI 10.1006/jabr.1998.7711
D. J. Benson, Representations and cohomology. II, Cambridge Studies in Advanced Mathematics, vol. 31, Cambridge University Press, Cambridge, 1991. Cohomology of groups and modules. MR 1156302
D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1994/95), 196–205, electronic. MR 1362976
D. J. Benson, The nucleus, and extensions between modules for a finite group, Representations of Algebras, Proceedings of the Ninth International Conference (Beijing 2000), Beijing Normal University Press, 2002.
D. J. Benson, Jon F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules. II, Math. Proc. Cambridge Philos. Soc. 120 (1996), no. 4, 597–615. MR 1401950, DOI 10.1017/S0305004100001584
D. J. Benson, Jon F. Carlson, and Jeremy Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), no. 1, 59–80. MR 1450996, DOI 10.4064/fm-153-1-59-80
D. J. Benson, J. F. Carlson, and G. R. Robinson, On the vanishing of group cohomology, J. Algebra 131 (1990), no. 1, 40–73. MR 1054998, DOI 10.1016/0021-8693(90)90165-K
Jon F. Carlson, Varieties for cohomology with twisted coefficients, Acta Math. Sin. (Engl. Ser.) 15 (1999), no. 1, 81–92. MR 1701134, DOI 10.1007/s10114-999-0061-9
Jon F. Carlson, The thick subcategory generated by the trivial module, Infinite length modules (Bielefeld, 1998) Trends Math., Birkhäuser, Basel, 2000, pp. 285–296. MR 1789221
Jon F. Carlson and Geoffrey R. Robinson, Varieties and modules with vanishing cohomology, Math. Proc. Cambridge Philos. Soc. 116 (1994), no. 2, 245–251. MR 1281544, DOI 10.1017/S0305004100072558
David H. Collingwood and William M. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Mathematics Series, Van Nostrand Reinhold Co., New York, 1993. MR 1251060
Jon F. Carlson, Daniel K. Nakano, and Karl M. Peters, On the vanishing of extensions of modules over reduced enveloping algebras, Math. Ann. 302 (1995), no. 3, 541–560. MR 1339926, DOI 10.1007/BF01444507
Eric M. Friedlander and Brian J. Parshall, Support varieties for restricted Lie algebras, Invent. Math. 86 (1986), no. 3, 553–562. MR 860682, DOI 10.1007/BF01389268
Eric M. Friedlander and Brian J. Parshall, Geometry of $p$-unipotent Lie algebras, J. Algebra 109 (1987), no. 1, 25–45. MR 898334, DOI 10.1016/0021-8693(87)90161-X
Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270. MR 1427618, DOI 10.1007/s002220050119
James E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs, vol. 43, American Mathematical Society, Providence, RI, 1995. MR 1343976, DOI 10.1090/surv/043
Jens Carsten Jantzen, Representations of algebraic groups, Pure and Applied Mathematics, vol. 131, Academic Press, Inc., Boston, MA, 1987. MR 899071
Daniel K. Nakano, Brian J. Parshall, and David C. Vella, Support varieties for algebraic groups, J. Reine Angew. Math. 547 (2002), 15–49. MR 1900135, DOI 10.1515/crll.2002.049
Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI 10.2307/1970770
Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270. MR 1427618, DOI 10.1007/s002220050119
Eric M. Friedlander and Andrei Suslin, Cohomology of finite group schemes over a field, Invent. Math. 127 (1997), no. 2, 209–270. MR 1427618, DOI 10.1007/s002220050119