Varieties of elementary abelian Lie algebras and degrees of modules

Chang, Hao; Farnsteiner, Rolf

doi:10.1090/ert/559

Previous volume | This volume | All volumes | Previous article | Next article

Varieties of elementary abelian Lie algebras and degrees of modules

Authors: Hao Chang and Rolf Farnsteiner
Journal: Represent. Theory 25 (2021), 90-141
MSC (2020): Primary 17B50, 16G10
DOI: https://doi.org/10.1090/ert/559
Published electronically: February 11, 2021
MathSciNet review: 4214336
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $(\mathfrak {g},[p])$ be a restricted Lie algebra over an algebraically closed field $k$ of characteristic $p\!\ge \!3$. Motivated by the behavior of geometric invariants of the so-called $(\mathfrak {g},[p])$-modules of constant $j$-rank ($j \in \{1,\ldots ,p\!-\!1\}$), we study the projective variety $\mathbb {E}(2,\mathfrak {g})$ of two-dimensional elementary abelian subalgebras. If $p\!\ge \!5$, then the topological space $\mathbb {E}(2,\mathfrak {g}/C(\mathfrak {g}))$, associated to the factor algebra of $\mathfrak {g}$ by its center $C(\mathfrak {g})$, is shown to be connected. We give applications concerning categories of $(\mathfrak {g},[p])$-modules of constant $j$-rank and certain invariants, called $j$-degrees.

References

Maurice Auslander, Idun Reiten, and Sverre O. Smalø, Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics, vol. 36, Cambridge University Press, Cambridge, 1997. Corrected reprint of the 1995 original. MR 1476671
David J. Benson, Representations of elementary abelian $p$-groups and vector bundles, Cambridge Tracts in Mathematics, vol. 208, Cambridge University Press, Cambridge, 2017. MR 3585474
Daniel Bissinger, Representations of constant socle rank for the Kronecker algebra, Forum Math. 32 (2020), no. 1, 23–43. MR 4048452, DOI https://doi.org/10.1515/forum-2018-0143
Brian D. Boe, Daniel K. Nakano, and Emilie Wiesner, $\rm Ext^1$-quivers for the Witt algebra $W(1,1)$, J. Algebra 322 (2009), no. 5, 1548–1564. MR 2543622, DOI https://doi.org/10.1016/j.jalgebra.2009.05.043
Jean-Marie Bois, Rolf Farnsteiner, and Bin Shu, Weyl groups for non-classical restricted Lie algebras and the Chevalley restriction theorem, Forum Math. 26 (2014), no. 5, 1333–1379. MR 3334032, DOI https://doi.org/10.1515/forum-2011-0145
Vitalij M. Bondarenko and Iryna V. Lytvynchuk, The representation type of elementary abelian $p$-groups with respect to the modules of constant Jordan type, Algebra Discrete Math. 14 (2012), no. 1, 29–36. MR 3052319
Jon F. Carlson, The variety of an indecomposable module is connected, Invent. Math. 77 (1984), no. 2, 291–299. MR 752822, DOI https://doi.org/10.1007/BF01388448
Jon F. Carlson, Eric M. Friedlander, and Julia Pevtsova, Modules of constant Jordan type, J. Reine Angew. Math. 614 (2008), 191–234. MR 2376286, DOI https://doi.org/10.1515/CRELLE.2008.006
Jon F. Carlson, Eric M. Friedlander, and Julia Pevtsova, Elementary subalgebras of Lie algebras, J. Algebra 442 (2015), 155–189. MR 3395058, DOI https://doi.org/10.1016/j.jalgebra.2014.10.015
Jon F. Carlson, Eric M. Friedlander, and Andrei Suslin, Modules for $\Bbb Z/p\times \Bbb Z/p$, Comment. Math. Helv. 86 (2011), no. 3, 609–657. MR 2803855, DOI https://doi.org/10.4171/CMH/236
Roger W. Carter, Finite groups of Lie type, Wiley Classics Library, John Wiley & Sons, Ltd., Chichester, 1993. Conjugacy classes and complex characters; Reprint of the 1985 original; A Wiley-Interscience Publication. MR 1266626

H. Chang, Varieties of elementary Lie algebras, Ph.-D. Thesis, University of Kiel, Kiel, Germany, 2016.

Hao Chang and Rolf Farnsteiner, Finite group schemes of $p$-rank $\leqslant 1$, Math. Proc. Cambridge Philos. Soc. 166 (2019), no. 2, 297–323. MR 3903120, DOI https://doi.org/10.1017/S0305004117000834

Ho-Jui Chang, Über Wittsche Lie-Ringe, Abh. Math. Sem. Hansischen Univ. 14 (1941), 151–184 (German). MR 5100, DOI https://doi.org/10.1007/BF02940743

M. Demazure and P. Gabriel, Groupes Algébriques, Masson, Paris/North Holland, Amsterdam, 1970.

S. P. Demuškin, Cartan subalgebras of the simple Lie $p$-algebras $W_{n}$ and $S_{n}$, Sibirsk. Mat. Ž. 11 (1970), 310–325 (Russian). MR 0262310

S. P. Demuškin, Cartan subalgebras of simple non-classical Lie $p$-algebras, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 915–932 (Russian). MR 0327854

Karin Erdmann, Blocks of tame representation type and related algebras, Lecture Notes in Mathematics, vol. 1428, Springer-Verlag, Berlin, 1990. MR 1064107

Rolf Farnsteiner, Derivations and central extensions of finitely generated graded Lie algebras, J. Algebra 118 (1988), no. 1, 33–45. MR 961324, DOI https://doi.org/10.1016/0021-8693%2888%2990046-4

Rolf Farnsteiner, Varieties of tori and Cartan subalgebras of restricted Lie algebras, Trans. Amer. Math. Soc. 356 (2004), no. 10, 4181–4236. MR 2058843, DOI https://doi.org/10.1090/S0002-9947-04-03476-2

Rolf Farnsteiner, Tameness and complexity of finite group schemes, Bull. Lond. Math. Soc. 39 (2007), no. 1, 63–70. MR 2303520, DOI https://doi.org/10.1112/blms/bdl006

Rolf Farnsteiner, Jordan types for indecomposable modules of finite group schemes, J. Eur. Math. Soc. (JEMS) 16 (2014), no. 5, 925–989. MR 3210958, DOI https://doi.org/10.4171/JEMS/452

Rolf Farnsteiner, Representations of finite group schemes and morphisms of projective varieties, Proc. Lond. Math. Soc. (3) 114 (2017), no. 3, 433–475. MR 3653236, DOI https://doi.org/10.1112/plms.12010

Rolf Farnsteiner and Andrzej Skowroński, Classification of restricted Lie algebras with tame principal block, J. Reine Angew. Math. 546 (2002), 1–45. MR 1900992, DOI https://doi.org/10.1515/crll.2002.043

Eric M. Friedlander and Julia Pevtsova, Generalized support varieties for finite group schemes, Doc. Math. Extra vol.: Andrei A. Suslin sixtieth birthday (2010), 197–222. MR 2804254

Meinolf Geck, An introduction to algebraic geometry and algebraic groups, Oxford Graduate Texts in Mathematics, vol. 10, Oxford University Press, Oxford, 2003. MR 2032320

Ulrich Görtz and Torsten Wedhorn, Algebraic geometry I, Advanced Lectures in Mathematics, Vieweg + Teubner, Wiesbaden, 2010. Schemes with examples and exercises. MR 2675155

Sebastian Herpel and David I. Stewart, On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations, Doc. Math. 21 (2016), 1–37. MR 3465106

Sebastian Herpel and David I. Stewart, Maximal subalgebras of Cartan type in the exceptional Lie algebras, Selecta Math. (N.S.) 22 (2016), no. 2, 765–799. MR 3477335, DOI https://doi.org/10.1007/s00029-015-0199-5

G. Hochschild, Cohomology of restricted Lie algebras, Amer. J. Math. 76 (1954), 555–580. MR 63361, DOI https://doi.org/10.2307/2372701

Randall R. Holmes, Simple restricted modules for the restricted Hamiltonian algebra, J. Algebra 199 (1998), no. 1, 229–261. MR 1489362, DOI https://doi.org/10.1006/jabr.1997.7172

James E. Humphreys, Algebraic groups and modular Lie algebras, Memoirs of the American Mathematical Society, No. 71, American Mathematical Society, Providence, R.I., 1967. MR 0217075

J. E. Humphreys, Projective modules for ${\rm SL}(2,$ $q)$, J. Algebra 25 (1973), 513–518. MR 399241, DOI https://doi.org/10.1016/0021-8693%2873%2990097-5

J. E. Humphreys, Symmetry for finite dimensional Hopf algebras, Proc. Amer. Math. Soc. 68 (1978), 143–148.

James E. Humphreys, Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975. Graduate Texts in Mathematics, No. 21. MR 0396773

J. E. Humpphreys, Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics 29. Cambridge University Press, 1990.

Jens Carsten Jantzen, Modular representations of reductive Lie algebras, J. Pure Appl. Algebra 152 (2000), no. 1-3, 133–185. Commutative algebra, homological algebra and representation theory (Catania/Genoa/Rome, 1998). MR 1783993, DOI https://doi.org/10.1016/S0022-4049%2899%2900142-5

Jens Carsten Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, American Mathematical Society, Providence, RI, 2003. MR 2015057

Jens Carsten Jantzen, Nilpotent orbits in representation theory, Lie theory, Progr. Math., vol. 228, Birkhäuser Boston, Boston, MA, 2004, pp. 1–211. MR 2042689

A. I. Kostrikin, Strong degeneracy of simple Lie $p$-algebras, Dokl. Akad. Nauk SSSR 150 (1963), 248–250. MR 0148718

Paul Levy, George McNinch, and Donna M. Testerman, Nilpotent subalgebras of semisimple Lie algebras, C. R. Math. Acad. Sci. Paris 347 (2009), no. 9-10, 477–482 (English, with English and French summaries). MR 2576893, DOI https://doi.org/10.1016/j.crma.2009.03.015

Gunter Malle and Donna Testerman, Linear algebraic groups and finite groups of Lie type, Cambridge Studies in Advanced Mathematics, vol. 133, Cambridge University Press, Cambridge, 2011. MR 2850737

J. S. Milne, Algebraic groups, Cambridge Studies in Advanced Mathematics, vol. 170, Cambridge University Press, Cambridge, 2017. The theory of group schemes of finite type over a field. MR 3729270

Yang Pan, Varieties of elementary subalgebras of submaximal rank in type A, J. Lie Theory 28 (2018), no. 1, 57–70. MR 3683371

Julia Pevtsova and Jim Stark, Varieties of elementary subalgebras of maximal dimension for modular Lie algebras, Geometric and topological aspects of the representation theory of finite groups, Springer Proc. Math. Stat., vol. 242, Springer, Cham, 2018, pp. 339–375. MR 3901167, DOI https://doi.org/10.1007/978-3-319-94033-5_14

A. A. Premet, Inner ideals in modular Lie algebras, Vestsī Akad. Navuk BSSR Ser. Fīz.-Mat. Navuk 5 (1986), 11–15, 123 (Russian, with English summary). MR 876665

A. A. Premet, Lie algebras without strong degeneration, Mat. Sb. (N.S.) 129(171) (1986), no. 1, 140–153 (Russian); English transl., Math. USSR-Sb. 57 (1987), no. 1, 151–164. MR 830100, DOI https://doi.org/10.1070/SM1987v057n01ABEH003060

A. A. Premet, Absolute zero divisors in Lie algebras over a perfect field, Dokl. Akad. Nauk BSSR 31 (1987), no. 10, 869–872, 956 (Russian, with English summary). MR 920913

Alexander Premet, Nilpotent commuting varieties of reductive Lie algebras, Invent. Math. 154 (2003), no. 3, 653–683. MR 2018787, DOI https://doi.org/10.1007/s00222-003-0315-6

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 https://doi.org/10.2307/1970770

Khalid Rian, Extensions of the Witt algebra and applications, J. Algebra Appl. 10 (2011), no. 6, 1233–1259. MR 2864573, DOI https://doi.org/10.1142/S0219498811005154

G. Seligman, The complete reducibility of certain modules, Mimeographed, Yale University, 1961.

G. B. Seligman, Modular Lie algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 40, Springer-Verlag New York, Inc., New York, 1967. MR 0245627

Daniel Simson, On representation types of module subcategories and orders, Bull. Polish Acad. Sci. Math. 41 (1993), no. 2, 77–93 (1994). MR 1414754

Serge Skryabin, Tori in the Melikian algebra, J. Algebra 243 (2001), no. 1, 69–95. MR 1851654, DOI https://doi.org/10.1006/jabr.2001.8849

T. A. Springer, Linear algebraic groups, Progress in Mathematics, vol. 9, Birkhäuser, Boston, Mass., 1981. MR 632835

Robert Steinberg, Torsion in reductive groups, Advances in Math. 15 (1975), 63–92. MR 354892, DOI https://doi.org/10.1016/0001-8708%2875%2990125-5

Helmut Strade, Simple Lie algebras over fields of positive characteristic. I, De Gruyter Expositions in Mathematics, vol. 38, Walter de Gruyter & Co., Berlin, 2004. Structure theory. MR 2059133

Helmut Strade and Rolf Farnsteiner, Modular Lie algebras and their representations, Monographs and Textbooks in Pure and Applied Mathematics, vol. 116, Marcel Dekker, Inc., New York, 1988. MR 929682

Andrei Suslin, Eric M. Friedlander, and Christopher P. Bendel, Infinitesimal $1$-parameter subgroups and cohomology, J. Amer. Math. Soc. 10 (1997), no. 3, 693–728. MR 1443546, DOI https://doi.org/10.1090/S0894-0347-97-00240-3

Andrei Suslin, Eric M. Friedlander, and Christopher P. Bendel, Support varieties for infinitesimal group schemes, J. Amer. Math. Soc. 10 (1997), no. 3, 729–759. MR 1443547, DOI https://doi.org/10.1090/S0894-0347-97-00239-7

Jared Warner, Rational points and orbits on the variety of elementary subalgebras, J. Pure Appl. Algebra 219 (2015), no. 8, 3355–3371. MR 3320224, DOI https://doi.org/10.1016/j.jpaa.2014.10.019