Algebra & Number Theory

Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras

Dmitri I. Panyushev

Full-text: Open access

Abstract

Let σ1 and σ2 be commuting involutions of a connected reductive algebraic group G with g=Lie(G). Let

g = i , j = 0 , 1 g @ i j

be the corresponding 2×2-grading. If {α,β,γ}={01,10,11}, then [,] maps g@α×gβ into gγ, and the zero fiber of this bracket is called a σ-commuting variety. The commuting variety of g and commuting varieties related to one involution are particular cases of this construction. We develop a general theory of such varieties and point out some cases, when they have especially good properties. If GGσ1 is a Hermitian symmetric space of tube type, then one can find three conjugate pairwise commuting involutions σ1, σ2, and σ3=σ1σ2. In this case, any σ-commuting variety is isomorphic to the commuting variety of the simple Jordan algebra associated with σ1. As an application, we show that if J is the Jordan algebra of symmetric matrices, then the product map J×JJ is equidimensional, while for all other simple Jordan algebras equidimensionality fails.

Article information

Source
Algebra Number Theory, Volume 7, Number 6 (2013), 1505-1534.

Dates
Received: 19 September 2012
Accepted: 24 January 2013
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.ant/1513730036

Digital Object Identifier
doi:10.2140/ant.2013.7.1505

Mathematical Reviews number (MathSciNet)
MR3107571

Zentralblatt MATH identifier
1304.14061

Subjects
Primary: 14L30: Group actions on varieties or schemes (quotients) [See also 13A50, 14L24, 14M17]
Secondary: 17B08: Coadjoint orbits; nilpotent varieties 17B40: Automorphisms, derivations, other operators 17C20: Simple, semisimple algebras 22E46: Semisimple Lie groups and their representations

Keywords
semisimple Lie algebra commuting variety Cartan subspace quaternionic decomposition nilpotent orbit Jordan algebra

Citation

Panyushev, Dmitri I. Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras. Algebra Number Theory 7 (2013), no. 6, 1505--1534. doi:10.2140/ant.2013.7.1505. https://projecteuclid.org/euclid.ant/1513730036


Export citation

References

  • L. V. Antonyan, “\cyr O klassifikacii odnorodnykh e1lementov $\Z_2$-graduirovannykh poluprostykh algebr Li”, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 2 (1982), 29–34. Translated as “Classification of homogeneous elements of $\Z_2$-graded semisimple Lie algebras” in Moscow Univ. Math. Bulletin, 37:2 (1982), 36–43.
  • M. Brion, “Invariants et covariants des groupes algébriques réductifs”, pp. 83–168 in Théorie des invariants et géométrie des variétés quotients (Monastir, 1996), edited by G. W. Schwarz and M. Brion, Travaux en Cours 61, Hermann, Paris, 2000.
  • M. Bulois, “Irregular locus of the commuting variety of reductive symmetric Lie algebras and rigid pairs”, Transform. Groups 16:4 (2011), 1027–1061.
  • E. B. Dynkin, “\cyr Poluprostye podalgebry poluprostykh algebr Li”, Mat. Sbornik $($N.S.$)$ 30(72) (1952), 349–462. Translated as “Semisimple subalgebras of semisimple Lie algebras”, pp. 111–244 in Five papers on algebra and group theory by E. B. Dynkin et al., Amer. Math. Soc. Transl. $(2)$ 6, Amer. Math. Soc., Providence, RI, 1957 $($see also [pp. 175–308]?$)$.
  • E. B. Dynkin, Selected papers of E. B. Dynkin with commentary, edited by A. A. Yushkevich et al., Amer. Math. Soc., Providence, RI, 2000.
  • S. Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics 80, Academic Press, New York, 1978.
  • W. H. Hesselink, “Singularities in the nilpotent scheme of a classical group”, Trans. Amer. Math. Soc. 222 (1976), 1–32.
  • V. G. Kac, “Some remarks on nilpotent orbits”, J. Algebra 64:1 (1980), 190–213.
  • I. L. Kantor, “\cyr Klassifikatsiya neprivodimykh tranzitivno-differentsial'nykh grupp”, Dokl. Akad. Nauk SSSR 158:6 (1964), 1271–1274. Translated as “Classification of irreducible transitively differential groups” in Sov. Math. Dokl. 5 (1965), 1404–1407.
  • A. Kollross, “Exceptional $\Z_2\times \Z_2$-symmetric spaces”, Pacific J. Math. 242:1 (2009), 113–130.
  • B. Kostant and S. Rallis, “Orbits and representations associated with symmetric spaces”, Amer. J. Math. 93 (1971), 753–809.
  • H. Kraft and C. Procesi, “On the geometry of conjugacy classes in classical groups”, Comment. Math. Helv. 57:4 (1982), 539–602.
  • D. I. Panyushev, “Complexity and nilpotent orbits”, Manuscripta Math. 83:3-4 (1994), 223–237.
  • D. I. Panyushev, “The Jacobian modules of a representation of a Lie algebra and geometry of commuting varieties”, Compositio Math. 94:2 (1994), 181–199.
  • D. I. Panyushev, “On the conormal bundle of a $G$-stable subvariety”, Manuscripta Math. 99:2 (1999), 185–202.
  • D. I. Panyushev, “Commuting involutions and degenerations of isotropy representations”, Transform. Groups 18:2 (2013), 507–537.
  • D. I. Panyushev and O. Yakimova, “Symmetric pairs and associated commuting varieties”, Math. Proc. Cambridge Philos. Soc. 143:2 (2007), 307–321.
  • A. Premet, “Nilpotent commuting varieties of reductive Lie algebras”, Invent. Math. 154:3 (2003), 653–683.
  • R. W. Richardson, “Commuting varieties of semisimple Lie algebras and algebraic groups”, Compositio Math. 38:3 (1979), 311–327.
  • R. W. Richardson, “An application of the Serre conjecture to semisimple algebraic groups”, pp. 141–151 in Algebra, Carbondale 1980: Lie algebras, group theory, and partially ordered algebraic structures (Carbondale, IL, 1980), edited by R. K. Amayo, Lecture Notes in Math. 848, Springer, Berlin, 1981.
  • R. W. Richardson, “On orbits of algebraic groups and Lie groups”, Bull. Austral. Math. Soc. 25:1 (1982), 1–28.
  • R. W. Richardson, “Normality of $G$-stable subvarieties of a semisimple Lie algebra”, pp. 243–264 in Algebraic groups (Utrecht, 1986), edited by A. M. Cohen et al., Lecture Notes in Math. 1271, Springer, Berlin, 1987.
  • J. Tits, “Une classe d'algèbres de Lie en rélation avec les algèbres de Jordan”, Nederl. Akad. Wetensch. Proc. $($A$)$ 65 (1962), 530–535.
  • W. V. Vasconcelos, Arithmetic of blowup algebras, London Mathematical Society Lecture Note Series 195, Cambridge University Press, 1994.
  • M. Vergne, “Instantons et correspondance de Kostant–Sekiguchi”, C. R. Acad. Sci. Paris $($I$)$ Math. 320:8 (1995), 901–906.
  • È. B. Vinberg, “\cyr Gruppa Veĭlya graduirovannoĭ algebry Li”, Izv. Akad. Nauk SSSR Ser. Mat. 40:3 (1976), 488–526. Translated as in Math. USSR-Izv. 10 (1976), 463–495.
  • È. B. Vinberg, “\cyr Klassifikatsiya odnorodnykh nil'potentnykh e1lementov poluprostoĭ graduirovannoĭ algebry Li”, Trudy Sem. Vektor. Tenzor. Anal. 19 (1979), 155–177. Translated as “Classification of homogeneous nilpotent elements of a semisimple graded Lie algebra” in Selecta Math. Sov. 6 (1987), 15–35.
  • È. B. Vinberg, “Short ${\rm SO}_3$-structures on simple Lie algebras and associated quasielliptic planes”, pp. 243–270 in Lie groups and invariant theory, edited by È. B. Vinberg, Amer. Math. Soc. Transl. $(2)$ 213, Amer. Math. Soc., Providence, RI, 2005.