## Algebra & Number Theory

### Vanishing of trace forms in low characteristics

Skip Garibaldi

#### Abstract

Every finite-dimensional representation of an algebraic group $G$ gives a trace symmetric bilinear form on the Lie algebra of $G$. We give criteria in terms of root system data for the existence of a representation such that this form is nonzero or nondegenerate. As a corollary, we show that a Lie algebra of type $E8$ over a field of characteristic 5 does not have a “quotient trace form”, answering a question posed in the 1960s.

#### Article information

Source
Algebra Number Theory, Volume 3, Number 5 (2009), 543-566.

Dates
Revised: 5 March 2009
Accepted: 6 April 2009
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.ant/1513797448

Digital Object Identifier
doi:10.2140/ant.2009.3.543

Mathematical Reviews number (MathSciNet)
MR2578888

Zentralblatt MATH identifier
1282.20052

Subjects
Primary: 20G05: Representation theory
Secondary: 17B50: Modular Lie (super)algebras 17B25: Exceptional (super)algebras

#### Citation

Garibaldi, Skip. Vanishing of trace forms in low characteristics. Algebra Number Theory 3 (2009), no. 5, 543--566. doi:10.2140/ant.2009.3.543. https://projecteuclid.org/euclid.ant/1513797448

#### References

• P. Bardsley and R. W. Richardson, “Étale slices for algebraic transformation groups in characteristic $p$”, Proc. London Math. Soc. $(3)$ 51:2 (1985), 295–317.
• R. Block, “Trace forms on Lie algebras”, Canad. J. Math. 14 (1962), 553–564.
• R. E. Block and H. Zassenhaus, “The Lie algebras with a nondegenerate trace form”, Illinois J. Math. 8 (1964), 543–549.
• N. Bourbaki, Lie groups and Lie algebras. Chapters 4–6, Springer, Berlin, 2002.
• R. W. Carter, Finite groups of Lie type. Conjugacy classes and complex characters, Wiley, New York, 1985.
• C. W. Curtis, “Representations of Lie algebras of classical type with applications to linear groups”, J. Math. Mech. 9 (1960), 307–326.
• M. Demazure and P. Gabriel, Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs, Masson & Cie, Éditeur, Paris, 1970. Avec un appendice Corps de classes local par Michiel Hazewinkel.
• E. B. Dynkin, “Semisimple subalgebras of semisimple Lie algebras”, Mat. Sbornik N.S. 30(72) (1952), 349–462 (3 plates).
• S. Garibaldi, “Orthogonal representations of twisted forms of $\rm SL\sb 2$”, Represent. Theory 12 (2008), 435–446.
• B. H. Gross and G. Nebe, “Globally maximal arithmetic groups”, J. Algebra 272:2 (2004), 625–642.
• G. Hiss, “Die adjungierten Darstellungen der Chevalley-Gruppen”, Arch. Math. $($Basel$)$ 42:5 (1984), 408–416.
• G. M. D. Hogeweij, “Almost-classical Lie algebras. I, II”, Nederl. Akad. Wetensch. Indag. Math. 44:4 (1982), 441–452, 453–460.
• J. E. Humphreys, Conjugacy classes in semisimple algebraic groups, Mathematical Surveys and Monographs 43, American Mathematical Society, Providence, RI, 1995.
• J. C. Jantzen, Representations of algebraic groups, 2nd ed., Mathematical Surveys and Monographs 107, American Mathematical Society, Providence, RI, 2003.
• J. C. Jantzen, “Nilpotent orbits in representation theory”, pp. 1–211 in Lie theory, edited by J.-P. Anker and B. Orsted, Progr. Math. 228, Birkhäuser, Boston, 2004.
• Y. Laszlo and C. Sorger, “The line bundles on the moduli of parabolic $G$-bundles over curves and their sections”, Ann. Sci. École Norm. Sup. $(4)$ 30:4 (1997), 499–525.
• O. Mathieu, “Classification des algèbres de Lie simples”, pp. 245–286 in Séminaire Bourbaki, 1998/99, Astérisque 266, Soc. Math. de France, Paris, 2000.
• W. G. McKay, J. Patera, and D. W. Rand, Tables of representations of simple Lie algebras, I: Exceptional simple Lie algebras, Université de Montréal Centre de Recherches Mathématiques, Montreal, 1990.
• A. Merkurjev, “Rost invariants of simply connected algebraic groups”, pp. 101–158 in Cohomological invariants in Galois cohomology, Univ. Lecture Ser. 28, Amer. Math. Soc., Providence, RI, 2003.
• A. Premet, “Irreducible representations of Lie algebras of reductive groups and the Kac–Weisfeiler conjecture”, Invent. Math. 121:1 (1995), 79–117.
• R. W. Richardson, Jr., “Conjugacy classes in Lie algebras and algebraic groups”, Ann. of Math. $(2)$ 86 (1967), 1–15.
• G. B. Seligman, Modular Lie algebras, Ergebnisse der Math. 40, Springer, 1967.
• T. A. Springer and R. Steinberg, “Conjugacy classes”, pp. 167–266 in Seminar on algebraic groups and related finite groups (Princeton, NJ, 1968/69), Lecture Notes in Mathematics 131, Springer, Berlin, 1970.
• R. Steinberg, “Automorphisms of classical Lie algebras”, Pacific J. Math. 11 (1961), 1119–1129. Reprinted as pp. 101–111 in his Collected Papers, American Math. Soc., Providence, 1997.
• R. Steinberg, “Générateurs, relations et revêtements de groupes algébriques”, pp. 113–127 in Colloque sur la Théorie des Groupes Algébriques (Bruxelles), Librairie Universitaire, Louvain, 1962. Reprinted as pp. 113–127 in his Collected Papers, American Math. Soc., Providence, 1997.
• R. Steinberg, “Representations of algebraic groups”, Nagoya Math. J. 22 (1963), 33–56. Reprinted as pp. 149–172 in his Collected Papers, American Math. Soc., Providence, 1997.
• R. Steinberg, Lectures on Chevalley groups, Yale University, New Haven, 1968.
• R. Steinberg, “Torsion in reductive groups”, Advances in Math. 15 (1975), 63–92. Reprinted as pp. 415-444 in his Collected Papers, American Math. Soc., Providence, 1997.
• H. Strade, Simple Lie algebras over fields of positive characteristic, I: Structure theory, Expositions in Mathematics 38, de Gruyter, Berlin, 2004.
• J. Tits, “Classification of algebraic semisimple groups”, pp. 33–62 in Algebraic groups and discontinuous subgroups (Boulder, CO, 1965), Amer. Math. Soc., Providence, R.I., 1966, 1966.