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Vanishing of trace forms in low characteristic, Algebra
 Number Theory
"... Abstract. A finitedimensional representation of an algebraic group G gives a trace symmetric bilinear form on the Lie algebra of G. We give a criterion in terms of the root system data for this form to vanish. As a corollary, we show that a Lie algebra of type E8 over a field of characteristic 5 do ..."
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Abstract. A finitedimensional representation of an algebraic group G gives a trace symmetric bilinear form on the Lie algebra of G. We give a criterion in terms of the root system data for this form to vanish. As a corollary, we show that a Lie algebra of type E8 over a field of characteristic 5 does not have a socalled “quotient trace form”, answering a question posed in the 1960s. Let G be an algebraic group over a field F, acting on a finitedimensional vector space V via a homomorphism ρ: G → GL(V). The differential dρ of ρ maps the Lie algebra Lie(G) of G into gl(V), and we put Trρ for the symmetric bilinear form Trρ(x, y): = trace(dρ(x)dρ(y)) for x, y ∈ Lie(G). We call Trρ a trace form of G. Such forms appear, for example, in the hypotheses for the JacobsonMorozov Theorem [Ca, 5.3.1]. We prove: Theorem A. Assume G is simply connected, split, and almost simple. Then the following are equivalent: (a) The characteristic of F is a torsion prime for G. (b) Every trace form of G is zero. The set of torsion primes for G is given by the following table, cf. e.g. [St75,
Simple finite group schemes and their infinitesimal deformations
, 811
"... We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classi ..."
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We show that the classification of simple finite group schemes over an algebraically closed field reduces to the classification of abstract simple finite groups and of simple restricted Lie algebras in positive characteristic. Both these two simple objects have been classified. We review this classification. Finally, we address the problem of determining the infinitesimal deformations of simple finite group schemes.
E8 HAS NO QUOTIENT TRACE FORM IN CHARACTERISTIC 5
, 712
"... Abstract. Since the 1960s, the question of whether or not a Lie algebra of type E8 over a field of characteristic 5 has a quotient trace form has been open. We close this gap in the literature by proving that it does not. Let L be a Lie algebra over a field F and fix a representation ρ of L. The map ..."
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Abstract. Since the 1960s, the question of whether or not a Lie algebra of type E8 over a field of characteristic 5 has a quotient trace form has been open. We close this gap in the literature by proving that it does not. Let L be a Lie algebra over a field F and fix a representation ρ of L. The map (x, y) ↦ → tr(ρ(x)ρ(y)) (x, y ∈ L) defines a symmetric bilinear form on L that we denote by trρ. It is called a trace form on L. It is invariant under L, so its radical R is an ideal in L. It induces a nondegenerate symmetric bilinear form trρ on the Lie algebra L: = L/R. Such a form is called a quotient trace form on L. Given a particular Lie algebra over some field F, one can ask if it has a quotient trace form, i.e., if it can be obtained as L in the notation of the preceding paragraph. For Lie algebras of type E8 over fields of characteristic 5, this question has been open since the 1960s, see e.g. [Bl 62, p. 554], [BlZ, p. 543], or [Se 67, p. 48]. We settle it here. Theorem. Let L be a Lie algebra of type E8 over a field of characteristic 5. Then there is no quotient trace form on L. Interest in quotient trace forms arose because every Lie algebra with such a form over an algebraically closed field of characteristic ̸ = 2, 3 is a direct sum of abelian Lie algebras, simple Lie algebras “of classical type”, 1 and certain algebras constructed in [Bl65, §2], see e.g. [Bl65, Th. 5.1]. Roughly speaking, we use lemmas due to Block to reduce to showing that the trace is zero for representations coming from algebraic groups of type E8. From this, it is easy to see that it suffices to consider only the Weyl modules, which are defined over Z. Leaning on the fact that a Lie algebra of type E8 is simple over every field [St61, 2.6], we note that the trace form is zero because 5 divides 60, the Dynkin index of E8. 1. Algebraic groups of type E8 We begin by recalling some material from [GN, §4]. 1.1. Let G be a split simple linear algebraic group over a field F, and write g for its Lie algebra. We fix a pinning for G and g, i.e., Zforms GZ and gZ of G and g respectively. We write κg for the unique symmetric bilinear form on gZ such that