Varieties of Elementary Subalgebras of Maximal Dimension for Modular Lie Algebras

Pevtsova, Julia; Stark, Jim

doi:10.1007/978-3-319-94033-5_14

Cited by 7 publications

(8 citation statements)

References 21 publications

(11 reference statements)

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“…However, an old result of Malcev says that in type E 8 the size of any abelian subset of Φ cannot be bigger than 36; see [Ma45]. It is not hard to see that this result of Malcev is still valid in our situation; see [PeS15] for detail. Thus G is not of type E 8 .…”

Section: The Uniqueness Of A(ḡ)mentioning

confidence: 70%

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A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

Premet

2017

Advances in Mathematics

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Let G be a simple algebraic group over an algebraically closed field of characteristic p > 0 and suppose that p is a very good prime for G. In this paper we prove that any maximal Lie subalgebra M of g = Lie(G) with rad(M ) = 0 has the form M = Lie(P ) for some maximal parabolic subgroup P of G. This means that Morozov's theorem on maximal subalgebras is valid under mild assumptions on G. We show that such assumptions are necessary by providing a counterexample to Morozov's theorem for groups of type E8 over fields of characteristic 5. Our proof relies on the main results and methods of the classification theory of finite dimensional simple Lie algebras over fields of prime characteristic. 2010 Mathematics Subject Classification. 17B45. Supported by the Leverhulme Trust (Grant RPG-2013-293).Proof. Replacing L by its p-closure in g if need be we may assume that L is a restricted subalgebra of g. We use induction on the dimension of G. Suppose the statement holds for all standard reductive k-groups of dimension ≤ n (this is obviously true when n = 1). Now suppose dim(G) = n + 1. By Engel's theorem, the Lie algebra [L, L] is nilpotent. If [L, L] = 0 then nil(L) = 0. If [L, L] = 0 then L = L s ⊕ L n where L s is a toral subalgebra of g and L n = nil(L). If L = L s then L ⊆ Lie(T ) for some maximal torus T of G; see [Hum67, Theorem 13.3]. So we are done in this case.Thus we may assume that n := nil(L) = 0. Then z(n) = 0. As [L, L] ⊆ nil(L) by our assumption on L, the adjoint action of L induces a representation of the abelian Lie algebra L/[L, L] on z(n). So there exists a nonzero e ∈ z(n) such that [L, e] ⊆ ke. As e is a nilpotent element of g it admits a cocharacter λ : k × → G optimal in the sense of the Kempf-Rousseau theory. Let P be the parabolic subgroup associated with λ and p = Lie(P ). Then p = i≥0 g(λ, i). By [Pre03, Theorem 2.3], one can choose λ in such a way that e ∈ g(λ, 2) and g e ⊆ p. Furthermore, [g(λ, i), e] = g(λ, i + 2) for all i ≥ 0. Taking i = 0 we find h 0 ∈ g(λ, 0) with [h 0 , e] = e. This implies thatBy our induction assumption, the image of L in the Levi subalgebra g(λ, 0) ∼ = p/nil(p) of g is contained in a Borel subalgebra of g(λ, 0), say b. Since the inverse image of b under the canonical homomorphism p ։ g(λ, 0) is a Borel subalgebra of g, this accomplishes the induction step of our proof.2.4. We denote by O min the minimal nonzero nilpotent orbit in g. It consists of all nonzero e ∈ g with the property that [e, [e, g]] = ke. Corollary 2.3. Let M be a maximal Lie subalgebra of g and denote by N the nilradical of M . Suppose N = 0 and let R be any Lie subalgebra of M whose derived ideal [R, R] consists of nilpotent elements of g. Then the centraliser c g (N ) is an ideal of M and there exists e ∈ c g (N ) ∩ O min such that [R, e] ⊆ ke. Proof. Since N is nilpotent we have that 0 = z(N ) ⊆ c g (N ). Therefore, c g (N ) is a nonzero Lie subalgebra of g. If x ∈ M , c ∈ c g (N ) and n ∈ N then [[x, c], n] = [x, [c, n]] − [c, [x, n]] = 0. So [M, c g (N )] ⊆ c g (N ) which implies that M := M + ...

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Section: The Uniqueness Of A(ḡ)mentioning

confidence: 70%

“…If p = 11 then dim S (p−1) ≥ 44 and if p = 7 then dim S (p−1) ≥ 14. Applying [Ma45] and [PeS15] and arguing as in (3.21) we now observe that cases (v) and (vi) of Proposition 2.7 are impossible.…”

Section: Further Remarks and Observationsmentioning

confidence: 90%

A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

Premet

2017

Advances in Mathematics

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“…Since this is an open condition and there are, up to conjugacy, at most two commutative nil subalgebras of 𝔤 of maximal dimension, the subvariety 𝐺 ⋅ 𝔪 𝑟 is an irreducible component of 𝐶 𝑟 ( (𝔤)) of dimension 𝑟 dim 𝔪 + dim 𝐺 − dim 𝑁 𝐺 (𝔪). For 𝔤 of type 𝐴, the commutative nil subalgebras of maximal dimension were studied in [6]; the remaining types have been dealt with in the subsequent paper [30], generalizing results of Malcev in characteristic zero [22]. Subsets 𝐺 ⋅ 𝔪 𝑟 for various classical 𝔤 were also investigated by the second author in [25].…”

Section: Commuting Varietiesmentioning

confidence: 99%

“…Note that in this case 𝔪 has dimension 𝑛(𝑛 + 1)∕2, and this also equals the codimension (in Sp 2𝑛 ) of the normalizer of 𝔪. In general we make the following (see [30] for background on the variety 𝔼(𝑑, 𝔤)):…”

Section: Commuting Varietiesmentioning

confidence: 99%

Commuting varieties and cohomological complexity theory

Lévy

Ngo

Šivic

2022

Journal of London Math Soc

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In this paper we determine, for all r$r$ sufficiently large, the irreducible component(s) of maximal dimension of the variety of commuting r$r$‐tuples of nilpotent elements of frakturgln$\mathfrak {gl}_n$. Our main result is that in characteristic ≠2,3$\ne 2,3$, this nilpotent commuting variety has dimension (r+1)⌊n24⌋$(r+1)\lfloor \frac{n^2}{4}\rfloor$ for n⩾4$n\geqslant 4$, r⩾7$r\geqslant 7$. We use this to find the dimension of the (ordinary) r$r$th commuting varieties of frakturgln$\mathfrak {gl}_n$ and fraktursln$\mathfrak {sl}_n$ for the same range of values of r$r$ and n$n$. Our principal motivation is the connection between nilpotent commuting varieties and cohomological complexity of finite group schemes, which we exploit in the last section of the paper to obtain explicit values for complexities of a large family of modules over the r$r$th Frobenius kernel (prefixGLn)(r)$(\operatorname{GL}_n)_{(r)}$. These results indicate an inequality between the complexities of a rational G$G$‐module M$M$ when restricted to G(r)$G_{(r)}$ or to Gfalse(Fprfalse)$G(\mathbb {F}_{p^r})$; we subsequently establish this inequality for every simple algebraic group G$G$ defined over an algebraically closed field of good characteristic, significantly extending the main theorem of Lin and Nakano, Inventiones Mathematicae, 138 (1999), 85–101.

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“…They also exist for any simple group G (not necessarily simply connected) so long as the surjective morphism G sc → G from the simply-connected cover is separable. In such a case, we follow Pevtsova and Stark in calling p a separably good prime for the root datum of G [14]. The only situation in which good does not always imply separably good is in type A.…”

Section: Introductionmentioning

confidence: 99%

Unipotent elements and generalized exponential maps

Sobaje

2018

Advances in Mathematics

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Let G be a simple and simply connected algebraic group over an algebraically closed field k of characteristic p > 0. Assume that p is good for the root system of G and that the covering map Gsc → G is separable. In previous work we proved the existence of a (not necessarily unique) Springer isomorphism for G that behaved like the exponential map on the resticted nullcone of G.In the present paper we give a formal definition of these maps, which we call 'generalized exponential maps.' We provide an explicit and uniform construction of such maps for all root systems, demonstrate their existence over Z (p) , and give a complete parameterization of all such maps. One application is that this gives a uniform approach to dealing with the "saturation problem" for a unipotent element u in G, providing a new proof of the known result that u lies inside a subgroup of CG(u) that is isomorphic to a truncated Witt group. We also develop a number of other explicit and new computations for g and for G. This paper grew out of an attempt to answer a series of questions posed to us by P. Deligne, who also contributed several of the new ideas that appear here.

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Varieties of Elementary Subalgebras of Maximal Dimension for Modular Lie Algebras

Cited by 7 publications

References 21 publications

A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

A modular analogue of Morozov's theorem on maximal subalgebras of simple Lie algebras

Commuting varieties and cohomological complexity theory

Unipotent elements and generalized exponential maps

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