On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations

Herpel, Sebastian; Stewart, DI

doi:10.4171/dm/525

Cited by 9 publications

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“…Note that is smooth and connected because so are the . Under some extra assumptions such as: the smoothness of all normalisers of all subspaces of (which is ensured under very strict conditions on , as described in [HS16, Theorem A]); the smoothness of ; the third point of the above lemma also allows us to conclude that is generated by the as a restricted -Lie algebra. Indeed one only needs to obtain the inclusion .…”

Section: -Infinitesimal Saturation and Proof Of Theorem 11mentioning

confidence: 99%

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Integration questions in separably good characteristics

Jeannin¹

2023

Compositio Math.

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Let $G$ be a reductive group over an algebraically closed field $k$ of separably good characteristic $p>0$ for $G$ . Under these assumptions, a Springer isomorphism $\phi : \mathcal {N}_{\mathrm {red}}(\mathfrak {g}) \rightarrow \mathcal {V}_{\mathrm {red}}(G)$ from the nilpotent scheme of $\mathfrak {g}$ to the unipotent scheme of $G$ always exists and allows one to integrate any $p$ -nilpotent element of $\mathfrak {g}$ into a unipotent element of $G$ . One should wonder whether such a punctual integration can lead to an integration of restricted $p$ -nil $p$ -subalgebras of $\mathfrak {g}= \operatorname {Lie}(G)$ . We provide a counter-example of the existence of such an integration in general, as well as criteria to integrate some restricted $p$ -nil $p$ -subalgebras of $\mathfrak {g}$ (that are maximal in a certain sense). This requires the generalisation of the notion of infinitesimal saturation first introduced by Deligne and the extension of one of his theorems on infinitesimally saturated subgroups of $G$ to the previously mentioned framework.

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Section: -Infinitesimal Saturation and Proof Of Theorem 11mentioning

confidence: 99%

“…the smoothness of all normalisers of all subspaces of (which is ensured under very strict conditions on , as described in [HS16, Theorem A]);…”

Section: -Infinitesimal Saturation and Proof Of Theorem 11mentioning

confidence: 99%

Integration questions in separably good characteristics

Jeannin¹

2023

Compositio Math.

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Complete reducibility for Lie subalgebras and semisimplification

Bate,

Böhm,

Martin

et al. 2023

European Journal of Mathematics

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Let G be a connected reductive linear algebraic group over a field k. Using ideas from geometric invariant theory, we study the notion of G-complete reducibility over k for a Lie subalgebra $${\mathfrak {h}}$$ h of the Lie algebra $${\mathfrak {g}} = \hbox {Lie} (G)$$ g = Lie ( G ) of G and prove some results when $${\mathfrak {h}}$$ h is solvable or $$\textrm{char}\hspace{0.55542pt}(k)= 0$$ char ( k ) = 0 . We introduce the concept of a k-semisimplification$${\mathfrak {h}}'$$ h ′ of $${\mathfrak {h}}$$ h ; $${\mathfrak {h}}'$$ h ′ is a Lie subalgebra of $${\mathfrak {g}}$$ g associated to $${\mathfrak {h}}$$ h which is G-completely reducible over k. This is the Lie algebra counterpart of the analogous notion for subgroups studied earlier by the first, third and fourth authors. As in the subgroup case, we show that $${\mathfrak {h}}'$$ h ′ is unique up to $$\textrm{Ad}\hspace{0.55542pt}(G(k))$$ Ad ( G ( k ) ) -conjugacy in $${\mathfrak {g}}$$ g . Moreover, we prove that the two concepts are compatible: for H a closed subgroup of G and $$H'$$ H ′ a k-semisimplification of H, the Lie algebra $$\textrm{Lie}\hspace{0.55542pt}(H')$$ Lie ( H ′ ) is a k-semisimplification of $$\textrm{Lie}\hspace{0.55542pt}(H)$$ Lie ( H ) .

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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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On the smoothness of normalisers, the subalgebra structure of modular Lie algebras, and the cohomology of small representations

Cited by 9 publications

References 25 publications

Integration questions in separably good characteristics

Integration questions in separably good characteristics

Complete reducibility for Lie subalgebras and semisimplification

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

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