The Simple Lie p-Algebras of Rank Two

Block, Richard E.; Wilson, Robert Lee

doi:10.2307/1971340

Cited by 39 publications

(208 citation statements)

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“…For example, this is stated in [13,Lemma 1.8.3] under the blanket assumption of that paper that p > 7, but the proof given there is seen to be valid for p > 3. (In particular, one ingredient of that proof, namely [61, Corollary 2], was originally proved for p > 5; however, it is now a special case of more general results in [9] or [54] which assume only p > 3.…”

Section: Hamiltonian Algebras and Block Algebrasmentioning

confidence: 99%

“…In [20] we also constructed thin Lie algebras with second diamond in degree 2q -1 and all diamonds of finite types, as loop algebras of certain Block algebras. The Block algebras used in [20] are actually isomorphic with algebras of Albert and Frank (being simple Block algebras with G = G o , see Section 3, and according to known results, for example [13,Lemma 1.8.3]), but were presented there in a different basis. As we have mentioned earlier, the original motivation for the present paper was finding an explicit identification of those algebras with Hamiltonian algebras H{2 : n; <o 2 ) and describing a corresponding cyclic grading [7] Gradings of non-graded Hamiltonian Lie algebras 405 of the latter.…”

Section: Introductionmentioning

confidence: 99%

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Gradings of non-graded Hamiltonian Lie algebras

Caranti¹,

Mattarei²

2005

J. Aust. Math. Soc.

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A thin Lie algebra is a Lie algebra graded over the positive integers satisfying a certain narrowness condition. We describe several cyclic grading of the modular Hamiltonian Lie algebras H(2: n; (02) (of dimension one less than a power of p) from which we construct infinite-dimensional thin Lie algebras. In the process we provide an explicit identification of H(2: n;0)2) with a Block algebra. We also compute its second cohomology group and its derivation algebra (in arbitrary prime characteristic).2000 Mathematics subject classification: primary 17B50; secondary 17B70, 17B56, 17B65.

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Section: Hamiltonian Algebras and Block Algebrasmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gradings of non-graded Hamiltonian Lie algebras

Caranti¹,

Mattarei²

2005

J. Aust. Math. Soc.

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“…Then Urgau{o> ac* ^t *s a weakly closed set of nilpotent transformations on G_, so that by [J,Theorem 3.1'] or [BW,Theorem 1.10.1] there exists a nonzero vector in G , annihilated by ady. The space of all vectors annihilated by ady is a G0-submodule of G_, , and so is all of G_, .…”

Section: Some General Lemmasmentioning

confidence: 99%

Simple Lie algebras of characteristic 𝑝 with dependent roots

Benkart¹,

Osborn²

1990

Trans. Amer. Math. Soc.

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Abstract.We investigate finite dimensional simple Lie algebras over an algebraically closed field F of characteristic p > 7 having a Cartan subalgebra H whose roots are dependent over F . We show that H must be one-dimensional or for some root a relative to H there is a 1-section L such that the core of L is a simple Lie algebra of Cartan type H(2 : m : 4>) or W(\ : n) for some n > 1 . The results we obtain have applications to studying the local behavior of simple Lie algebras and to classifying simple Lie algebras which have a Cartan subalgebra of dimension less than p -2 .

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“…If one refers to these algebras as Lie algebras of Cartan type also, then one has the generalized Kostrikin-Safarevic conjecture-every simple Lie algebra over an algebraically closed field of characteristic p > 5 is classical or Cartan type. There is much evidence to support this conjecture: [MS,Kap,Kac2,Kac3,Wl,W2,W3,BW1,BO]. A major breakthrough came with the result which was announced in 1984 [BW2,BW3,W4] and which appeared in 1988 [BW4] that a simple restricted Lie algebra is classical or Cartan type if p > 1.…”

Section: Introductionmentioning

confidence: 99%

Contributions to the Classification of Simple Modular Lie Algebras

Benkart¹,

Osborn²,

Strade³

1994

Transactions of the American Mathematical Society

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Abstract.We develop results directed towards the problem of classifying the finite-dimensional simple Lie algebras over an algebraically closed field of characteristic p > 1. A 1-section of such a Lie algebra relative to a torus T of maximal absolute toral rank possesses a unique subalgebra maximal with respect to having a composition series with factors which are abelian or classical simple. In this paper we show that the sum Q of those compositionally classical subalgebras is a subalgebra. This extends to the general case a crucial step in the classification by Block and Wilson of the restricted simple Lie algebras. We derive properties of the filtration which can be constructed using Q and obtain structural information about the 1-sections and 2-sections of Q relative to T . We further classify all those algebras in which Q is solvable.

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The Simple Lie p-Algebras of Rank Two

Cited by 39 publications

References 0 publications

Gradings of non-graded Hamiltonian Lie algebras

Gradings of non-graded Hamiltonian Lie algebras

Simple Lie algebras of characteristic 𝑝 with dependent roots

Contributions to the Classification of Simple Modular Lie Algebras

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