Thin Groups of Prime-Power Order and Thin Lie Algebras

Caranti, A.; Mattarei, Sandro; Newman, M. F.; Scoppola, Carlo Maria

doi:10.1093/qmath/47.3.279

Cited by 33 publications

(84 citation statements)

References 9 publications

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“…over a field F is said to be thin (according to [CMNS96]) if dim(L 1 ) = 2 and the following covering property holds: L i+1 = [u, L 1 ] for every 0 = u ∈ L i , for all i 1.…”

Section: Introductionmentioning

confidence: 99%

Thin Lie algebras with diamonds of finite and infinite type

Avitabile

Mattarei

2005

Journal of Algebra

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We investigate a class of infinite-dimensional modular Lie algebras, graded over the positive integers, in which every homogeneous component has dimension one or two. We identify these Lie algebras with loop algebras of certain simple Lie algebras of Hamiltonian type. These Lie algebras are not finitely presented, but certain central extensions of them are, and we give an explicit construction for the latter

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“…over a field F is said to be thin (according to [CMNS96]) if dim(L 1 ) = 2 and the following covering property holds: L i+1 = [u, L 1 ] for every 0 = u ∈ L i , for all i 1.…”

Section: Introductionmentioning

confidence: 99%

Thin Lie algebras with diamonds of finite and infinite type

Avitabile

Mattarei

2005

Journal of Algebra

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“…Knowledge of the second cohomology group of H(2 : n;o> 2 ) sheds light on these particular extensions of the algebras of Albert-Frank-Shalev, as we illustrate below. [23] Gradings of non-graded Hamiltonian Lie algebras 421…”

Section: We Say That a Finite-dimensional Lie Algebra L Admits A Nonsmentioning

confidence: 99%

“…Nevertheless, Shalev's algebras occupy a unique place in the description of the graded Lie algebras of maximal class 402 A. Caranti and S. Mattarei [4] follow for more details.) The arguments of [23] have been extended in [4,20] to show that the second diamond in an infinite-dimensional thin Lie algebra (or one of finite dimension large enough) can only occur in degree 3, 5, q or 2q -1, for some power q of the characteristic p of the underlying field. It follows from [23] that there are, up to isomorphism and with the possible exception of very small characteristics, one or two (depending on the ground field) infinite-dimensional thin Lie algebras with second diamond in degree 3 and no diamond in degree 4, and one with second diamond in degree 5.…”

Section: Introductionmentioning

confidence: 99%

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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“…(Pro-)p-groups of width two and obliquity zero have been introduced in [7,6] under the name of thin groups. The corresponding notion for graded Lie algebras has been introduced in [14].…”

Section: Introductionmentioning

confidence: 99%

“…Now it is proved in [14,3,12] that if there are other diamonds, the second one can only occur in weight 3, 5, q, or 2q − 1, where q is a power of the characteristic of the underlying field. The case when the second diamond occurs in weight 3 or 5 has been investigated in [14,18]; see also [30]. (Here and in the following one has to assume the characteristic of the underlying field to be big enough.)…”

Section: Introductionmentioning

confidence: 99%