Lie algebras, decomposable into a sum of an Abelian and a nilpotent subalgebra

Петравчук, А. П.

doi:10.1007/bf01061314

Cited by 22 publications

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“…In view of the above result it is natural to get results relating the degree of nilpotence of g * with the degree of solvability of g. In this direction we mention that it was proved recently by Andrada-Barberis-Dotti-Ovando [1] (see also [12]) that g is 2-step solvable if it is sum of two abelian subalgebras.…”

Section: G Is Solvable If G * Is Nilpotentmentioning

confidence: 93%

Lie algebras with complex structures having nilpotent eigenspaces

Santos¹,

Martín

2011

Proyecciones (Antofagasta)

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Let (g, [·, ·]) be a Lie algebra with an integrable complex structure J. The ±i eigenspaces of J are complex subalgebras of g C isomorphic to the algebraWe consider here the case where these subalgebras are nilpotent and prove that the original (g, [·, ·]) Lie algebra must be solvable. We consider also the 6-dimensional case and determine explicitly the possible nilpotent Lie algebras (g, [ * ] J ). Finally we produce several examples illustrating different situations, in particular we show that for each given s there exists g with complex structure J such that (g, [ * ] J ) is s-step nilpotent. Similar examples of hypercomplex structures are also built.

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Section: G Is Solvable If G * Is Nilpotentmentioning

confidence: 93%

Lie algebras with complex structures having nilpotent eigenspaces

Santos¹,

Martín

2011

Proyecciones (Antofagasta)

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“…In [1], [3], [4], [5], a similar statement is proved under additional restrictions on the nilpotency index of one summand (with fewer restrictions on the characteristic of the ground field). Note that the theorem is no longer true when p = 2 (an appropriate counter example has been constructed in [4]).…”

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confidence: 83%

“…Note that the theorem is no longer true when p = 2 (an appropriate counter example has been constructed in [4]). We make an essential use of Weisfeiler's results [6] on Lie algebras with a solvable maximal subalgebra which dictates a restriction upon the characteristic of the ground field.…”

mentioning

confidence: 99%

A Lie algebra that can be written as a sum of two nilpotent subalgebras is solvable

Зусманович¹

1991

Mathematical Notes of the Academy of Sciences of the USSR

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In 1963 O. Kegel raised the following question: is a Lie ring written as a sum of two nilpotent subrings solvable? Recently, Kostrikin [1] brought a renewed attention to this question in the case of finite-dimensional algebras over a field. The question is easily solved in the affirmative in the case of characteristic zero (cf.[1] or [2]). The purpose of this note is to prove the following theorem.Theorem. Over a field of characteristic p > 5, a finite-dimensional Lie algebra written as a sum of two nilpotent subalgebras, is solvable., a similar statement is proved under additional restrictions on the nilpotency index of one summand (with fewer restrictions on the characteristic of the ground field). Note that the theorem is no longer true when p = 2 (an appropriate counter example has been constructed in [4]). We make an essential use of Weisfeiler's results [6] on Lie algebras with a solvable maximal subalgebra which dictates a restriction upon the characteristic of the ground field.We turn to the proof of the theorem. We may assume the ground field K to be algebraically closed. Let L be a counter example to the theorem having the least possible dimension, let L = A + B be the corresponding decomposition into a sum of nilpotent subalgebras, where dim A ≤ dim B. The following lemma can be easily deduced from the minimality of the counter example (cf.[1])., then L 0 is solvable. In particular, L possesses a solvable maximal subalgebra.The following result has been proved in [6]: a maximal solvable subalgebra in a simple Lie algebra determines a long filtration in it. A closer analysis of this proof enables us to replace the simplicity condition by semisimplicity. Indeed, a semisimple Lie algebra with a solvable maximal subalgebra possesses a unique ideal by Block's theorem [7]. The proof of Theorem 1.2.2 in [6] may be repeated for such algebras almost verbatim. But the proof of Theorem 1.5.1 relies only on the conclusion of Theorem 1.2.2 and never makes use of the simplicity of the algebra.Let L 0 be a maximal solvable subalgebra of L containing A (it exists due to Lemma 1). By the above, we can apply Theorem 2.1.3 in [6] which states that for the filtrationby the subalgebra L 0 , the associated graded algebra will have the form grL = G = S ⊗ O m + D, where S is a simple Lie algebra isomorphic to sl 2 (K) or to the Zassenhaus algebra W 1 (n), O m is the algebra of truncated polynomials in m variables, D is a solvable subalgebra of W m = Der(O m ) such that O m contains no D-invariant ideals. The grading is given as follows:where S = i≥−1 e i is the standard grading. Henceforth, this particular grading is meant each time we refer to S ⊗ O m + D as a graded algebra.

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“…(O1) Let g be a real Lie algebra admitting an abelian complex structure. Then g is 2-step solvable [17].…”

Section: Abelian Complex Structures On Lie Algebrasmentioning

confidence: 99%

“…If G is a connected Lie group with Lie algebra g, by left translating J one obtains a complex manifold (G, J) such that left multiplication is holomorphic and, in the bi-invariant case, also right multiplication is holomorphic, which implies that (G, J) is a complex Lie group.Our concern here will be the case when J is abelian. In this case the Lie algebra has abelian commutator, thus it is 2-step solvable (see [17]). However, its nilradical need not be abelian (see Remark 8).…”

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confidence: 99%

Classification of abelian complex structures on 6-dimensional Lie algebras

Andrada

Barberis

Dotti

2011

Journal of the London Mathematical Society

128

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We classify the 6-dimensional Lie algebras that can be endowed with an abelian complex structure and parametrize, on each of these algebras, the space of such structures up to holomorphic isomorphism. IntroductionLet g be a Lie algebra, J be an endomorphism of g such that J 2 = −I, and let g 1,0 be the i-eigenspace of J in g C := g ⊗ R C. When g 1,0 is a complex subalgebra, we say that J is integrable; when g 1,0 is abelian, we say that J is abelian; and when g 1,0 is a complex ideal, we say that J is bi-invariant. We note that a complex structure on a Lie algebra cannot be both abelian and bi-invariant, unless the Lie bracket is trivial. If G is a connected Lie group with Lie algebra g, by left translating J one obtains a complex manifold (G, J) such that left multiplication is holomorphic and, in the bi-invariant case, also right multiplication is holomorphic, which implies that (G, J) is a complex Lie group.Our concern here will be the case when J is abelian. In this case the Lie algebra has abelian commutator, thus it is 2-step solvable (see [17]). However, its nilradical need not be abelian (see Remark 8). Abelian complex structures have interesting applications in hyper-Kähler with torsion geometry (see [6]). It has been shown in [9] that the Dolbeault cohomology of a nilmanifold with an abelian complex structure can be computed algebraically. Also, deformations of abelian complex structures on nilmanifolds have been studied in [10].Of importance, when studying complex structures on a Lie algebra g, is the ideal g J := g + Jg constructed from algebraic and complex data. We say that the complex structure J is proper when g J is properly contained in g. Any complex structure on a nilpotent Lie algebra is proper [19]. The 6-dimensional nilpotent Lie algebras carrying complex structures were classified in [19], and those carrying abelian complex structures were classified in [12].There is only one 2-dimensional non-abelian Lie algebra, the Lie algebra of the affine motion group of R, denoted by aff(R). It carries a unique complex structure, up to equivalence, which turns out to be abelian. The 4-dimensional Lie algebras admitting abelian complex structures were classified in [20]. Each of these Lie algebras, with the exception of aff(C), the realification of the Lie algebra of the affine motion group of C, has a unique abelian complex structure up to equivalence. On aff(C) there is a 2-sphere of abelian complex structures, but only two equivalence classes distinguished by J being proper or not. Furthermore, aff(C) is equipped with a natural bi-invariant complex structure. In dimension 6 it turns out that, as a consequence of our results, some of the Lie algebras equipped with abelian complex structures are of the form aff(A), where A is a 3-dimensional commutative associative algebra. Mathematics Subject Classification 17B30 (primary), 53C15 (secondary).The authors were partially supported by CONICET, ANPCyT and SECyT-UNC (Argentina). n 2 . Furthermore, for 0 r, s n 2 , if J r is equivalent to J s , by comparin...

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Lie algebras, decomposable into a sum of an Abelian and a nilpotent subalgebra

Cited by 22 publications

References 2 publications

Lie algebras with complex structures having nilpotent eigenspaces

Lie algebras with complex structures having nilpotent eigenspaces

A Lie algebra that can be written as a sum of two nilpotent subalgebras is solvable

Classification of abelian complex structures on 6-dimensional Lie algebras

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