Classification of 6-dimensional nilpotent Lie algebras over fields of characteristic not 2

Graaf, Willem A. de

doi:10.1016/j.jalgebra.2006.08.006

Cited by 175 publications

(229 citation statements)

References 9 publications

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“…First of all, it is easy to see that (12) holds for the pair (g, 0), (h, 0), for all g, h ∈ g. Now we prove that (12) …”

Section: Unified Products For Lie Algebrasmentioning

confidence: 71%

“…One of the special cases that we introduce in Example 2.3 is called twisted product, the terminology being borrowed from Hopf algebra theory. The two Lie algebras g and V involved in the construction of the twisted product are connected by a classical 2-cocycle f : V × V → g and plays a key role in the classification of all 6-dimensional nilpotent Lie algebras [12]. Apart from the twisted product, we show in Section 3 that both the classical crossed product and bicrossed product of Lie algebras appear as special cases of the unified product.…”

Section: Extending Structures Problem Let G Be a Lie Algebra And E Amentioning

confidence: 96%

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Extending structures for Lie algebras

Agore

Militaru

2013

Monatsh Math

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Abstract. Let g be a Lie algebra, E a vector space containing g as a subspace. The paper is devoted to the extending structures problem which asks for the classification of all Lie algebra structures on E such that g is a Lie subalgebra of E. A general product, called the unified product, is introduced as a tool for our approach. Let V be a complement of g in E: the unified product g ♮ V is associated to a system (⊳, ⊲, f, {−, −}) consisting of two actions ⊳ and ⊲, a generalized cocycle f and a twisted Jacobi bracket {−, −} on V . There exists a Lie algebra structure [−, −] on E containing g as a Lie subalgebra if and only if there exists an isomorphism of Lie algebras (E, [−, −]) ∼ = g ♮ V . All such Lie algebra structures on E are classified by two cohomological type objects which are explicitly constructed. The first one H 2 g (V, g) will classify all Lie algebra structures on E up to an isomorphism that stabilizes g while the second object H 2 (V, g) provides the classification from the view point of the extension problem. Several examples that compute both classifying objects H 2 g (V, g) and H 2 (V, g) are worked out in detail in the case of flag extending structures.

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“…First of all, it is easy to see that (12) holds for the pair (g, 0), (h, 0), for all g, h ∈ g. Now we prove that (12) …”

Section: Unified Products For Lie Algebrasmentioning

confidence: 71%

Section: Extending Structures Problem Let G Be a Lie Algebra And E Amentioning

confidence: 96%

Extending structures for Lie algebras

Agore

Militaru

2013

Monatsh Math

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“…Here, we only include one example. Example Let

p \neq 2

and let

g = \{([], \begin{matrix} 0 & x_{1} & \frac{x_{2}}{2} & - \frac{x_{3}}{2} & x_{5} \\ 0 & 0 & x_{1} & 0 & x_{4} \\ 0 & 0 & 0 & x_{1} & x_{3} \\ 0 & 0 & 0 & 0 & x_{2} \\ 0 & 0 & 0 & 0 & 0 \end{matrix}) : x_{1}, \dots, x_{5} \in frakturO\} .

Then

frakturg

is a Lie subalgebra of

{frakturn}_{5} false(frakturO false)

of nilpotency class 4, listed as

L_{5, 6}

(de Graaf's notation) in [, Table 3]. For sufficiently large

p

\begin{matrix} true \\ right Z_{g} (T) = & left false(+ q^{8} T^{7} - 3 q^{8} T^{6} + q^{8} T^{5} + q^{7} T^{6} + 2 q^{7} T^{5} - 2 q^{6} T^{5} - 2 q^{6} T^{4} - q^{5} T^{5} + 6 q^{5} T^{4} \\ left - 3 q^{4} T^{4} - 3 q^{4} T^{3} + 6 q^{3} T^{3} - q^{3} T^{2} - 2 q^{2} T^{3} - 2 q^{2} T^{2} + 2 q T^{2} + q T + \end{matrix}

…”

Section: Further Examplesmentioning

confidence: 99%

The average size of the kernel of a matrix and orbits of linear groups

Rossmann

2018

Proc. London Math. Soc.

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Let frakturO be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over frakturO, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of frakturO. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p‐adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro‐p groups.

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“…The classification of nilpotent Lie algebras of dimension ≤ 5 was obtained in [6] (see also [12,13] The rest of the classes g 7 , g 8 , g 9 are abelian.…”

Section: φ(X Y) = G(x φ(Y))mentioning

confidence: 99%