We study ideals of Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We present the first example of a three-step nilpotent Lie algebra which does not admit a Novikov ...structure. On the other hand we show that any free three-step nilpotent Lie algebra admits a Novikov structure. We study the existence question also for Lie algebras of triangular matrices. Finally we show that there are families of Lie algebras of arbitrary high solvability class which admit Novikov structures.
We study the matrix equation
XA
−
AX
=
X
p
in
M
n
(
K) for 1
<
p
<
n. It is shown that every matrix solution
X is nilpotent and that the generalized eigenspaces of
A are
X-invariant. For
A being a ...full Jordan block we describe how to compute all matrix solutions. Combinatorial formulas for
A
m
X
ℓ,
X
ℓ
A
m
and (
AX)
ℓ are given. The case
p
=
2 is a special case of the algebraic Riccati equation.
Let Ln(C) be the variety of complexn-dimensional Lie algebras. The groupGLn(C) acts on it via change of basis. An orbitO(μ) under this action consists of all structures isomorphic to μ. The aim of ...this paper is to give a complete classification of orbit closures of 4-dimensional Lie algebras, i.e., determining all μ∈O(λ)where λ∈L4(C). Starting with a classification of complex Lie algebras of dimensionn≤4, we study the behavior of several Lie algebra invariants under degeneration, i.e., under transition to the orbit closure. As a corollary, we will show that all degenerations in L3(C) can be realized via a one-parameter subgroup, but this is not the case in L4(C).
Affine actions on nilpotent Lie groups Burde, Dietrich; Dekimpe, Karel; Deschamps, Sandra
Forum mathematicum,
09/2009, Letnik:
21, Številka:
5
Journal Article
Recenzirano
Odprti dostop
To any connected and simply connected nilpotent Lie group N, one can associate its group of affine transformations Aff (N). In this paper, we study simply transitive actions of a given nilpotent Lie ...group G on another nilpotent Lie group N, via such affine transformations. We succeed in translating the existence question of such a simply transitive affine action to a corresponding question on the Lie algebra level. As an example of the possible use of this translation, we then consider the case where dim(G) = dim(N) ≤ 5. Finally, we specialize to the case of abelian simply transitive affine actions on a given connected and simply connected nilpotent Lie group. It turns out that such a simply transitive abelian affine action on N corresponds to a particular Lie compatible bilinear product on the Lie algebra 𝔫 of N, which we call an LR-structure.
We study the existence of post-Lie algebra structures on pairs of Lie algebras \((\mathfrak{g},\mathfrak{n})\), where one of the algebras is perfect non-semisimple, and the other one is abelian, ...nilpotent non-abelian, solvable non-nilpotent, simple, semisimple non-simple, reductive non-semisimple or complete non-perfect. We prove several non-existence results, but also provide examples in some cases for the existence of a post-Lie algebra structure. Among other results we show that there is no post-Lie algebra structure on \((\mathfrak{g},\mathfrak{n})\), where \(\mathfrak{g}\) is perfect non-semisimple, and \(\mathfrak{n}\) is \(\mathfrak{sl}_3(\mathbb{C})\). We also show that there is no post-Lie algebra structure on \((\mathfrak{g},\mathfrak{n})\), where \(\mathfrak{g}\) is perfect and \(\mathfrak{n}\) is reductive with a \(1\)-dimensional center.
We study {\em disemisimple} Lie algebras, i.e., Lie algebras which can be written as a vector space sum of two semisimple subalgebras. We show that a Lie algebra \(\mathfrak{g}\) is disemisimple if ...and only if its solvable radical coincides with its nilradical and is a prehomogeneous \(\mathfrak{s}\)-module for a Levi subalgebra \(\mathfrak{s}\) of \(\mathfrak{g}\). We use the classification of prehomogeneous \(\mathfrak{s}\)-modules for simple Lie algebras \(\mathfrak{s}\) given by Vinberg to show that the solvable radical of a disemisimple Lie algebra with simple Levi subalgebra is abelian. We extend this result to disemisimple Lie algebras having no simple quotients of type \(A\).
We give a translation from Russian into English of the article "In memory of Igor Dmitrievich Ado" written by A.V. Dorodnov and I.I. Sakhaev and published in Izv.\ Vyssh.\ Uchebn.\ Zaved.\ Mat.\ no. ...8, (1984), 87--88. It is an orbituary for I. D. Ado. A translation might be useful in general, and in particular for a possible Wikipedia entry of Ado's life in English. In the references we list all known \(12\) publications of I.D. Ado, taken from the article and the MATHSCINET of the AMS. The original orbituary only lists \(9\) publications.
We provide an infinite family of counterexamples to the conjecture of Zassenhaus on the solvability of the outer derivation algebra of a simple modular Lie algebra. In fact, we show that the simple ...modular Lie algebras \(H(2;(1,n))^{(2)}\) of dimension \(3^{n+1}-2\) in characteristic \(p=3\) do not have a solvable outer derivation algebra for all \(n\ge 1\). For \(n=1\) this recovers the known counterexample of \(\mathfrak{psl}_3(F)\). We show that the outer derivation algebra of \(H(2;(1,n))^{(2)}\) is isomorphic to \((\mathfrak{sl}_2(F)\ltimes V(2))\oplus F^{n-1}\), where \(V(2)\) is the natural representation of \(\mathfrak{sl}_2(F)\). We also study other known simple Lie algebras in characteristic three, but they do not yield a new counterexample.
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