We describe a new method to determine faithful representations of small dimension for a finite-dimensional nilpotent Lie algebra. We give various applications of this method. In particular we find a ...new upper bound on the minimal dimension of a faithful module for the Lie algebras being counterexamples to a well-known conjecture of J. Milnor.
We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space H 2(𝔤,k) for certain Lie algebras 𝔤. Among these Lie algebras are filiform ...CNLAs of dimension n ≤ 14. It turns out that there are many examples of CNLAs which admit a symplectic structure. A generalization of a sympletic structure is an affine structure on a Lie algebra.
We study Novikov algebras and Novikov structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov structure must be solvable. Conversely we present an example of a ...nilpotent two-step solvable Lie algebra without any Novikov structure. We construct Novikov structures on certain Lie algebras via classical
r-matrices and via extensions. In the latter case we lift Novikov structures on an abelian Lie algebra
a
and a Lie algebra
b
to certain extensions of
b
by
a
. We apply this to prove the existence of affine and Novikov structures on several classes of two-step solvable Lie algebras. In particular we generalize a well known result of Scheuneman concerning affine structures on three-step nilpotent Lie algebras.
An LR-structure on a Lie algebra 𝔤 is a bilinear product on 𝔤, satisfying certain commutativity relations, and which is compatible with the Lie product. LR-structures arise in the study of simply ...transitive affine actions on Lie groups. In particular one is interested in the question which Lie algebras admit a complete LR-structure. In this paper we show that a Lie algebra admits a complete LR-structure if and only if it admits any LR-structure.
We describe various methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods to finding bounds ...for the smallest dimension
μ
(
g
)
of a faithful
g
-module for some nilpotent Lie algebras
g
. In particular, we introduce an infinite family of filiform nilpotent Lie algebras
f
n
of dimension
n over
Q
and conjecture that
μ
(
f
n
)
>
n
+
1
holds.
We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present ...a \(2\)-step solvable Lie algebra without any Novikov structure. We use extensions and classical \(r\)-matrices to construct Novikov structures on certain classes of solvable Lie algebras.
We study sympathetic Lie algebras, namely perfect and complete Lie algebras. They arise among other things in the study of adjoint Lie algebra cohomology. This is motivated by a conjecture of ...Pirashvili, which says that a non-trivial finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several results on sympathetic Lie algebras and the adjoint Lie algebra cohomology of Lie algebras in general, using the Hochschild-Serre formula. For certain semidirect products we obtain explicit results for the adjoint cohomology.
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