arXiv subjects
Publications and source records attributed to Dietrich Burde.
An LR-structure on a Lie algebra is a bilinear product, 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 consider the variety of pre-Lie algebra structures on a given n-dimensional vector space. The group GL_n(K) acts on it, and we study the closure of the orbits with respect to the Zariski topology. This leads to the definition of pre-Lie algebra degenerations. We give fundamental results on such degenerations, including invariants and necessary degeneration criteria. We demonstrate the close relationship to Lie algebra degenerations. Finally we classify all orbit closures in the variety of complex 2-dimensional pre-Lie algebras.
We describe three methods to determine a faithful representation of small dimension for a finite-dimensional nilpotent Lie algebra over an arbitrary field. We apply our methods in finding bounds for the smallest dimension $μ(\Lg)$ of a faithful $\Lg$-module for some nilpotent Lie algebras $\Lg$. In particular, we describe an infinite family of filiform nilpotent Lie algebras $\Lf_n$ of dimension $n$ over $\Q$ and conjecture that $μ(\Lf_n) > n+1$. Experiments with our algorithms suggest that $μ(\Lf_n)$ is polynomial in $n$.
In the study of NIL-affine actions on nilpotent Lie groups we introduced so called LR-structures on Lie algebras. The aim of this paper is to consider the existence question of LR-structures, and to start a structure theory of LR-algebras. We show that any Lie algebra admitting an LR-structure is 2-step solvable. Conversely we find several classes of 2-step solvable Lie algebras admitting an LR-structure, but also classes not admitting such a structure. We study also ideals in LR-algebras, and classify low-dimensional real LR-algebras.
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) less than 6. 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 of N, which we call an LR-structure.
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 note that the inequalities $0.92 \frac{x}{\log(x)} <π(x)< 1.11 \frac{x}{\log(x)}$ do not hold for all $x\ge 30$, contrary to some references. These estimates on $π(x)$ came up recently in papers on algebraic number theory.
Degenerations, contractions and deformations of various algebraic structures play an important role in mathematics and physics. There are many different definitions and special cases of these notions. We try to give a general definition which unifies these notions and shows the connections among them. Here we focus on contractions of Lie algebras and algebraic groups.
We prove an explicit formula for the invariant $μ(\Lg)$ for finite-dimensional semisimple, and reductive Lie algebras $\Lg$ over $\C$. Here $μ(\Lg)$ is the minimal dimension of a faithful linear representation of $\Lg$. The result can be used to study Dynkin's classification of maximal reductive subalgebras of semisimple Lie algebras.
In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs in mathematical physics (QFT and renormalization theory), where the name pre-Lie algebras is used quite often. Already Cayley wrote about such algebras more than hundred years ago. Indeed, LSAs arise in many different areas of mathematics and physics. We attempt to give a survey of the fields where LSAs play an important role. Furthermore we study the algebraic theory of LSAs such as structure theory, radical theory, cohomology theory and the classification of simple LSAs. We also discuss applications to faithful Lie algebra representations.
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 $A^mX^{\ell},X^{\ell}A^m$ and $(AX)^{\ell}$ are given. The case $p=2$ is a special case of the algebraic Riccati equation.
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 2-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 2-step solvable Lie algebras. In particular we generalize a well known result of Scheuneman concerning affine structures on 3-step nilpotent Lie algebras.
Let N be a simply connected, connected real nilpotent Lie group of finite dimension n. We study subgroups $Γ$ in $\Aff (N)=N\rtimes \Aut (N)$ acting properly discontinuously and cocompactly on N. This situation is a natural generalization of the so-called affine crystallographic groups. We prove that for all dimensions $1\le n\le 5$ the generalized Auslander conjecture holds, i.e., that such subgroups are virtually polycyclic.
We study the varieties of Lie algebra laws and their subvarieties of nilpotent Lie algebra laws. We classify all degenerations of (almost all) five-step and six-step nilpotent seven-dimensional complex Lie algebras. One of the main tools is the use of trivial and adjoint cohomology of these algebras. In addition, we give some new results on the varieties of complex Lie algebra laws in low dimension.
We study Lie algebra prederivations. A Lie algebra admitting a non-singular prederivation is nilpotent. We classify filiform Lie algebras admitting a non-singular prederivation but no non-singular derivation. We prove that any 4-step nilpotent Lie algebra admits a non-singular prederivation. The results can be applied to construct affine structures on Lie algebras.
We study symplectic structures on characteristically nilpotent Lie algebras (CNLAs) by computing the cohomology space $H^2(\Lg,k)$ for certain Lie algebras $\Lg$. Among these Lie algebras are filiform CNLAs of dimension $n\le 14$. 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.