Leibniz algebras as non-associative algebras

In this paper we define the basic concepts for left or right Leibniz algebras and prove some of the main results. Our proofs are often variations of the known proofs and several results seem to be new.Comment: 34 pages; the false proof of the second Whitehead lemma for Leibniz algebras was omitted and some minor corrections were mad

Homological Topics in the Representation Theory of Restricted Lie Algebras

We present some recent developments in the application of homological methods to the representation theory of finite dimensional restricted Lie algebras

Existence of Solutions of the Classical Yang-Baxter Equation for a Real Lie Algebra

We characterize finite-dimensional Lie algebras over the real numbers for which the classical Yang-Baxter equation has a non-trivial skew-symmetric solution (resp. a non-trivial solution with invariant symmetric part). Equivalently, we obtain a characterization of those finite-dimensional real Lie algebras which admit a non-trivial (quasi-) triangular Lie bialgebra structure

Erratum to “Support Varieties and Representation Type of Small Quantum Groups”

Some of the general results in the paper require an additional hypothesis, such as quasitriangularity. Applications to specific types of Hopf algebras are correct, as some of these are quasitriangular, and for those that are not, the Hochschild support variety theory may be applied instead

On the Cohomology of Solvable Leibniz Algebras

This paper is a sequel to J. Feldvoss and F. Wagemann: On Leibniz cohomology, (2021), where we mainly consider semisimple Leibniz algebras. It turns out that the analogue of the Hochschild-Serre spectral sequence for Leibniz cohomology cannot be applied to many ideals, and therefore this spectral sequence seems not to be applicable for computing the cohomology of non-semi-simple Leibniz algebras. The main idea of the present paper is to use similar tools as developed by Farnsteiner for Hochschild cohomology (see R. Farnsteiner: On the cohomology of associative algebras and Lie algebras (1987) and R. Farnsteiner: On the vanishing of homology and cohomology groups of associative algebras (1988)) to work around this. Unfortunately, it does not seem to be possible to relate the cohomology of a Leibniz algebra directly to Hochschild cohomology as is the case for Lie algebras, but all the desired results can be obtained in a similar way. In particular, this enables us to generalize the vanishing theorems of Dixmier and Barnes for nilpotent and (super)solvable Lie algebras to Leibniz algebras. Moreover, we compute the cohomology of the one-dimensional Lie algebra with values in an arbitrary Leibniz bimodule and show that it is periodic with period two. As a consequence, we prove the Leibniz analogue of a non-vanishing theorem of Dixmier. Although not needed in full for the aforementioned results, we prove a Fitting lemma for Leibniz bimodules that might be useful elsewhere

Ein Beitrag Zur Klassifikation Von Koppelkurven (A Contribution to the Classification of Coupler Curves)

A study of the coupler curves of four-bar and Watt mechanisms is undertaken either theoretically or based on a computer program that is integrated in a CAD-System. This leads even in the case of four-far linkages to some new insights concerning the existence of nodes and cusps. The possible real singularities of non-degenerate coupler curves of the Watt mechanism are determined and by means of our computer program, we demonstrate their difficult structure

On the Number of Simple Modules of a Supersolvable Restricted Lie Algebra

Around 1939 Hans Zassenhaus started to study the representation theory of finite-dimensional Lie algebras over a field of prime characteristic [18, 19]. Since then a lot of progress has been made but nevertheless a classification of the simple modules (up to isomorphism) is not known for many classes of Lie algebras. In the papers cited above Zassenhaus gave such a classification for nilpotent Lie algebras. About 30 years later parts of this were extended to supersolvable Lie algebras by Veisfeiler and Kac (see [17, Sect. 2]). More generally, they introduced a partition of the (infinite) set of isomorphism classes of simple modules into (infinitely many) finite sets, namely, the sets of isomorphism classes of simple modules with a fixed p-character (see [17, Sect. 1]). In particular, for a nilpotent restricted Lie algebra the number of isomorphism classes of simple modules with a fixed p-character is known (see [15, Satz 6]). A fundamental question that still remains open is to determine this number for more general classes of Lie algebras. The aim of this paper is to develop an approach for attacking this problem in the case of supersolvable (restricted) Lie algebras. In the following we will describe the contents of the paper in more detail