Communications in Mathematics |

Editors: Friedrich Wagemann & Ivan Kaygorodov

We prove that every local derivation on a complex semisimplefinite-dimensional Leibniz algebra is a derivation.

The Lie algebra of infinitesimal isometries of a Riemannian manifold containsat most two commutative ideals. One coming from the horizontal nullity space ofthe Nijenhuis tensor of the canonical connection, the other coming from theconstant vectors fields independent of the Riemannian metric.

This paper is devoted to several new results concerning (standard) octonionpolynomials. The first is the determination of the roots of all right scalarmultiples of octonion polynomials. The roots of left multiples are alsodiscussed, especially over fields of characteristic not 2. We then turn tostudy the dynamics of monic quadratic real octonion polynomials, classifyingthe fixed points into attracting, repelling and ambivalent, and concluding witha discussion on the behavior of pseudo-periodic points.

This paper introduces two new algorithms for Lie algebras over finite fieldsand applies them to the investigate the known simple Lie algebras of dimensionat most $20$ over the field $\mathbb{F}_2$ with two elements. The firstalgorithm is a new approach towards the construction of $\mathbb{Z}_2$-gradingsof a Lie algebra over a finite field of characteristic $2$. Using this, weobserve that each of the known simple Lie algebras of dimension at most $20$over $\mathbb{F}_2$ has a $\mathbb{Z}_2$-grading and we determine theassociated simple Lie superalgebras. The second algorithm allows us to computeall subalgebras of a Lie algebra over a finite field. We apply this to computethe subalgebras, the maximal subalgebras and the simple subquotients of theknown simple Lie algebras of dimension at most $16$ over $\mathbb{F}_2$ (withthe exception of the $15$-dimensional Zassenhaus algebra).

In this paper, we study the generalized derivation of a Lie sub-algebra ofthe Lie algebra of polynomial vector fields on $\mathbb{R}^n$ where $n\geq1$,containing all constant vector fields and the Euler vector field, under someconditions on this Lie sub-algebra.

The purpose of this paper is to study Lie-Rinehart superalgebras overcharacteristic zero fields, which are consisting of a supercommutativeassociative superalgebra $A$ and a Lie superalgebra $L$ that are compatible ina certain way. We discuss their structure and provide a classification in smalldimensions. We describe all possible pairs defining a Lie-Rinehart superalgebrafor $\dim(A)\leq 2$ and $\dim(L)\leq 4$. Moreover, we construct a cohomologycomplex and develop a theory of formal deformations based on formal powerseries and this cohomology.

A Rota-Baxter Leibniz algebra is a Leibniz algebra$(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T :\mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dualrepresentation of Rota-Baxter Leibniz algebras. Next, we define a cohomologytheory of Rota-Baxter Leibniz algebras. We also study the infinitesimal andformal deformation theory of Rota-Baxter Leibniz algebras and show that ourcohomology is deformation cohomology. Moreover, We define an abelian extensionof Rota-Baxter Leibniz algebras and show that equivalence classes of suchextensions are related to the cohomology groups.

We define a homology theory for pre-crossed modules that specifies to rackhomology in the case when the pre-crossed module is freely generated by a rack.

Let $\mathfrak{n}$ be a maximal nilpotent subalgebra of a simple complex Liealgebra with root system $\Phi$. A subset $D$ of the set $\Phi^+$ of positiveroots is called a rook placement if it consists of roots with pairwisenon-positive scalar products. To each rook placement $D$ and each map $\xi$from $D$ to the set $\mathbb{C}^{\times}$ of nonzero complex numbers one cannaturally assign the coadjoint orbit $\Omega_{D,\xi}$ in the dual space$\mathfrak{n}^*$. By definition, $\Omega_{D,\xi}$ is the orbit of $f_{D,\xi}$,where $f_{D,\xi}$ is the sum of root covectors $e_{\alpha}^*$ multiplied by$\xi(\alpha)$, $\alpha\in D$. (In fact, almost all coadjoint orbits studied atthe moment have such a form for certain $D$ and $\xi$.) It follows from theresults of AndrĂ¨ that if $\xi_1$ and $\xi_2$ are distinct maps from $D$ to$\mathbb{C}^{\times}$ then $\Omega_{D,\xi_1}$ and $\Omega_{D,\xi_2}$ do notcoincide for classical root systems $\Phi$. We prove that this is true if$\Phi$ is of type $G_2$, or if $\Phi$ is of type $F_4$ and $D$ is orthogonal.