Nilpotent Ones Research Articles

Central extensions of some solvable Leibniz superalgebras

This work is devoted to the study of central extensions of some solvable Leibniz superalgebras. We show that a solvable Leibniz superalgebra with non-null center can be obtained by central extension of other solvable ones of lower dimensions. Moreover, we describe the central extensions for the maximal solvable Lie superalgebras with nilradical which neither characteristically nilpotent in non-split case nor do not involve characteristically nilpotent ones as a term in split case.Additionally, we apply two different procedures to the null-filiform Leibniz superalgebra and the model filiform Lie superalgebra. On the first one, we compute its central extensions and then study the maximal solvable extension of the superalgebras obtained. However, on the second procedure, we consider first its maximal solvable superalgebra and then study its central extensions. Finally, we compare the results obtained at the end of the two procedures.

On the Deepest Fallacy in the History of Mathematics: The Denial of the Postulate about the Approximation Nature of a Simple-Iteration Method and Iterative Derivation of Cramer’s Formulas

Contrary to the opinion about approximation nature of a simple-iteration method, the exact solution of a system of linear algebraic equations (SLAE) in a finite number of iterations with a stationary matrix is demonstrated. We present a theorem and its proof that confirms the possibility to obtain the finite process and imposes the requirement for the matrix of SLAE. This matrix must be unipotent, i.e. all its eigenvalues to be equal to 1. An example of transformation of SLAE given analytically to the form with a unipotent matrix is presented. It is shown that splitting the unipotent matrix into identity and nilpotent ones results in Cramer’s analytical formulas in a finite number of iterations.

The status of polycyclic group-based cryptography: A survey and open problems

AbstractPolycyclic groups are natural generalizations of cyclic groups but with more complicated algorithmic properties. They are finitely presented and the word, conjugacy, and isomorphism decision problems are all solvable in these groups. Moreover, the non-virtually nilpotent ones exhibit an exponential growth rate. These properties make them suitable for use in group-based cryptography, which was proposed in 2004 by Eick and Kahrobaei [

FILIFORM LIE ALGEBRAS OF DIMENSION 8 AS DEGENERATIONS

For each complex 8-dimensional filiform Lie algebra we find another nonisomorphic Lie algebra that degenerates to it. Since this is already known for nilpotent Lie algebras of rank ≥ 1, only the characteristically nilpotent ones should be considered.

Groups with Finite Outer Automizers

We describe generalised soluble or nilpotent groups G in which, for all H ≤ G, the group of outer automorphisms induced on H by elements of G via conjugation (that is, the factor N G (H)/HC G (H)) is finite. It turns out that such groups are abelian-by-finite; the nilpotent ones are centre-by-finite. Also groups in which the same condition is imposed on abelian subgroups only are considered.

The accessible sets of quadratic free nilpotent control systems

We consider the problem of describing the accessible set of some simple control systems, the free nilpotent ones. We review what is known about linear free nilpotent control sys- tems and the simplest quadratic free nilpotent control system. In these cases the accessible sets are cell complexes and the boundary cells are completely known. We are particularly interested in a quadratic free nilpotent system that is a extension of the famous system of Fuller. In this case the accessible sets are harder to describe because there are boundary points that can only be reached by chattering bang bang trajectories. A bang bang trajectory is chattering if it has an infinite number of switches in finite time. We oer some insights and conjectures about the accessible sets of the extended Fuller system.

COMPLEXIFICATIONS OF HOLOMORPHIC ACTIONS AND THE BERGMAN METRIC

Let G=(ℝ,+) act by biholomorphisms on a Stein manifold X which admits the Bergman metric. We show that X can be regarded as a G-invariant domain in a "universal" complex manifold X* on which the complexification [Formula: see text] of G acts. The analogous result holds for actions of a larger class of real Lie groups containing, e.g. abelian and certain nilpotent ones. For holomorphic actions of such groups on Stein manifolds, necessary and sufficient conditions for the existence of X* are given.

Induced local actions on taut and Stein manifolds

Let G = ( R , + ) G=({\mathbb {R}},+) act by biholomorphisms on a taut manifold X X . We show that X X can be regarded as a G G -invariant domain in a complex manifold X ∗ X^{*} on which the universal complexification ( C , + ) ({\mathbb {C}},+) of G G acts. If X X is also Stein, an analogous result holds for actions of a larger class of real Lie groups containing, e.g., abelian and certain nilpotent ones. In this case the question of Steinness of X ∗ X^{*} is discussed.

Vaught’s conjecture and the Glimm-Effros property for Polish transformation groups

We extend the original Glimm-Effros theorem for locally compact groups to a class of Polish groups including the nilpotent ones and those with an invariant metric. For this class we thereby obtain the topological Vaught conjecture.

Synthetic Differential Supergeometry

Three branches of modern mathematics makesomething of infinitesimals per se, namely, nonstandardanalysis, synthetic differential geometry, andsupergeometry. The first is concerned exclusively withinvertible infinitesimals, whereas the second deals mainlywith nilpotent ones. Both of the former two are engagedexclusively in commuting or bosonic infinitesimals,while the third treats anticommuting or fermionic ones, leading to so-called noncommutativemathematics. the unification of the first two approacheswas nicely discussed by Moerdijk and Reyes, but theunification of the second and the third seems to remain open. The principal objective of this paper isto fill the gap, arguing that a super version ofmicrolinear space, dubbed “supermicrolinearspace,” is a natural generalization ofsupermanifold, just as the synthetic concept of microlinearspace is replacing the classical concept of smoothmanifold. The central result of the paper is that thegraded tangency of a supermicrolinear space forms a Lie superalgebra, while it is well known thatthe tangency of a microlinear space (i.e., its totalityof vector fields) forms a Lie algebra.

Polycyclic-by-finite groups admit a bounded-degree polynomial structure

For a polycyclic-by-finite group $\Gamma$ , of Hirsch length $h$ , an affine (resp. polynomial) structure is a representation of $\Gamma$ into ${\rm Aff}({\Bbb R}^{h})$ (resp. ${\rm P}({\Bbb R}^h)$ , the group of polynomial diffeomorphisms) letting $\Gamma$ act properly discontinuously on ${\Bbb R}^{h}$ . Recently it was shown by counter-examples that there exist groups $\Gamma$ (even nilpotent ones) which do not admit an affine structure, thus giving a negative answer to a long-standing question of John Milnor. We prove that every polycyclic-by-finite group $\Gamma$ admits a polynomial structure, which moreover appears to be of a special (“simple”) type (called ”canonical”) and, as a consequence of this, consists entirely of polynomials of a bounded degree. The construction of this polynomial structure is a special case of an iterated Seifert Fiber Space construction, which can be achieved here because of a very strong and surprising cohomology vanishing theorem.

On the 𝑝-adic completions of nonnilpotent spaces

This paper deals with the p p -adic completion F p ∞ X {F_{p\infty }}X developed by Bousfield-Kan for a space X X and prime p p . A space X X is called F p {F_p} -good when the map X → F p ∞ X X \to {F_{p\infty }}X is a mod - p \bmod \text {-}p homology equivalence, and called F p {F_p} -bad otherwise. General examples of F p {F_p} -good spaces are established beyond the usual nilpotent or virtually nilpotent ones. These include the polycyclic-by-finite spaces. However, the wedge of a circle with a sphere of positive dimension is shown to be F p {F_p} -bad. This provides the first example of an F p {F_p} -bad space of finite type and implies that the p p -profinite completion of a free group on two generators must have nontrivial higher mod - p \bmod \text {-}p homology as a discrete group. A major part of the paper is devoted to showing that the desirable properties of nilpotent spaces under the p p -adic completion can be extended to the wider class of p p -seminilpotent spaces.

Fourier Inversion for Unipotent Invariant Integrals

Consider G a semisimple Lie group and $\Gamma \subseteq G$ a discrete subgroup such that ${\text {vol(}}G/\Gamma ) < \infty$. An important problem for number theory and representation theory is to find the decomposition of ${L^2}(G/\Gamma )$ into irreducible representations. Some progress in this direction has been made by J. Arthur and G. Warner by using the Selberg trace formula, which expresses the trace of a subrepresentation of ${L^2}(G/\Gamma )$ in terms of certain invariant distributions. In particular, measures supported on orbits of unipotent elements of G occur. In order to obtain information about representations it is necessary to expand these distributions into Fourier components using characters of irreducible unitary representations of G. This problem is solved in this paper for real rank $G = 1$. In particular, a relationship between the semisimple orbits and the nilpotent ones is made explicit generalizing an earlier result of R. Rao.

Fourier inversion for unipotent invariant integrals

Consider G a semisimple Lie group and Γ ⊆ G \Gamma \, \subseteq \,G a discrete subgroup such that vol( G / Γ ) &gt; ∞ {\text {vol(}}G/\Gamma )\, &gt; \,\infty . An important problem for number theory and representation theory is to find the decomposition of L 2 ( G / Γ ) {L^2}(G/\Gamma ) into irreducible representations. Some progress in this direction has been made by J. Arthur and G. Warner by using the Selberg trace formula, which expresses the trace of a subrepresentation of L 2 ( G / Γ ) {L^2}(G/\Gamma ) in terms of certain invariant distributions. In particular, measures supported on orbits of unipotent elements of G occur. In order to obtain information about representations it is necessary to expand these distributions into Fourier components using characters of irreducible unitary representations of G. This problem is solved in this paper for real rank G = 1 G\, =\, 1 . In particular, a relationship between the semisimple orbits and the nilpotent ones is made explicit generalizing an earlier result of R. Rao.