Profinite just infinite residually solvable Lie algebras

Finite dimensional Lie algebras are semi-direct product of the semi-simple Lie algebras and solvable Lie algebras. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. It is extremely important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this article, a new natural connection between the set of complex analytic isolated hypersurface singularities and the set of finite dimensional solvable (nilpotent) Lie algebras has been constructed. We construct finite dimensional solvable (nilpotent) Lie algebras naturally from isolated hypersurface singularities. These constructions help us to understand the solvable (nilpotent) Lie algebras from the geometric point of view. Moreover, it is known that the classification of nilpotent Lie algebras in higher dimensions ($\gt 7$) remains to be a vast open area. There are one-parameter families of non-isomorphic nilpotent Lie algebras (but no two-parameter families) in dimension seven. Dimension seven is the watershed of the existence of such families. It is well-known that no such family exists in dimension less than seven, while it is hard to construct one-parameter family in dimension greater than seven. In this article, we construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $11$ (resp. $10$) from $\tilde{E}_7$ singularities and show that the weak Torelli-type theorem holds. We shall also construct an one-parameter family of solvable (resp. nilpotent) Lie algebras of dimension $12$ (resp. $11$) from $\tilde{E}_8$ singularities and show that the Torelli-type theorem holds. Moreover, we investigate the numerical relation between the dimensions of the new Lie algebras and Yau algebras. Finally, the new Lie algebras arising from fewnomial isolated singularities are also computed.

The Levi theorem tells us that every finite-dimensional Lie algebra is the semi-direct product of a semi-simple Lie algebra and a solvable Lie algebra. Brieskorn gave the connection between simple Lie algebras and simple singularities. Simple Lie algebras have been well understood, but not the solvable (nilpotent) Lie algebras. Therefore, it is important to establish connections between singularities and solvable (nilpotent) Lie algebras. In this paper, we give a new connection between nilpotent Lie algebras and nilradicals of derivation Lie algebras of isolated complete intersection singularities. As an application, we obtain the correspondence between the nilpotent Lie algebras of dimension less than or equal to 7 and the nilradicals of derivation Lie algebras of isolated complete intersection singularities with modality less than or equal to 1. Moreover, we give a new characterization theorem for zero-dimensional simple complete intersection singularities.

Nilpotent and solvable lie algebras

In this paper the authors propose a new approach to the study of weight systems. Instead of considering graphs whose vertices correspond to the generators of a Lie algebra (as for Cartan subalgebras in the semisimple case), the authors consider the whole weight system. The purpose is to extract information about the weight system from the geometry of the weights. The considerations are restricted to the case where a torus of derivations induces a decomposition of a nilpotent Lie algebra g into one-dimensional weight spaces, none of which is associated with the zero weight. The paper is structured as follows: In Section 2 the most important facts of weight systems of nilpotent Lie algebras and the root system associated to solvable Lie algebras are recalled. In Section 3 the authors formulate their conditions on the weight systems and analyze the consequences of these conditions on the structure of the weight system. They also define associated weight graphs and deduce their elementary geometrical properties. This provides a characterization of the three-dimensional Heisenberg Lie algebra in terms of trees. Section 4 is devoted to the study of certain subgraphs of a weight graph which can be used to reconstruct the weight system from the weight graph. If r is a semidirect product of g and a torus T these subgraphs determine bounds for the solvability class of r . In Section 5 these results are applied to obtain a geometrical proof of the nonexistence of two-step solvable rigid Lie algebras.

The class of antinilpotent Lie algebras closely related to the problem of constructing solutions with constant coefficients for the Yang-Mills equation is considered. A complete description of the antinilpotent Lie algebras is given. A Lie algebra is said to be antinilpotent if any of its nilpotent subalgebras is Abelian. The Yang-Mills equation with coefficients in a Lie algebra L has nontrivial solutions with constant coefficients if and only if the Lie algebra L is not antinilpotent. In Theorem 1, a description of all semisimple real antinilpotent Lie algebras is given. In Theorem 2, the problem of describing the antinilpotent Lie algebras is completely reduced to the case of semisimple Lie algebras (treated in Theorem 1) and solvable Lie algebras. The description of solvable antinilpotent Lie algebras is given in Theorem 3.

We determine fundamental systems of invariants for complex solvable rigid Lie algebras having nonsplit nilradicals of characteristic sequence (3,1,…,1), these algebras being the natural followers of solvable algebras having Heisenberg nilradicals. A special case of this allows us to obtain a criterion to determine the number of functionally independent invariants of rank one subalgebras of (real or complex) solvable Lie algebras. Finally, we give examples of the inverse procedure, obtaining fundamental systems of an algebra starting from rank one subalgebras, and a criterion for the nonexistence of nontrivial invariants.

We described all transposed Poisson algebra structures on oscillator Lie algebras, i.e. on one-dimensional solvable extensions of the (2n+1) -dimensional Heisenberg algebra; on solvable Lie algebras with naturally graded filiform nilpotent radical; on (n+1) -dimensional solvable extensions of the (2n+1) -dimensional Heisenberg algebra; and on n-dimensional solvable extensions of the n-dimensional algebra with trivial multiplication. We also answered one question on transposed Poisson algebras early posted in a paper by Beites, Ferreira and Kaygorodov. Namely, we found that the semidirect product of sl2 and irreducible module gives a finite-dimensional Lie algebra with non-trivial 12 -derivations, but without non-trivial transposed Poisson structures.

This paper investigates the role of solvable and nilpotent Lie algebras in the domains of cryptography and steganography, emphasizing their potential in enhancing security protocols and covert communication methods. In the context of cryptography, we explore their application in public-key infrastructure, secure data verification, and the resolution of commutator-based problems that underpin data protection strategies. In steganography, we examine how the algebraic properties of solvable Lie algebras can be leveraged to embed confidential messages within multimedia content, such as images and video, thereby reinforcing secure communication in dynamic environments. We introduce a key exchange protocol founded on the structural properties of solvable Lie algebras, offering an alternative to traditional number-theoretic approaches. The proposed Lie Exponential Diffie–Hellman Problem (LEDHP) introduces a novel cryptographic challenge based on Lie group structures, offering enhanced security through the complexity of non-commutative algebraic operations. The protocol utilizes the non-commutative nature of Lie brackets and the computational difficulty of certain algebraic problems to ensure secure key agreement between parties. A detailed security analysis is provided, including resistance to classical attacks and discussion of post-quantum considerations. The algebraic complexity inherent to solvable Lie algebras presents promising potential for developing cryptographic protocols resilient to quantum adversaries, positioning these mathematical structures as candidates for future-proof security systems. Additionally, we propose a method for secure message embedding using the Lie algebra in combination with frame deformation techniques in animated objects, offering a novel approach to steganography in motion-based media.

CHAPTER 2 - NILPOTENT AND SOLVABLE LIE ALGEBRAS

In this paper, we study a Lie algebra equipped by an exact symplectic structure. This condition implies that the Lie algebra has even dimension. The research aims to identify and to contruct 2-step solvable exact symplectic Lie algebras of low dimension with explicit formulas for their one-forms and symplectic forms. For case of four-dimensional, we found that only one class among three classes is 2-step solvable exact symplectic Lie algebra. Furthermore, we also give more examples for case six and eight dimensional of Lie algebras with exact symplectic forms which is included 2-step solvable exact sympletic Lie algebras. Moreover, it is well known that a 2-step solvable Lie algebra equipped by an exact symplectic form is nothing but it is called a 2-step solvable Frobenius Lie algebra.

Cohomologically rigid solvable Lie algebras with a nilradical of arbitrary characteristic sequence

Given two complex Banach spaces $X_1$ and $X_2$, a tensor product $X_1\mathbin{\widetilde{\otimes}} X_2$ of $X_1$ and $X_2$ in the sense of [14], two complex solvable finite-dimensional Lie algebras $L_1$ and $L_2$, and two representations $\varrho_i\colo

The aim of the paper is to provide a characterization criterion of exponential solvable Frobenius Lie algebras (having open coadjoint orbits), in terms of primitive ideals of the associated enveloping algebra. In the case of complex solvable Lie algebras, we also show that an algebraic adjoint orbit is open if and only if the associated primitive ideal through the Dixmier map is trivial.

Blocks and modules for Whittaker pairs

Deformations and rigidity in varieties of Lie algebras