We study rigidity questions for pairs of Lie algebras [Formula: see text] admitting a post-Lie algebra structure. show that if is semisimple and arbitrary, then we have in the sense must be isomorphic. The proof uses result on decomposition as direct vector space sum two subalgebras. hence isomorphic to text]. This solves some open existence structures prove additional results text], where comp...

Every finite-dimensional Lie algebra is a semi-direct product of a solvable Lie algebra and a semisimple Lie algebra. Classifying the solvable Lie algebras is difficult, but the semisimple Lie algebras have a relatively easy classification. We discuss in some detail how the representation theory of the particular Lie algebra sl2 tightly controls the structure of general semisimple Lie algebras,...

This paper is devoted to deformation theory of "anabelian" representations of the absolute Galois group landing in outer automorphism group of the algebraic fundamental group of a hyperbolic smooth curve defined over a number-field. In the first part of this paper, we obtained several universal deformations for Lie-algebra versions of the above representation using the Schlessinger criteria for...

Let G be a finite group and p a prime number. The Plesken Lie algebra is a subalgebra of the complex group algebra C[G] and admits a directsum decomposition into simple Lie algebras. We describe finite-field versions of the Plesken Lie algebra via traditional and computational methods. The computations motivate our conjectures on the general structure of the modular Plesken Lie algebra.

This work is an investigation into the structure and properties of supersymmetric hypermatrix Lie algebra generated by elements of the dihedral group D3. It is based on previous work on the subject of supersymmetric Lie algebra (Schreiber, 2012). In preview work I used several new algebraic tools; namely cubic hypermatrices (including special arrangements of such hypermatrices) and I obtained a...

We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T∗-extension of a nilpotent algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic Lie algebra by the one-dimensional Lie algebra...

We construct the entire generalized Kac-Moody Lie algebra as a quotient of the positive part of another generalized Kac-Moody Lie algebra. The positive part of a generalized Kac-Moody Lie algebra can be constructed from representations of quivers using Ringel's Hall algebra construction. Thus we give a direct realization of the entire generalized Kac-Moody Lie algebra. This idea arises from the...

We prove some commutation relations for a 3-graded Lie algebra, i.e., a Z-graded Lie algebra whose nonzero homogeneous elements have degrees −1, 0 or 1, over a field K. In particular, we examine the free 3-graded Lie algebra generated by an element of degree −1 and another of degree 1. We show that if K has characteristic zero, such a Lie algebra can be realized as a Lie algebra of matrices ove...

We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov’s generalized coherent states. On the other...

We study the existence problem for Novikov algebra structures on finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov algebra is necessarily solvable. Conversely we present a 2-step solvable Lie algebra without any Novikov structure. We use extensions and classical r-matrices to construct Novikov structures on certain classes of solvable Lie algebras.