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Functional bases of second-order differential invariants of the Euclid, Poincar\'e, Galilei, conformal, and projective algebras are constructed. The results obtained allow us to describe new classes of nonlinear many-dimensional invariant…
The Casimir operators of a Lie algebra are in one-to-one correspondence with the symmetric invariant tensors of the algebra. There is an infinite family of Casimir operators whose members are expressible in terms of a number of primitive…
In this paper, we classify solvable quadratic Lie algebras up to dimension 6. In dimensions smaller than 6, we use the Witt decomposition given in \cite{Bou59} and a result in \cite{PU07} to obtain two non-Abelian indecomposable solvable…
The main purpose of this paper is to study the finite-dimensional solvable Lie algebras described in its title, which we call {\em minimal non-${\mathcal N}$}. To facilitate this we investigate solvable Lie algebras of nilpotent length $k$,…
An algebraic algorithm is developed for computation of invariants ('generalized Casimir operators') of general Lie algebras over the real or complex number field. Its main tools are the Cartan's method of moving frames and the knowledge of…
We consider Lie algebras admitting an ad-invariant metric, and we study the problem of uniqueness of the ad-invariant metric up to automorphisms. This is a common feature in low dimensions, as one can observe in the known classification of…
We consider finite-dimensional complex Lie algebras admitting a periodic derivation, i.e., a nonsingular derivation which has finite multiplicative order. We show that such Lie algebras are at most two-step nilpotent and give several…
In this paper various representations of the exceptional Lie algebra G_2 are investigated in a purely algebraic manner, and multi-boson/multi-fermion realizations are obtained. Matrix elements of the master representation, which is defined…
We initiate the investigation of the projective varieties $\mathbb E(r,\mathfrak g)$ of elementary subalgebras of dimension $r$ of a ($p$-restricted) Lie algebra $\mathfrak g$ for various $r \geq 1$. These varieties $\mathbb E(r,\mathfrak…
We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is…
Nongraded infinite-dimensional Lie algebras appeared naturally in the theory of Hamiltonian operators, the theory of vertex algebras and their multi-variable analogues. They play important roles in mathematical physics. This survey article…
Let V be a finite dimensional complex superspace and G a simple (or a ``close'' to simple) Lie superalgebra of matrix type, i.e., a Lie subsuperalgebra in GL(V). Under the classical invariant theory for G we mean the description of…
The classification of complex of real finite dimensional Lie algebras which are not semi simple is still in its early stages. For example the nilpotent Lie algebras are classified only up to the dimension 7. Moreover, to recognize a given…
We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations…
The aim of this note is to prove various general properties of a generalization of the full module of first order differential operators on a commutative ring - a $\operatorname{D}$-Lie algebra. A $\operatorname{D}$-Lie algebra $\tilde{L}$…
This is a short presentation of some classical results on finite dimensional complex Lie algebras (classification of nilpotent Lie algebras, deformations and perturbations, contractions and rigidity). We present some applications to…
Minimal unitary representation of $SO(d,2)$ and its deformations describe all the conformally massless fields in $d$ dimensional Minkowskian spacetimes. In critical dimensions these spacetimes admit extensions with twistorial coordinates…
Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, \textit{i.e.}, classes of differential equations that are in correspondence with their Lie point symmetries. In particular,…
We prove that for any known Lie algebra $\frak{g}$ having none invariants for the coadjoint representation, the absence of invariants is equivalent to the existence of a left invariant exact symplectic structure on the corresponding Lie…
The class of $\D$-locally nilpotent algebras (introduced in the paper) is a wide generalization of the algebras of differential operators on commutative algebras. Examples includes all the rings $\CD (A)$ of differential operators on…