A Riemannian manifold is called Einstein if it has constant Ricci curvature. Here we present an overview of recent results about left invariant Einstein metrics on compact simple Lie groups, which are not naturally reductive. We also explain methods, techniques and tools.

We express the Hall-Littlewood polynomials, up to a multiplicative factor, as expectations of Schur functions whose indexing partition is a coordinatewise randomisation of the original partition by a particular sequence of explicit correlated random variables. We also treat the case of the spin Hall-Littlewood polynomials by first writing them as randomisations of the Hall-Littlewood polynomials and generalise this framework to a class of symmetric functions that include the Jing and Milne polynomials.

New classes of Lie–Hamilton systems are obtained from the six-dimensional fundamental representation of the rank-three real symplectic Lie algebra. The ansatz is based on a recently proposed procedure for constructing higher-dimensional Lie–Hamilton systems through the representation theory of Lie algebras. As applications of the procedure, we study a time-dependent electromagnetic field and several types of coupled oscillators. The irreducible embedding of the special unitary Lie algebra on three dimensions into the symplectic algebra is also considered, yielding Lie–Hamilton systems arising from the sum of the quark and antiquark three-dimensional representations of the special unitary algebra, which are applied in the construction of \(t\)-dependent coupled systems. In addition, \(t\)-independent constants of the motion are obtained explicitly for all these Lie–Hamilton systems, which allows the derivation of a nonlinear superposition rule.

In this paper we will show a relation between class of Lie algebras given by eigenvalue problem and left-symmetric algebras. We also define multiplication on linear spaces originating from the eigenvalue problem.

The aim of the paper is to provide a conceptual theoretical framework based on symmetry analysis of differential equations in order to deal on the invariance properties of a physical perturbed fractional PDE. Also with the help of the concept of non-linear self-adjointness and the definition of conserved vector, approximate conservation laws are obtained without approximate Lagrangians by using their approximate symmetries. In order to apply the presented theory, we implement Lie symmetry analysis and concept of non-linear self-adjointness to reach the exact solutions, perturbed and unperturbed symmetries and conservation laws of converted equation. This indicates a need to introduce and extend the methodology of symmetry analysis of differential equations on the system of the kind of both fractional and perturbed form.