In ‎this ‎paper, ‎we ‎study ‎the ‎relation ‎between ‎the ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎with ‎the ‎lattice theory. ‎We ‎introduce ‎the ‎concepts ‎of ‎the ‎lattice ‎of ‎soft ‎sets, ‎full ‎soft ‎sets ‎and ‎soft ‎Lie ‎algebras ‎and next, we ‎verify ‎some ‎properties ‎of ‎them. We ‎prove ‎that ‎the ‎lattice ‎of ‎the ‎soft ‎sets ‎on ‎a fixed parameter set is isomorphic to the power set of a ...

We report on recent results of the authors concerning calculations of quantum invariants of Seifert 3–manifolds. These results include a derivation of the Reshetikhin–Turaev invariants of all oriented Seifert manifolds associated with an arbitrary complex finite dimensional simple Lie algebra, and a determination of the asymptotic expansions of these invariants for lens spaces. Our results are ...

We introduce an algorithm to decide isomorphism between tensors. The uses the Lie algebra of derivations a tensor compress space in which search takes place so-called densor space. To make method practicable we give polynomial-time solve generalization module for common class modules. As consequence, show that testing is polynomial time tensors whose derivation algebras are classical and spaces...

The gauge invariant observables of the closed bosonic string are quantized in four space-time dimensions by constructing their quantum algebra in a manifestly covariant approach, respecting all symmetries of the classical observables. The quantum algebra is the kernel of a derivation on the universal envelopping algebra of an infinitedimensional Lie algebra. The search for Hilbert space represe...

Abstract. Let F be a field of characteristic not 2 and assume all algebras are over F. We establish several conjugacy theorems for the special linear Lie algebra sl2 over an F -algebra which is a UFD. We find the structure of the full automorphism group as well as conditions for when two such algebras are isomorphic. We also study the structure of their derivation algebras. Often we work with m...

Let k[X]:= k[x 0 , x 1 ,. .. , x n ] be a polynomial algebra over a field k of characteristic zero. We offer an algorithm for calculation of kernel of Weitzenbök derivation d(x i) = x i−1 ,. .. , d(x 0) = 0, i = 1. .. n that is based on an analogue of the well known Casimir elements of finite dimensional Lie algebras. By using this algorithm, the kernel is calculated in the case n < 7.

We apply the theory of infinitesimal transformations for study a family 1+1 fifth-order partial differential equations which have been proposed before description multiple kink solutions. In this analysis we perform complete classification Lie symmetries and one-dimensional optimal system. The results are applied derivation similarity solutions in particular find travel-wave scaling show that k...

We present a detailed derivation of the renormalization group equations for two dimensional electromagnetic Coulomb gases whose charges lie on a triangular lattice (magnetic charges) and its dual (electric charges). The interactions between the charges involve both angular couplings and a new electromagnetic potential. This motivates the denomination of “elastic” Coulomb gas. Such elastic Coulo...

Introduction. Let VecR be the Lie algebra of smooth vector fields on R. In this talk we will survey some results concerning the action of VecR on modules of differential operators. To begin with, let σ be the two-sided action of VecR on the associative algebra Diff R of smooth differential operators on R: σ(X)T := X ◦ T − T ◦X. This is a derivation action which preserves the order filtration Di...