Let $L$ be a finite-dimensional Lie algebra. We say a subalgebra $H$ of $L$ is permutably complemented in $L$ if there is a subalgebra $K$ of $L$ such that $L=H+K$ and $Hcap K=0$. Also, if every subalgebra of $L$ is permutably complemented in $L$, then $L$ is called completely factorisable. In this article, we consider the influence of these concepts on the structure of a Lie algebra, in partic...

0. Introduction. In this article we show how to construct hierarchies of partial differential equations and their soliton-type solutions from the vertex operator representations of toroidal Lie algebras. Soliton theory was given a new impetus when it was linked with the representation theory of infinite-dimensional Lie algebras in the works of Sato [S], Date-Jimbo-Kashiwara-Miwa [DJKM] and Drin...

We consider the Calogero–Degasperis–Ibragimov–Shabat depending on the local variables and on the integral of the only local conserved density of the equation in question. The resulting Lie algebra of these symmetries turns out to be a central extension of that of local symmetries.

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...

We find a remarkable subalgebra of higher symmetries of the elliptic Euler-Darboux equation. To this aim we map such equation into its hyperbolic analogue already studied by Shemarulin. Taking into consideration how symmetries and recursion operators transform by this complex contact transformation, we explicitly give the structure of this Lie algebra and prove that it is finitely generated. Fu...

We define a new homotopy algebraic structure, that we call braided $L_\infty$-algebra, and use it to systematically construct class of noncommutative field theories, theories. Braided theories have gauge symmetries which realize Lie algebra, whose Noether identities are inhomogeneous extensions the classical identities, do not act on solutions equations. Drinfel'd twist deformation quantization...

It is well known that provided scalar (1+1)-dimensional evolution equation possesses the infinitedimensional commutative Lie algebra of time-independent non-classical symmetries, it is either linearizable or integrable via inverse scattering transform [1, 2]. The standard way to prove the existence of such algebra is to construct the recursion operator [2]. But Fuchssteiner [3] suggested an alt...

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 ...

According to the Ringel-Green Theorem([G],[R1]), the generic composition algebra of the Hall algebra provides a realization of the positive part of the quantum group. Furthermore, its Drinfeld double can be identified with the whole quantum group([X],[XY]), in which the BGP-reflection functors coincide with Lusztig’s symmetries. We first assert the elements corresponding to exceptional modules ...