Gardner's deformation of the Krasil'shchik-Kersten system

The classical problem of construction of Gardner's deformations for infinite-dimensional completely integrable systems of evolutionary partial differential equations (PDE) amounts essentially to finding the recurrence relations between the integrals of motion. Using the correspondence between the zero-curvature representations and Gardner deformations for PDE, we construct a Gardner's deformation for the Krasil'shchik-Kersten system. For this, we introduce the new nonlocal variables in such a way that the rules to differentiate them are consistent by virtue of the equations at hand and second, the full system of Krasil'shchik-Kersten's equations and the new rules contains the Korteweg-de Vries equation and classical Gardner's deformation for it. PACS: 02.30.Ik, 02.30,Jr, 02.40.-k, 11.30.-jComment: 7th International workshop "Group analysis of differential equations and integrable systems" (15-19 June 2014, Larnaca, Cyprus), 19 page

On the (non)removability of spectral parameters in Z2Z_2-graded zero-curvature representations and its applications

We generalise to the $\mathbb{Z}_2$-graded set-up a practical method for inspecting the (non)removability of parameters in zero-curvature representations for partial differential equations (PDEs) under the action of smooth families of gauge transformations. We illustrate the generation and elimination of parameters in the flat structures over $\mathbb{Z}_2$-graded PDEs by analysing the link between deformation of zero-curvature representations via infinitesimal gauge transformations and, on the other hand, propagation of linear coverings over PDEs using the Fr\"olicher--Nijenhuis bracket.Comment: 38 pages, accepted to Acta Appl. Mat

Non-Abelian Lie algebroids over jet spaces

We associate Hamiltonian homological evolutionary vector fields --which are the non-Abelian variational Lie algebroids' differentials-- with Lie algebra-valued zero-curvature representations for partial differential equations.Comment: Conf. "Nonlinear Mathematical Physics: 20 Years of JNMP" (Nordfjordeid, Norway, 2013); solution of Problem 12.8 from the book IHES/M-12-13 by the first author. --- Accepted for publication in JNMP (2014), 26 pages (3 figures

Gardner's deformations as generators of new integrable systems

We re-address the problem of construction of new infinite-dimensional completely integrable systems on the basis of known ones, and we reveal a working mechanism for such transitions. By splitting the problem's solution in two steps, we explain how the classical technique of Gardner's deformations facilitates -- in a regular way -- making the first, nontrivial move, in the course of which the drafts of new systems are created (often, of hydrodynamic type). The other step then amounts to higher differential order extensions of symbols in the intermediate hierarchies (e.g., by using the techniques of Dubrovin et al. [1,2] and Ferapontov et al. [3,4]).Comment: Accepted to Proc. Int. workshop 'Physics and Mathematics of Nonlinear Phenomena' (June 22-29, 2013; Gallipoli (LE), Italy), 6 page

Cubic Dirac operator for Uq(sl2)U_q(\mathfrak{sl}_2)

We construct the $q$-deformed Clifford algebra of $\mathfrak{sl}_2$ and study its properties. This allows us to define the $q$-deformed noncommutative Weil algebra for $U_q(\mathfrak{sl}_2)$ and the corresponding cubic Dirac operator. In the classical case it was done by Alekseev and Meinrenken. We compute the spectrum of the cubic element on finite-dimensional and Verma modules of $U_q(\mathfrak{sl}_2)$ and the corresponding Dirac cohomology.Comment: 14 page

Computing symmetries and recursion operators of evolutionary super-systems using the SsTools environment

At a very informal but practically convenient level, we discuss the step-by-step computation of nonlocal recursions for symmetry algebras of nonlinear coupled boson-fermion $N=1$ supersymmetric systems by using the SsTools environment.Comment: 18 pages, accepted to Nonlinear Systems and Their Remarkable Mathematical Structures. (N.Euler ed) CRC Press, Boca Raton FL, US

Restricted simple Lie (super)algebras in characteristic 33

We give explicit formulas proving restrictedness of the following Lie (super)algebras: known exceptional simple vectorial Lie (super)algebras in characteristic 3, deformed Lie (super)algebras with indecomposable Cartan matrix, and (under certain conditions) their simple subquotients over an algebraically closed field of characteristic 3, as well as one type of the deformed divergence-free Lie superalgebras with any number of indeterminates in any characteristic.Comment: the final version, as publishe