Related papers: Classical $W_3^{(2)}$ algebra and its Miura map

We give a construction of completely integrable 4-dimensional Hamiltonian systems with cubic Hamilton functions. Applying to the corresponding pairs of commuting quadratic Hamiltonian vector fields the so called Kahan-Hirota-Kimura…

The WDVV equations of associativity in 2-d topological field theory are completely integrable third order Monge-Amp\`ere equations which admit bi-Hamiltonian structure. The time variable plays a distinguished role in the discussion of…

We show that 2D periodic operators with local and perpendicular defects form an algebra. We provide an algorithm of finding spectrum for such operators. While the continuous spectral components can be computed by simple algebraic operations…

Using classical double G of a Lie algebra g equipped with a classical R-operator we define two sets of mutually commuting functions with respect to the initial Lie-Poisson bracket on g* and its extensions. We consider in details examples of…

We obtain a new free field realization of $N=2$ super $W_{3}$ algebra using the technique of quantum hamiltonian reduction. The construction is based on a particular choice of the simple root system of the affine Lie superalgebra…

We study the classical version of supersymmetric $W$-algebras. Using the second Gelfand-Dickey Hamiltonian structure we work out in detail $W_2$ and $W_3$-algebras.

In this dissertation the Weyl-Wigner approach is presented as a map between functions on a real cartesian symplectic vector space and a set of operators on a Hilbert space, to analyse some aspects of the relations between quantum and…

We study the topological structure of the quotient of $SU(3)\times SU(3)$ by diagonal conjugation. This is the simplest nontrivial example for the classical reduced configuration space of chromodynamics on a spatial lattice in the…

The operators of fractional calculus come in many different types, which can be categorised into general classes according to their nature and properties. We conduct a formal study of the class known as weighted fractional calculus and its…

Before we proposed an algebraic technics for the Hamiltonian approach to the evolution systems of partial differential equations, including systems with constraints. Here we further develop this approach and present the defining system of…

We consider the two-dimensional model of W3-gravity within Lagrangian quantization methods for general gauge theories. We use the Batalin-Vilkovisky formalism to study the arbitrariness in the realization of the gauge algebra. We obtain a…

In the present paper, we introduce a method to construct two variable biorthogonal polynomial families with the help of one variable biorthogonal and orthogonal polynomial families. By using this new technique, we define 2D Hermite…

There is developed a differential-algebraic approach to studying the representations of commuting differentiations in functional differential rings under nonlinear differential constraints. An example of the differential ideal with the only…

We describe a few properties of the non semi-simple associative algebra H = M_3 + (M_{2|1}(Lambda2))_0, where Lambda2 is the Grassmann algebra with two generators. We show that H is not only a finite dimensional algebra but also a (non…

We study a class of nonlinear PDEs that admit the same bi-Hamiltonian structure as WDVV equations: a Ferapontov-type first-order Hamiltonian operator and a homogeneous third-order Hamiltonian operator in a canonical Doyle--Potemin form,…

Consider a Hamiltonian action of a compact connected Lie group $G$ on an aspherical symplectic manifold $(M,\omega)$. Under some assumptions on $(M,\omega)$ and the action, D. A. Salamon conjectured that counting gauge equivalence classes…

When both Hamiltonian operators of a bi-Hamiltonian system are pure differential operators, we show that the generalized Kupershmidt deformation (GKD) developed from the Kupershmidt deformation in \cite{kd} offers an useful way to construct…

In this work, we introduce the ${\mathbb Z}_3$-graded differential algebra, denoted by $\Omega(\widetilde{\rm GL}_q(2))$, treated as the ${\mathbb Z}_3$-graded quantum de Rham complex of ${\mathbb Z}_3$-graded quantum group $\widetilde{\rm…

We treat two quite different problems related to changes of complex structures on K\"ahler manifolds by using global geometric method. First, by using operators from Hodge theory on compact K\"ahler manifold, we present a closed explicit…

This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect…