We regard the shearlet group as a semidirect product group and show that its standard representation is,typically, a quasiregu- lar representation. As a result we can characterize irreducible as well as square-integrable subrepresentations of the shearlet group.

We consider the Lax representation of the new two-component coupled integrable system recently discovered by the author. Connection of the hierarchy of infinitely many Lax pairs with each other is presented.

Let (M,ω) be a Kähler manifold. An integrable function φ on M is called ω-plurisubharmonic if the current ddφ ∧ ω is positive. We prove that φ is ωplurisubharmonic if and only if φ is subharmonic on all q-dimensional complex subvarieties. We prove that a ωplurisubharmonic function is q-convex, and admits a local approximation by smooth, ω-plurisubharmonic functions. For any closed subvariety Z ...

A higher dimensional analogue of the dispersionless KP hierarchy is introduced. In addition to the two-dimensional “phase space” variables (k, x) of the dispersionless KP hierarchy, this hierarchy has extra spatial dimensions compactified to a two (or any even) dimensional torus. Integrability of this hierarchy and the existence of an infinite dimensional of “additional symmetries” are ensured ...

We study differential-difference equation of the form d dx t(n+ 1, x) = f(t(n, x), t(n + 1, x), d dx t(n, x)) with unknown t(n, x) depending on continuous and discrete variables x and n. Equation of such kind is called Darboux integrable, if there exist two functions F and I of a finite number of arguments x, {t(n ± k, x)}k=−∞, { d dxk t(n, x) }∞ k=1 , such that DxF = 0 and DI = I, where Dx is ...

Renormalization of twist-three operators and integrable lattice models. Abstract We address the problem of solution of the QCD three-particle evolution equations which govern the Q 2-dependence of the chiral-even quark-gluon-quark and three-gluon correlators contributing to a number of asymmetries at leading order and the transversely polarized structure function g 2 (x Bj). The quark-gluon-qua...

Hilbert space representations of the cross product ∗-algebras of the Hopf ∗-algebra Uq(su2) and its module ∗-algebras O(Sqr) of Podleś spheres are investigated and classified by describing the action of generators. The representations are analyzed within two approaches. It is shown that the Hopf ∗-algebra O(SUq(2)) of the quantum group SUq(2) decomposes into an orthogonal sum of projective Hopf...

A function f ∈ C(Ω) is holomorphic on Ω, if it satisfies the CauchyRiemann equations: ∂̄f = ∑n k=1 ∂f ∂z̄k dz̄k = 0 in Ω. Denote the set of holomorphic functions on Ω by H(Ω). The Bergman projection, B0, is the orthogonal projection of square-integrable functions onto H(Ω)∩L2(Ω). Since the Cauchy-Riemann operator, ∂̄ above, extends naturally to act on higher order forms, we can as well define Bergm...

Hilbert space representations of the cross product ∗-algebras of the Hopf ∗-algebra Uq(su2) and its module ∗-algebras O(Sqr) of Podleś spheres are investigated and classified by describing the action of generators. The representations are analyzed within two approaches. It is shown that the Hopf ∗-algebra O(SUq(2)) of the quantum group SUq(2) decomposes into an orthogonal sum of projective Hopf...

A regular gradient-holonomic approach to studying the Lax type integrability of Ablowitz–Ladik hierarchy nonlinear integrable discrete dynamical systems in vertex operator representation is presented. The relationship Lie-algebraic scheme analyzed and connection with τ-function discussed.