Severin Pošta

Representations of the cyclically symmetric q-deformed algebra soq(3)

An algebra homomorphism ψ from the nonstandard q-deformed (cyclically symmetric) algebra Uq(so3) to the extension Uq(sl2) of the Hopf algebra Uq(sl2) is constructed. Not all irreducible representations (IR) of Uq(sl2) can be extended to representations of Uq(sl2). Composing the homomorphism ψ with irreducible representations of Uq(sl2) we obtain representations of Uq(so3). Not all of these representations of Uq(so3) are irreducible. Reducible representations of Uq(so3) are decomposed into irreducible components. In this way we obtain all IR of Uq(so3) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so3 when q → 1.

Representations of the cyclically symmetric q-deformed algebra Uq(so3)

An algebra homomorphism ψ from the nonstandard q-deformed (cyclically symmetric) algebra Uq(so3) to the extension Ûq(sl2) of the Hopf algebra Uq(sl2) is constructed. Not all irreducible representations (IR) of Uq(sl2) can be extended to representations of Ûq(sl2). Composing the homomorphism ψ with irreducible representations of Ûq(sl2) we obtain representations of Uq(so3). Not all of these representations of Uq(so3) are irreducible. Reducible representations of Uq(so3) are decomposed into irreducible components. In this way we obtain all IR of Uq(so3) when q is not a root of unity. A part of these representations turn into IR of the Lie algebra so3 when q → 1.

On the classification of irreducible finite-dimensional representations of Uq′(so3) algebra

In an earlier work [M. Havlicek et al., J. Math. Phys. 40, 2135 (1999)] we defined for any finite dimension five nonequivalent irreducible representations of the nonstandard deformation Uq′(so3) of the Lie algebra so3 where q is not a root of unity [for each dimension only one of them (called classical) admits limit q→1]. In the first part of this paper we show that any finite-dimensional irreducible representation is equivalent to some of these representations. In the case qn=1 we derive new Casimir elements of Uq′(so3) and show that a dimension of any irreducible representation is not higher than n. These elements are Casimir elements of Uq′(som) for all m and even of Uq′(isom+1) due to Inonu–Wigner contraction. According to the spectrum of one of the generators, the representations are found to belong to two main disjoint sets. We give full classification and explicit formulas for all representations from the first set (we call them nonsingular representations). If n is odd, we have full classificatio...

Group classification of variable coefficient generalized Kawahara equations

An exhaustive group classification of variable coefficient generalized Kawahara equations is carried out. As a result, we derive new variable coefficient nonlinear models admitting Lie symmetry extensions. All inequivalent Lie reductions of these equations to ordinary differential equations are performed. We also present some examples on the construction of exact and numerical solutions.

Nonlinear superposition formulas based on imprimitive group action

Systems of nonlinear ordinary differential equations are constructed for which the general solution is expressed algebraically in terms of a finite number of particular solutions. The equations and the corresponding nonlinear superposition formula are based on a nonlinear action of the Lie group SL(N,C) on a homogeneous space M. The isotropy group of the origin of this space is a nonmaximal parabolic subgroup of SL(N,C). Such equations can occur as Backlund transformations for soliton equations on flag manifolds.

Center of quantum algebra Uq′( so 3)

We analyze the structure of the center of the quantum algebra Uq′( so 3). This structure, as expected, depends substantially on the deformation parameter q. When q is not a root of unity, there exists only one Casimir element, which is the deformation of ordinary Casimir element known from so3. The center has a structure of a ring of polynomials of one variable. When qn = 1, there are three more Casimir elements of the form of polynomials in algebra generators. Uq′( so 3) can be seen as a finite dimensional module over the commutative ring of polynomials in these three Casimir elements. Knowing three-parametrical family of n-dimensional irreducible representations, we prove that the dimension of this module is n3. All four Casimir elements are no longer algebraically independent. We present the explicit description of the central variety, that is polynomial dependence between these four Casimir elements. This dependence differs for n = 2m + 1, n = 2(2m + 1) and n = 4m.

Equivalence groupoid of a class of variable coefficient Korteweg–de Vries equations

We classify the admissible transformations in a class of variable coefficient Korteweg--de Vries equations. As a result, full description of the structure of the equivalence groupoid of the class is given. The class under study is partitioned into six disjoint normalized subclasses. The widest possible equivalence group for each subclass is found which appears to be generalized extended in five cases. Ways for improvement of transformational properties of the subclasses are proposed using gaugings of arbitrary elements and mapping between classes. The group classification of one of the subclasses is carried out as an illustrative example.

Three-variable exponential functions of the alternating group

A new class of special functions of three real variables, based on the alternating subgroup of the permutation group S3, is studied. These functions are used for Fourier-like expansion of digital data given on lattice of any density and general position. Such functions have only trivial analogs in one and two variables; a connection to the E-functions of C3 is shown. Continuous interpolation of the three-dimensional data is studied and exemplified.

Enhanced group classification of Benjamin–Bona–Mahony–Burgers equations

Abstract A class of the Benjamin–Bona–Mahony–Burgers (BBMB) equations with time-dependent coefficients is investigated with the Lie symmetry point of view. The set of admissible transformations of the class is described exhaustively. The complete group classification is performed using the method of mapping between classes. The derived Lie symmetries are used to reduce BBMB equations to ordinary differential equations. Some exact solutions are constructed.

Group Analysis of Generalized Fifth-Order Korteweg–de Vries Equations with Time-Dependent Coefficients

We perform enhanced Lie symmetry analysis of generalized fifth-order Korteweg–de Vries equations with time-dependent coefficients. The corresponding similarity reductions are classified and some exact solutions are constructed.