In an earlier work ͓M. Havlíček et al., J. Math. Phys. 40, 2135 ͑1999͔͒ we defined for any finite dimension five nonequivalent irreducible representations of the nonstandard deformation U q Ј(so 3 ) of the Lie algebra so 3 where q is not a root of unity ͓for each dimension only one of them ͑called classical͒ admits limit q→1͔. 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 q n ϭ1 we derive new Casimir elements of U q Ј(so 3 ) and show that a dimension of any irreducible representation is not higher than n. These elements are Casimir elements of U q Ј(so m ) for all m and even of U q Ј(iso mϩ1 ) due to Inönu-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 representations͒. If n is odd, we have full classification also for the remaining singular case with the exception of a finite number of representations.

An algebra homomorphism ψ from the nonstandard q-deformed (cyclically symmetric) algebra U q (so 3 ) to the extensionÛ q (sl 2 ) of the Hopf algebra U q (sl 2 ) is constructed. Not all irreducible representations of U q (sl 2 ) can be extended to representations ofÛ q (sl 2 ). Composing the homomorphism ψ with irreducible representations ofÛ q (sl 2 ) we obtain representations of U q (so 3 ). Not all of these representations of U q (so 3 ) are irreducible. Reducible representations of U q (so 3 ) are decomposed into irreducible components. In this way we obtain all irreducible representations of U q (so 3 ) when q is not a root of unity. A part of these representations turns into irreducible representations of the Lie algebra so 3 when q → 1. Representations of the other part have no classical analogue. Using the homomorphism ψ it is shown how to construct tensor products of finite dimensional representations of U q (so 3 ). Irreducible representations of U q (so 3 ) when q is a root of unity are constructed. Part of them are obtained from irreducible representations ofÛ q (sl 2 ) by means of the homomorphism ψ.

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.

No abstract

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.