Search

In this paper, we give an explicit construction of the unitary irreducible representations of the Poincare groups in 2, 3, and 4 space-time dimensions on Hilbert spaces associated with the Schrodinger representation of the Weyl algebra for n = 1, 2, and 3, respectively. Our method of constructing the representations uses extension and localization of the enveloping algebras associated with these Weyl algebras and the Poincare algebras.

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...

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.

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.