We consider Nambu-Poisson 3-algebras on three dimensional manifolds M3, such as the Euclidean 3-space R, the 3-sphere S as well as the 3-torus T . We demonstrate that in the Clebsch-Monge gauge, the Lie algebra of volume preserving diffeomorphisms SDiff(M3) is identical to the Nambu-Poisson algebra on M3. Moreover the fundamental identity for the Nambu 3-bracket is just the commutation relation...

E6 is an attractive group for unification model building. However, the complexity of a rank 6 group makes it non-trivial to write down the structure of higher dimensional operators in an E6 theory in terms of the states labeled by quantum numbers of the Standard Model gauge group. In this paper, we show the results of our computation of the Clebsch-Gordan coefficients for the products of the 27...

— A certain class of unitary representations of Uq(sl(2;R)) has the property of being simultanenously a representation of Uq̃(sl(2;R)) for a particular choice of q̃(q). Faddeev has proposed to unify the quantum groups Uq(sl(2;R)) and Uq̃(sl(2;R)) into some enlarged object for which he has coined the name “modular double”. We study the R-operator, the co-product and the Haar-measure for the modular...

This paper presents theoretical analysis and software implementation for real harmonics on the special orthogonal group. Noncommutative harmonic complex-valued functions group has been studied extensively. However, it is customary to treat as a case of complex analysis, there have limited results developed specifically real-valued functions. Here, we develop set explicit formulas irreducible un...

This paper shows how to reduce a Hamiltonian system on the cotangent bundle of a Lie group to a Hamiltonian system in the dual of the Lie algebra of a semidirect product. The procedure simplifies. unifies, and extends work of Greene, Guillemin, Holm, Holmes, Kupershmidt, Marsden, Morrison, Ratiu, Sternberg and others. The heavy top, compressible fluids, magnetohydrodynamics, elasticity, the Max...

We will derive both quaternion and octonion algebras as the Clebsch-Gordan algebras based upon the su(2) Lie algebra by considering angular momentum spaces of spin one and three. If we consider both spin 1 and 1 2 states, then the same method will lead to the Lie-super algebra osp(1,2). Also, the quantum generalization of the method is discussed on the basis of the quantum group suq(2). * Paper...

For optimal control problems involving isoperimetric constraints, by using a two-norm approach, a new sufficiency theorem for a proper strong minimum is obtained. It is applicable to processes that satisfy the Legendre-Clebsch necessary condition but its strict version is not imposed, that is, the processes may be singular. The conditions are expressed explicitly in terms of the second variatio...

It is one of the challenging problems to construct an efficient quantum algorithm which can compute the Jones’ polynomial for any knot or link obtained from closure or capping of a n-strand braid. We recapitulate the construction of braid-group Bn representations from vertex models. We perform orthogonal transformation involving quantum Clebsch-Gordan coefficient matrix on the qubit basis to ob...

We show the existence of three infinite sets of independent, conserved charges in three dimensional inviscid fluid dynamics, with a density dependent potential term. Out of these, terms in two sets are in involution. The structures of the conserved currents in Clebsch variable representation are polynomial in nature. Relativistic generalization of the free system and the Batalin-Tyutin extensio...

Article history: Received 13 June 2008 Received in revised form 29 May 2009 Accepted 2 June 2009 Available online 6 June 2009 Extending a geometric construction due to Sederberg and to Bajaj, Holt, and Netravali, an algorithm is presented for parameterizing a nonsingular cubic surface by polynomials of degree three. The fact that such a parametrization exists is classical. The present algorithm...