In this paper, prime IDEAL bundles and semi-prime and irreducible IDEAL bundles of a LIE algebra bundle are defined and their relation with prime IDEAL bundles is studied.

In this paper we apply a geometric integrator to the problem of LIE-Poisson system for IDEAL compressible isentropic fluids (ICIF) numerically. Our work is based on the decomposition of the phase space, as the semidirect product of two infinite dimensional LIE groups. We have shown that the solution of (ICIF) stays in coadjoint orbit and this result extends a similar result for matrix group discussed in [6]. By using the coadjoint action of the LIE group on the dual of its LIE algebra to advance the numerical flow, we (as in [2]) devise methods that automatically stay on the coadjoint orbit. The paper concludes with a concrete example.

Schur proved that if the center of a group G has finite index, then the derived subgroup G′ is also finite. Moneyhun proved that if L is a LIE algebra such that dim(L/Z(L)) = n, then dim(L^2) ≤1/2n(n-1) In this paper, we extend the converse of Moneyhun’s theorem. Also, we prove a wellknown result of nilpotent LIE algebras by using a different technique

Using the action of a LIE group on a hypergroup, the notion of LIE hypergroup is defined. It is proved that tangent space of a LIE hypergroup is a hypergroup and that a differentiable map between two LIE hypergroup is good homomorphism if and only if its differential map is a good homomorphism. The action of a hypergroup on a set is defined. Using this notion, hypergroup bundle is introduced and some of its basic properties are investigated. In addition, some results on qutient hypergroups are given.

The largest class of hyperstructures is the one which satisfies the weak properties. We connect the theory of P-hopes, a large class of hyperoperations, with the LIE-Santilli admissibility used in Hardonic Mechanics. This can be achieved by a kind of Rees, sandwich hyperop-eration.

Providing an appropriate definition of a horizontal subbundle of a LIE algebroid will lead to construction of a better framework on LIE algebriods. In this paper, we give a new and natural definition of a horizontal subbundle using the prolongation of a LIE algebroid and then we show that any linear connection on a LIE algebroid generates a horizontal subbundle and vice versa. The same correspondence will be proved for any covariant derivative on a LIE algebroid.