tailieunhanh - Báo cáo " REPRESENTATIONS OF THE REAL DIAMOND LIE GROUP "

We present explicit formulas representations of the real diamond Lie algebra obtained from the normal polarization on K-orbits. From this we have list irreducible unitary representations of the real diamond Lie group that is coincide with the representations via Fedosov deformation quantisation. Here the computations are more simple for use star-product. | VNU. JOURNAL OF SCIENCE Mathematics - Physics. N03 - 2005 REPRESENTATIONS OF THE REAL DIAMOND LIE GROUP Nguyen Viet Hai Department of Mathematics Hai Phong University Abstract. We present explicit formulas representations of the real diamond Lie algebra obtained from the normal polarization on K-orbits. From this we have list irreducible unitary representations of the real diamond Lie group that is coincide with the representations via Fedosov deformation quantisation. Here the computations are more simple for use star-product. 1. Introduction The method of orbits discovered in the pioneering works of Kirillov 8 is a universal base for performing harmonic analysis on homogeneous spaces and for constructing new methods of integrating linear differential equations. We give the explicit form of the representations of the real diamond Lie group which is coincide the representations obtained from the deformation quantisation see 4 5 . The representations first appeared in the noncommutative integration method of linear differential equations as a quantum analogue of the noncommutative Mishchenko-Fomenko integration method for finite-dimensional Hamiltonian systems 9 . Quantum groups are group Hopf algebras . replace C -algebras by special Hopf algebras of functions . It is therefore interesting to ask whether we could describe quantum groups as some repeated extensions of some kind quantum strata of coadjoint orbits We are attempting to give a positive answer to this question. It is not yet completely described but we obtained a reasonable answer. Let us describe the main ingredients of our approach. Let us in few words describe the structure of the paper. We introduce on K-orbits in 2 Darboux coordinates normal polarization on K-orbits and the notion of the quantization of K-orbits in 3. Then Hamiltonian functions in canonical coordinates of the co-adjoint orbit ƠỆ the operators Ạ1 which define the representations the representations of the real diamond Lie .