Representations, Realizations And Applications Of Multi-parameter Non-linearly Deformed So(3), So(4) Algebra

Symmetry plays an important roll when we investigate the world, and the math tool which study such property is also a important branch in modern mathematics。In the middle of1980s, quantum group arose from the exactly solvable problem studied by inverse scattering method in quantum field theory and statistical mechanics, and from then on, various deformed Lie algebra has been a research hotspots. Study on the irre-ducible representation, indecomposable representation and the boson realization of those deformed Lie algebra are of great importance.In this thesis, we studied multi-parameter non-linearly deformed so(3) algebra Rq,r(v) as well as multi-parameter non-linear deformed so(4) algebra Rq,q’r,r’(v,v’) and non-linearly so(4) algebra so(4)F. We discussed their structures, representations and boson real-izations, at the same time, we studied the connection between deformation function of deformation mapping and the structure function appeared in the commutation results.We used the method similar to Cartan-Weyl basis to study the irreducible repre-sentation of Rq,r(v) and Rq,q’r,r’(v,v’), the indecomposable representation using master repre-sentation and the inducible representation on various quotient spaces, we also give the Dyson-like boson realization. By the general Schwinger realization, we calculate in detail the single-boson, single-inverse-boson and double-boson(include a pair of boson with a pair of inverse boson and double inverse boson) realization, which are all Hostein-Primakoff-like. By discussing these realization, we find that the phase part given in polar decomposition remain the same as non-deformed, while the length part varies with dif-ferent deformation. Finally we studyed the application of non-linearly deformed so(3) algebra in two-dimension superintegrable system.On the other hand, we discussed the connection between deformation algebra and Lie algebra. Through studying the coupling of two Rq,r(v), we conclude the rules which should be satisfied when defining the commutation relation, furthermore, we spread this conclusion to the coupling problem of several deformed algebra, which is not limited to Rq,r(v). By such kind of coupling, we get a kind of deformed algebra like semisimple Lie algebra. When we studied the deformation function of high-rank Lie algebra, we found two different way which were different generalization of the case in so(3):if the deformation function depend on the Cartan subalgebra, the deformed algebra would get q-ommutator, this kind of deformation could be viewed as the generalization of Rq,r, while if the deformation function depend on the Casimir operator of non-deformed subalgebra, the commutation results would contain structure functions, such type could be viewed as the generalization of R(v).At the same time, through the study on the deformation mapping of so(3),so(4), we found the origin of some multi-parameter non-linear deformed so(3),so(4) algebra, and we formally discussed the connection between the origin of deformed algebra and the deformation mapping of Lie algebra.