多参数非线性变形so(3)，so(4)代数的表示、实现及其应用
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摘要
对称性在人们对世界的认识当中始终起着不可或缺的作用,而研究这一性质的数学工具群论也是近代数学中的一个重要分支。在上世纪80年代中期,量子群在应用反散射方法研究量子场论和统计物理中的精确可解模型中被人们提出,在此之后,各式各样的变形代数变成了物理学家们研究的热点之一。这些变形代数的不可分表示,不可约表示和振子实现等等问题有着重要的意义。
本文讨论了多参数非线性变形so(3)代数Rq,r(v)以及多参数非线性变形so(4)代数Rq,q'r,r1(v,v'),和非线性变形代数so(4)F的结构,表示和玻色子实现,并对变形映射中的变形函数和对易结果中的结构函数之间的关联进行了讨论。
一方面,本文采用类似Cartan-Weyl基的办法研究了Rq,r(v)和Rq,q',r,r'(v,v')的不可约表示,用主表示和商空间上的诱导表示对它们的不可分表示进行了研究,通过主表示讨论了各种不变子空间上的子表示及其对应的商空间上的诱导表示,并给出了与主表示、商空间上的诱导表示相应的所需不同数目的Dyson型玻色子实现。采用推广的Schwinger玻色子实现,详细计算了多参数非线性变形so(3)代数的单玻色子实现、单逆玻色子实现以及用两对玻色子(一对玻色子、一对逆玻色子,两对逆玻色子)给出的双玻色子实现,这些实现都是Hostein-Primakoff型的。通过这些玻色子实现的具体形式,本文对多参数非线性变形so(3)代数的极式分解作了讨论,发现极式分解的角向部分在变形下总是保持不变。最后讨论了非线性变形so(3)代数在二维可积系统中的应用。
另一方面,本文讨论了变形代数与Lie代数的关联关系。讨论了两个Rq,r(v)“耦合”时,相互之间的对易关系应该遵循的规则,并将结论推广到了多个变形代数相互“耦合”,给出了此时应该遵循的“耦合”规则,通过这种方式得到的变形代数类似于半单Lie代数；将so(3)到,Rq,r(v)的变形映射向高维Lie代数进行推广,得到了两种不同的推广方式：变形函数是单Lie代数的Cartan子代数的函数,变形后的代数一般具有q-变形的对易方式,可以看作是Rq,r%；r向高维的推广,变形函数是单Lie代数的子代数的Casimir算符的函数,变形后的代数一般具有包含结构函数的对易结果,可以看作R(v)向高维的推广。
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.
本文讨论了多参数非线性变形so(3)代数Rq,r(v)以及多参数非线性变形so(4)代数Rq,q'r,r1(v,v'),和非线性变形代数so(4)F的结构,表示和玻色子实现,并对变形映射中的变形函数和对易结果中的结构函数之间的关联进行了讨论。
一方面,本文采用类似Cartan-Weyl基的办法研究了Rq,r(v)和Rq,q',r,r'(v,v')的不可约表示,用主表示和商空间上的诱导表示对它们的不可分表示进行了研究,通过主表示讨论了各种不变子空间上的子表示及其对应的商空间上的诱导表示,并给出了与主表示、商空间上的诱导表示相应的所需不同数目的Dyson型玻色子实现。采用推广的Schwinger玻色子实现,详细计算了多参数非线性变形so(3)代数的单玻色子实现、单逆玻色子实现以及用两对玻色子(一对玻色子、一对逆玻色子,两对逆玻色子)给出的双玻色子实现,这些实现都是Hostein-Primakoff型的。通过这些玻色子实现的具体形式,本文对多参数非线性变形so(3)代数的极式分解作了讨论,发现极式分解的角向部分在变形下总是保持不变。最后讨论了非线性变形so(3)代数在二维可积系统中的应用。
另一方面,本文讨论了变形代数与Lie代数的关联关系。讨论了两个Rq,r(v)“耦合”时,相互之间的对易关系应该遵循的规则,并将结论推广到了多个变形代数相互“耦合”,给出了此时应该遵循的“耦合”规则,通过这种方式得到的变形代数类似于半单Lie代数；将so(3)到,Rq,r(v)的变形映射向高维Lie代数进行推广,得到了两种不同的推广方式：变形函数是单Lie代数的Cartan子代数的函数,变形后的代数一般具有q-变形的对易方式,可以看作是Rq,r%；r向高维的推广,变形函数是单Lie代数的子代数的Casimir算符的函数,变形后的代数一般具有包含结构函数的对易结果,可以看作R(v)向高维的推广。
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.
引文
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