Symmetry in Noncommutative Geometry and its String Theory Origin

研究结果总结：结果表明，Hopf代数作为弦理论的对称结构和非平凡b场背景的存在引起的算子代数的变形结构可以统一在“扭转量子化”下。我们提出了“扭转量子化”，并将其应用于Hopf代数中扭转元素的弦世界表量子化公式中。将算子代数因量化而产生的变形和b场背景的变形构造在同一层次的单次变形中。这意味着在弦水平上，包含弦理论中微分同构的Hopf代数，在量子化和b场背景的双重作用下，被变形为包含变形微分同构的量子群代数。

Summary of the results of the research : It became apparent that the structure of the Hopf algebra as a symmetry of the string theory and the structure of the deformation of the operator algebra caused by the existence of the non-trivial B-field background can be unified under the 'twist quantization', which is proposed by us and applied for the formulation of the string world sheet quantization by the twist element of the Hopf algebra, in the way that the deformation of operator algebra occurs due to the quantization and the deformation by B-field background are constructed in the single deformation of the same level. It means that in the string level, the Hopf algebra which contains the diffeomorphism in string theory, is deformed by both quantization and B-field background into the quantum group algebra including the deformed diffeomorphism.