Symmetry in Noncommutative Geometry and its String Theory Origin

研究结果总结：显然，在我们提出并应用于由Hopf代数的扭转元素表示的弦世界单量子化形式下，作为弦理论对称性的Hopf代数的结构和由于非平凡B场背景的存在而引起的算符代数的形变结构可以统一起来，因为算符代数的形变是由于量子化而发生的，而B场背景的形变是在同一水平的单一形变中构造的。这意味着在弦水平上，包含弦理论中的微分同胚的Hopf代数被量子化和B-场背景变形为包含变形的微分同胚的量子群代数。

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.