在這篇論文我們會透過實四維空間中的 Moyal 乘積經由 U(1)作用下來定義 一個實三維空間中的 ∗ 乘積。這個我們定的 ∗ 乘積與既有在 su(2)的對偶李代數 上的 Gutt 乘積同為實三維空間中的 ∗ 乘積在算數上有不同結果。我們將證明他 們是等價的並且構造出等價算子。

In this thesis, we define a star product on R^3 which is associated to the Moyal product on R^4. We also recall the Gutt product defined on the dual of a Lie algebra. Under the isomorphism between the dual of su(2) and R^3, we construct an equivalence operator between these two star products.

1 Introduction . . . . . . . . . . . .1
2 Star product on R^3 . . . . . . . . . . . .3
3 Gutt prodcut. . . . . . . . . . . . 7
4 Equivalence . . . . . . . . . . . .10
4.1 Equivalence in the homological language . . . . . . . . . . . . 10
4.2 Weyl product ........................... 12
4.3 Construction of the equivalence operator . . . . . . . . . . . . 17
A Smoothness of the star product . . . . . . . . . . . .25

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