| Canonical basis of quantum group plays an important role in quantum group and its repre-sentation theory, and is a main tool to study quantum group and its representation. But, it is not easy to determine Canonical basis elements of quantum group, Canonical basis elements are explicitly described only for few quantum groups so far. This paper is devoted to give some polynomial elements with three-dimensional support in Canonical basis of quantum group for type A4.The paper is divided into four chapters. Chapter 1 explains the background of quantum group and its Canonical basis, and the source of problems in this paper. Chapter 2 recalls the definition of quantum group and the algebraic definition of Canonical basis of quantum group. Chapter 3 sum-marizes the known results about Canonical basis of quantum group, including type An, Bn, G2, Dn. Chapter 4 gives the main theorem and its proof. |
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