Gr（o|¨）bner-Shirshov Basis And The Minimal Projective Resolution Of U_q⁺（G₂）

In this thesis, we will construct a minimal projective resolution of the positive part Uq(G2) of the quantized enveloping algebra Uq(G2)-Let A be a augmented algebra over a field k. When we consider many problems in homological algebra, we often need to construct a projective resolution of the trivial A-module k. It is well-known that the bar construction is easy to construct but it is too big to using for computation; and the minimal projective resolution is very good to using for computation, but, in general, it is very difficult to construct. In order to get a new resolution such that it is easy to construct and good enough to using for computation, Anick constructed a resolution between the bar resolution and the minimal projective resolution in. Nowadays, his construction is called Anick resolution. In homological algebra, we often use Acink resolution to compute some homological invariants of the trivial module k. This is an important role of Anick resolution.It is well-known that we can define the extension of two modules easily, but it is difficult to compute the extensions of two given modules. There is a close relationship between the projective resolution and the computation of the extension. So construct a projective resolution (or Anick resolution) by using the Grobner-Shirshov basis theory first, and then computing the extension by using this projective resolution is an meaningful work. In this thesis, on the basis of the Grobner-Shirshov basis of quantized enveloping algebra Uq{G2) given in, we construct an Ancik resolution of the positive part Uq+(G-2) of the quantized enveloping algebra Uq(G2). Then, by using the graded Jacobson radical of Uq+(G2), we "optimize" the Anick resolution into a minimal projective resolution. Finally, as an application, we compute the global dimension of Uq+(G2).