The Minimal Projective Resolution Of U⁺_q（D₄） And The Gelfand-Kirillov Dimension Of U_q（D₄）

In this thesis we consider two problems. Firstly, we constructed the ?rst three steps of a minimal projective resolution of the positive part U+q(D4) of the quantized enveloping algebra Uq(D4) of type D4. Let k be a ?eld and let A be an associative augmented kalgebra. When we consider many algebraic problems, we need to have projective resolution of k as an A-module. It is well-known that the bar resolution is easy to get but it is too large for use, and the minimal projective resolution, on the other hand, is very good for use but it is often very expensive. So in [1], Anick constructed an intermediate resolution which is suitable for many problems in homological algebra and nowadays this resolution called the Anick resolution. One of the important role of the Anick resolution in homological algebra is that we can use it to compute a projective resolution of the trivial module k. The main di?culty in constructing the Anick resolution is the computation of the set of chains.In the ?rst part of this thesis, ?rst by using the Gr¨obner-Shirshov basis for the positive part U+q(D4) of the quantized enveloping algebra Uq(D4) given in [2], we constructed the?rst three steps of the Anick resolution, then using the properties of the positively graded algebra, we “optimized” the above three steps of the Anick resolution into the ?rst three steps of the minimal projective resolution. Furthermore, by using the number of the elements in the set of n-chains, we give an upper bound of the global dimension of U+q(D4).Secondly, we computed the Gelfand-Kirillov dimension of the quantized enveloping algebra Uq(D4). In general, for noncommutative algebras, the classical Krull dimension is not a very useful tool, because it is de?ned by using chains of two-sided prime ideals.Fortunately, for ?nitely generated k-algebras, we have the Gelfand-Kirillov dimension which is a far better invariant, and which, moreover, coincides with the Krull dimension in the commutative case. The Gelfand-Kirillov dimension measures the the asymptotic rate of growth of algebras and provides important structural information, so this invariant has become one of the standard tools in the study of ?nitely generated in?nite dimensional algebras. But in general, the Gelfand-Kirillov dimension is extremely hard to compute.In the second part of this thesis, by using the method given in [3] and the Gr¨obnerShirshov basis given in [2], we compute the Gelfand-Kirillov dimension GKdim(Uq(D4))of the quantized enveloping algebra Uq(D4).