The Gelfand-Kirillov Dimension Of Quantized Enveloping Algebra Of Type G₂

In general, for noncommutative algebras, the classical Krull dimension is not a very useful tool, because it is defined by using chains of two-sided prime ideals. Fortunately, for finitely generated k-algebras, we have the Gelfand-Krillov dimension which is a far better invariant, and which, moreover, coincides with the Krull dimension in the commutative case. The Gelfand-Krillov dimension measures 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 finitely generated infinite dimensional algebras. But in general, the Gelfand-Kirillov dimension is extremely hard to compute.In [1], the authors give a detailed discussion of the Gelfand-Kirillov dimension of finitely generated k-algebras and modules over them, and also introduce an algorithm to compute the Gelfand-Kirillov dimension of several classical and non-classical examples (in the context of enveloping algebras and quantum groups).In this paper by using the method in [1] and the Grobner-Shirshov basis given in [2], we compute the Gelfand-Kirillov dimension GKdim(Uq(G2)) of the quantized enveloping algebra Uq{G2).