The Skew-commutator Relations and Gröbner-Shirshov Bases of Quantum Group of Type C3

Abstract

In this paper, we give a Grobner-Shirshov basis of quantum group of type ℂ3 by using the Ringel-Hall algebra approach. For this, first we compute all skew-commutator relations between the isoclasses of indecomposable reprersentations of Ringel-Hall algebras of type ℂ3 by using an “inductive” method. Precisely, we do not use the traditional way of computing the skew-commutative relations, that is first compute all Hall polynomials then compute the corresponding skew-commutator relations; contrarily, we compute the “easier” skew-commutator relations which corresponding to those exact sequences with middile term indecomposable or the split exact sequences first, then “inductive” others from these “easier” ones and this in turn gives Hall polynomials as a byproduct. Then we prove that the set of these relations is closed under composition. So they constitutes a minimal Grobner-Shirshov basis of the positive part of quantum group of type ℂ3. Dually, we get a Grobner-Shirshov basis of the negative part of quantum group of type C3. And finally we give a Grobner-Shirshov basis for the whole quantum group of type ℂ3.