Gr(?)bner-Shirshov Basis Of Quantum Group Of Type G₂

The Gr¨obner basis theory for commutative algebras was introduced by Buchberger andit provides a solution to the reduction problem for commutative algebras. It gives analgorithm of computing a set of generators for a given ideal of a commutative ring whichcan be used to determine the reduced elements with respect to the relations given by theideal. Bergman generalized the Groebner basis theory to associative algebras by providingthe Diamond Lemma.The Gr¨obner basis theory for Lie algebras was developed by Shirshov. The key ingre-dient of the theory is the so-called Composition Lemma which characterizes the leadingterms of elements in the given ideal. Bokut noticed that Shirshov's method works forassociative algebras as well, and for this reason, Shirshov's theory for Lie algebras andtheir universal enveloping algebras is called the Gr¨obner-Shirshov basis theory. Bokut andMalcolmson developed the theory of Gr¨obner-Shirshov basis for the quantum envelopingalgebras, or the so-called quantum groups and explicitly constructed the Gr¨obner-Shirshovbasis for the quantum groups of type An for q8 = 1. The Gr¨obner-Shirshov basis for quan-tum groups of other types is not known.For constructing a PBW type basis for quantum groups, Ringel constucted a gener-ating sequence for Ringel-Hall algebras and some skew-commutator relations for thesegenerators using the Auslander-reiten theory. On the other hand, there is an isomor-phism between Ringel-Hall algebras and quantum groups. In this dissertation, using therelations and the isomorphism mentioned above, we got some skew-commutator relationsfor the positive part of quantum group of type G2, then to prove the set of these relationsis closed under compositions. So they compose a Gr¨obner-Shirshov basis of the positivepart of quantum group of type G2. Dually, we got a Gr¨obner-Shirshov basis of the nega-tive part of quantum group of type G2. Finally, we give a Gr¨obner-Shirshov basis of thequantum group of type G2.