Grobner-shirshov Basis Of The Quantum Group Of Type E₇

The reduction is an important problem in algebra and the Grobner-Shirshov basistheory provides a solution to the reduction problem for various kinds of algebras. But untilnow, there are only a few results about the Grobner-Shirshov basis of the quantum groups.For the quantum groups of type An,D4,B2,E6were given by Bokut, Malcolmson, Kong,Ren,Obul and Yunus, respectively. So it is meaningful to work out a Grobner-Shirshovbasis of quantum group of other types.For constructing a PBW type basis for quantum groups, Ringel constucted a gen-erating 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 algebra and the positive part of quantum group of same type.In this paper, using the relations and the isomorphism mentioned above, we gave all skewcommutator relations for the positive part of quantum group of type E7, then prove thatthe set of these relations is closed under composition. So the set is a Grobner-Shirshovbasis of the positive part of quantum group of type E7. Dually, we get a Grobner-Shirshovbasis of the negative part of quantum group of type E7and by using the result in [12] wegave a Grobner-Shirshov basis of the quantum group of type E7.