Gr（?）bner-Shirshov Basis Of Quantum Group Of Type F₄

In algebra presenting an algebraic structure by a set of generators and a set of re-lations between them is very common now and is a very efective method to define analgebraic structure. Here the"membership"problem is a fundamental problem and of-ten called the reduction problem. The Gr（?）bner-Shirshov basis theory provides a solutionto the reduction problem for various kinds of algebras. Being an associative algebra, the"membership"problem in quantum groups can also be solved by using the Gr（?）bner-Shirshov basis theory. Up to our knowledge, at the moment there are few results aboutthis. For the quantum groups of type A_n, B₂, G₂, D₄, E6_the Gr（?）bner-Shirshov basiswere given by Bokut, Malcolmson, Kong, Ren, Obul and Yunus, respectively. But for thequantum group of type F₄Gr（?）bner-Shirshov basis is not given, yet.For constructing a PBW type basis for quantum groups, Ringel constructed a gen-erating sequence for Ringel-Hall algebras and some skew commutator relations for thesegenerators using the representation theory of algebras, precisely, the Auslander-Reitentheory. On the other hand, there is an isomorphism between Ringel-Hall algebra and thepositive part of quantum group of same type. In this paper, using the relations and theisomorphism mentioned above, first we compute all skew-commutator relations for thepositive part of quantum group of type F₄by using an"inductive"method. Precisely,we do not use the traditional way of computing the skew-commutative relations, thatis first compute all Hall polynomials then compute the corresponding skew-commutator relations; contrarily, we computed the"easier"skew-commutator relations which corre-sponding to those exact sequences with middle term indecomposable or the split exactsequences first, then"inductively"computed the others from these"easier"ones andthis in turn gives Hall polynomials as a byproduct. Then we prove that the set of these re- lations is closed under composition. So they constitutes a minimal Gr（?）bner-Shirshov basisof the positive part of quantum group of type F₄. Dually, we get a Gr（?）bner-Shirshov basisof the negative part of quantum group of type F₄. And finally we give a Gr（?）bner-Shirshovbasis for the whole quantum group of type F₄.