Skew-commutator relations and Gröbner-Shirshov basis of quantum group of type F₄

Abstract

We give a Gröbner-Shirshov basis of quantum group of type F₄ by using the Ringel-Hall algebra approach. We compute all skew-commutator relations between the isoclasses of indecomposable representations of Ringel-Hall algebras of type F₄ 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 skewcommutator relations; instead, we compute the ‘easier’ skew-commutator relations which correspond to those exact sequences with middle term indecomposable or the split exact sequences first, then ‘deduce’ others from these ‘easier’ ones and this in turn gives Hall polynomials as a byproduct. Then using the composition-diamond lemma prove that the set of these relations constitute a minimal Gröbner-Shirshov basis of the positive part of the quantum group of type F₄. Dually, we get a Gröbner-Shirshov basis of the negative part of the quantum group of type F₄. And finally, we give a Gröbner-Shirshov basis for the whole quantum group of type F₄.

Keywords

Ringel-Hall algebra / root vector / indecomposable module / Gröbner-Shirshov basis

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