Root vectors in quantum groups

Abstract

The root vector defined in [L1-2] plays a fundamental role in quantum group theory. However even for some simple questions, such as the number of root vectors, the relations between root vectors, etc., we know little. There are several formulas concerned with the coproducts of root vectors in [AJS, KR, LS]. These formulas are important, but for many purposes it is inconvenient to use them, because these formulas in fact are not formulas in quantum group but in certain completions of quantum groups and are involved products of infinite sums. It seems also no explicit formula for the antipode of a root vector at hand. The arguments in the remarkable work [AJS] show that for a quantum group it is valuable to have formulas (in the quantum group) of coproducts and antipodes of root vectors. Therefore it is necessary to understand root vectors further. This paper is motivated by the work [AJS]. In this paper, we prove that for a root vector, certain presentation is unique (see Theorem 4.4 (ii) and Lemma 4.2). The uniqueness of the presentation is useful to prove that root vectors are linearly independent and can be used to get some explicit formulas concerned with root vectors, for example, coproduct formulas. The uniqueness of the presentation also can be used to count root vectors. Other known presentations of root vectors are not effective for these purposes. In this paper we also prove that for a root vector there exists a unique shortest element (in a reasonable sense) in the Weyl group attached to it (see Theorem 4.4 (iii) and Proposition 2.12 (i)). Using Theorem 4.4 (ii) and Proposition 4.8 we get an explicit formula for the coproduct of a root vector in a quantum group of type A. Unfortunately it is not easy to get such a formula for other types in general. The contents of the paper are as follows. In section 1 we recall some basic definitions and fix notations. We also list some formulas for later uses. In section 2 we prove some results about root systems. Some of them are needed in sections 4 and 5. For the possible generalizations of the results in section 4, we also consider infinite root systems. In section 3 we give several lemmas which are important for our proof of the main result in technique. Lemma 3.2 is originally proved for type A, D, E by Lusztig in [L3] based on the relations between quantum groups and