Root strings with two consecutive real roots
Yuly Billig, Arturo Pianzola · Tohoku Mathematical Journal · 1995
For a Kac-Moody Lie algebra we study pairs of real roots the sum of which is a real root.More precisely, we study in which way the existence of such pair of roots determines the existence of certain subroot system within the root system. Introduction.The study of pairs of real roots {y u γ 2 } of a Kac-Moody Lie algebra g whose sum is a real root was initiated by Morita in [3] and [4] (though [4] contains a mistake as pointed out in [5]).Morita put this information to good use to derive information about K 2 in the case of Kac-Moody groups.Morita looks at the case when = -1 and =a where a= 1, 2, 3. (There are also some results if a>3 but only under some strong assumptions on the Cartan matrix.)Morita assumes that y ί9 γ 2 are positive and that y^-y 2 is not a root (a Morita pair in our terminology).His key observation is that a determines the existence of certain entries in the corresponding Cartan matrix A of g (and hence that A somehow sheds information about the existence of such pairs of roots).Our own interest in this problem came out from trying to understand the nilpotency degree of certain subalgebras of g (Conjecture 1 below).We will deal with a above arbitrary and show how a determines a sequence of entries in A with certain properties. Notation and some basic facts about root systems of Kac-Moody Lie algebra.We begin by recalling some well-known objects related to Kac-Moody Lie algebras.Our running reference for this will be [9, Ch. 4,5].Most of this material is also covered in [1]. A=(A ij ) ijeI will throughout denote a generalizedCartan matrix.(The index set / is allowed to be infinite.)Let (I), 77, Π v ) be a realization of A. Thus As usual we set W= (r t I /e /> where r t : = r α ., re A = WΠ, 1991