A plactic algebra for semisimple Lie algebras P. Littelmann ∗ December 17, 2005 1 Introduction A plactic algebra can be thought of as a (non-commutative) model for the representation ring of a semisimple Lie algebra g. This algebra was introduced by Lascoux and Sch¨ utzenberger in [13], [18] in order to study the representation theory of GL n (C) and S n . This new tool enabled them for example to give the first rigouros proof of the Littlewood-Richardson rule to determine the decomposition of tensor products into direct sums of irreducible representations. Using a case by case analysis, such a plactic algebra has been constructed also for some other simple groups, see [1], [8], [19], [20], [21]. Recently, two constructions of isomorphic plactic algebras have been given for symmetrisable Kac-Moody algebras. From the point of view of quantum groups, this algebra is the algebra of crystal bases ([5], [6], [7], [16], [17], [19]). The second construction realizes this algebra as the algebra ZP of equivalence classes of paths in the space X Q of rational weights ([5], [14], [15]). For simplicity, assume that G is a simple, simply connected algebraic group. To give a description of ZP which is more in the spirit of the original work of Lascoux and Sch¨ utzenberger, let V = V λ 1 ⊕ ... ⊕ V λr be a faithful representation and let D be the associated set of L-S paths, i.e. D is a basis of the corresponding model of V in ZP . Let Z{D} be the free associative algebra generated by D. If λ = ∑ a ω ω is a dominant weight, then let |λ| denote the sum ∑ a ω . The canonical projection which maps a monomial to the concatenation: ψ : Z{D}→ ZP ,d 1 ··· d s → [d 1 ∗ ... ∗ d s ] is surjective. For N ∈ N denote by R N ⊂ Ker ψ the set R N := {d 1 ··· d s − c 1 ··· c r | ψ(d 1 ··· d s )= ψ(c 1 ··· c r ), r,s ≤ N }. Main Theorem A Fix m V ∈ N such that for every fundamental weight ω of G there exists an injection V ω  → V ⊗mω for some m ω ≤ m V . Let I ⊂ Z{D} be the two-sided ideal generated by ∗ This research has been partially supported by the Schweizerischer Nationalfonds 1