Bracket width of current Lie algebras

The length of an element z of a Lie algebra L is defined as the smallest number s needed to represent z as a sum of s brackets. The bracket width of L is defined as supremum of the lengths of its elements. Given a finite-dimensional simple Lie algebra g over an algebraically closed field k of characteristic zero, we study the bracket width of current Lie algebras L = g ⊗ A . We show that for an arbitrary A the bracket width is at most 2. For A = k [ [ t ] ] and A = k [ t ] we compute the bracket width for algebras isomorphic to sl n and sp 2 n .