The Lie algebra of derivations of a current Lie algebra

Abstract

Let be a field of characteristic zero, be a finite dimensional -Lie algebra and let A be a finite dimensional associative and commutative -algebra with unit. We describe the structure of the Lie algebra of derivations of the current Lie algebra , denoted by . Furthermore, we obtain the Levi decomposition of . As a consequence of the last result, if is the Heisenberg Lie algebra of dimension , we obtain a faithful representation of of the current truncated Heisenberg Lie algebra for all positive integer k.