The Lie algebra of derivations of a current Lie algebra

Abstract

Let K be a field of characteristic zero, g be a finite dimensional K-Lie algebra and let A be a finite dimensional associative and commutative K-algebra with unit. We describe the structure of the Lie algebra of derivations of the current Lie algebra (Formula presented.), denoted by Der(gA). Furthermore, we obtain the Levi decomposition of Der(gA. As a consequence of the last result, if hm is the Heisenberg Lie algebra of dimension 2m + 1, we obtain a faithful representation of Der(hm,k of the current truncated Heisenberg Lie algebra (Formula presented.) for all positive integer k.