The Lie algebra of derivations of a current Lie algebra

OCHOA ARANGO, JESÚS ALONSO; ROJAS, NADINA

COMMUNICATIONS IN ALGEBRA

TAYLOR & FRANCIS INC

Año: 2019 vol. 48 p. 625 - 637

0092-7872

egin{abstract}Let $mathbb{K}$ be a field of characteristic zero, $mathfrak{g}$ 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 derivation Lie algebraof the current Lie algebra$mathfrak{g}_A= mathfrak{g} otimes_{K} A$, denoted by$der(mathfrak{g}_A)$. Furthermore, we obtain the Levi decomposition of$der(mathfrak{g}_A)$.As a consequence of the last result, if$mathfrak{h}_m$ is the Heisenberg Lie algebra of dimension$2 m + 1$, we obtain a faithful representation of$der(mathfrak{h}_{m,k})$ of the currenttruncated Heisenberg Lie algebra$mathfrak{h}_{m,k}= mathfrak{h}_m otimes K[t]/ (t^{k + 1})$ for allpositive integer$k$. %