## The automorphisms of generalized Witt type Lie algebras.(English)Zbl 1038.17015

Let $$\partial=\frac{d}{dx}$$, $$F[x^{\pm}, e^{\pm x}]=F[x,x^{-1}, e^x,e^{-x}]$$, and let $$F[a_1,\ldots,a_n]$$ be a subalgebra of $$F[x^{\pm x},e^{\pm x}]$$ generated by $$a_1,\ldots, a_n$$. If $$F[a_1,\ldots,a_n]$$ is $$\partial$$-stable we put $$W[a_1,\ldots,a_n]=\{f\partial \mid f\in F[a_1,\ldots,a_n]\}$$. Then $$W[a_1,\ldots,a_n]$$ is a Lie algebra over $$F$$ with the usual product $[f\partial,g\partial]=f\partial\circ g\partial - g\partial\circ f\partial=(f(\partial g)-(\partial f)g)\partial\;\;(f,g\in F[a_1,\ldots,a_n]).$ In this paper, the authors show that the automorphism group of $$W[x,e^x]$$ is isomorphic to $$F^*\times F$$, while the automorphism group of $$F[x,e^{\pm x}]$$ is isomorphic to $$\mathbb Z/2\mathbb Z\ltimes (F^*\times F)$$.

### MSC:

 17B65 Infinite-dimensional Lie (super)algebras 17B40 Automorphisms, derivations, other operators for Lie algebras and super algebras

### Keywords:

automorphism; generalized Witt type Lie algebra
Full Text: