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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
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