Automorphisms of a Weyl-type algebra I
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Abstract
Every non-associative algebra $L${\hskip-0.015cm} corresponds to its symmetric semi-Lie algebra $L_{[,]}$ with respect to its commutator. It is an interesting problem whether the equality {\tiny$Aut_{non}(L){\hskip-0.03cm}={\hskip-0.03cm}Aut_{semi-Lie}(L)$} holds or not \cite{Al}, \cite{San}. We find the non-associative algebra automorphism groups $Aut_{non}$ $(\overline {WN_{0,0,1}}_ {[0,1,r_1,\ldots ,r_p]} )$ and $Aut_{semi-Lie}$ $(\overline {WN_{0,0,1}}_ {[0,1,r_1,\ldots ,r_p]} )$, where every automorphism of the automorphism groups is the composition of elementary maps \cite{CN}, \cite{CN1}, \cite{N}, \cite{Nam2}, \cite{NC}, \cite{NKW}, \cite{NW}. The results of the paper show that the ${\mathbf F}$-algebra automorphism groups of a polynomial ring and its Laurent extension make easy to find the automorphism groups of the algebras in the paper.
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