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There are nontrivial dualities and parallels between polynomial algebras and the Grassmann algebras (e.g., the Grassmann algebras are dual of polynomial algebras as quadratic algebras). This paper is an attempt to look at the Grassmann algebras at the angle of the Jacobian conjecture for polynomial algebras (which is the question/conjecture about the $$ Jacobian set– the set of all algebra endomorphisms of a polynomial algebra with the Jacobian $1$ – the Jacobian...

Let $L_n=K[x_1^{\pm 1},...,x_n^{\pm 1}]$ be a Laurent polynomial algebra over a field $K$ of characteristic zero, $W_n:={\mathrm{Der}}_K\left(L_n\right)$ the Lie algebra of $K$-derivations of the algebra $L_n$, the so-called Witt Lie algebra, and let $\mathrm{Vir}$ be the Virasoro Lie algebra which is a $1$-dimensional central extension of the Witt Lie algebra. The Lie algebras $W_n$ and $\mathrm{Vir}$ are infinite dimensional Lie algebras. We prove that the following isomorphisms of the groups of Lie algebra automorphisms hold: ${\mathrm{Aut}}_{\mathrm{Lie}}\left(\mathrm{Vir}\right)\simeq {\mathrm{Aut}}_{\mathrm{Lie}}\left(W_1\right)\simeq \{\pm 1\}⋉K^{*}$, and give a short proof that ${\mathrm{Aut}}_{\mathrm{Lie}}\left(W_n\right)\simeq {\mathrm{Aut}}_{\mathrm{K}-\mathrm{alg}}\left(L_n\right)\simeq {\mathrm{GL}}_n\left(\mathbb{Z}\right)⋉K^{*n}$.