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What are the automorphisms of $SL_n$ as an algebraic variety?

In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of $SL_n$ regarded as an algebraic variety over $k$. Assume that $\tau$ takes the unit element $e$ of $G$ to itself. Is it true that $\tau$ is an automorphism of $SL_n$ as an algebraic group over $k$?

EDIT: the answer is NO, see the comment of ACL. The edited question: Is it true that $\tau$ is an automorphism of $SL_n$ as an algebraic group over $k$ or the composition of an automorphism of $SL_n$ as an algebraic group with the inversion?

3 added 252 characters in body

What are the automorphisms of $SL_n$ as an algebraic variety?

In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of $SL_n$ regarded as an algebraic variety over $k$. Assume that $\tau$ takes the unit element $e$ of $G$ to itself. Is it true that $\tau$ is an isomorphism automorphism of $SL_n$ as an algebraic group over $k$?

EDIT: the answer is NO, see the comment of ACL. The edited question: Is it true that $\tau$ is an automorphism of $SL_n$ as an algebraic group over $k$ or the composition of an automorphism of $SL_n$ as an algebraic group with the inversion?

2 added 2 characters in body

What are the automorphisms of $SL_n$ as an algebraic variety?

In other words, let $k$ be an algebraically closed field of characteristic 0 (e.g e.g., $k=\mathbb{C}$). Let $\tau$ be an automorphism of $SL_n$ regarded as an algebraic variety over $k$. Assume that $\tau$ takes the unit element $e$ of $G$ to itself. Is it true that $\tau$ is an isomorphism of $SL_n$ as an algebraic group over $k$?

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