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We also show that every finite group and every torus appears as $Aut(X)$ for a suitable affine variety $X$, but that $Aut(X)$ cannot be isomorphic to a semisimple group. In fact, if $Aut(X)$ is finite dimensional and $X$ not isomorphic to the affine line $A^1$, then the connected component $Aut(X)^0$ is a torus.\n  Concerning the structure of $Aut(A^n)$ we p"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.06362","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2015-01-26T12:24:50Z","cross_cats_sorted":["math.GR"],"title_canon_sha256":"c93c9a8b8fd4bf93d6b57b4a6406ffccb21355d03e8bb9341c190ffdaedf3b40","abstract_canon_sha256":"ccc14b0177533ad258fd933da41aff4763e7cdd100c6f142292c8ac4d7a7c495"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:41.681343Z","signature_b64":"rPMplnYqwWycmdzzzLib3kskZQSm09UQI9GPnqCC7XXW7HNE5p39yLbAQZlmMFyTHCnHWl2eAcg25Xi2aQO2Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6973e7b9a1e7aa50a197b7fd434061bcd3e8caa1e4642b3966276f7bcb0755cb","last_reissued_at":"2026-05-18T02:28:41.680846Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:41.680846Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Automorphism Groups of Affine Varieties and a Characterization of Affine n-Space","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GR"],"primary_cat":"math.AG","authors_text":"Hanspeter Kraft","submitted_at":"2015-01-26T12:24:50Z","abstract_excerpt":"We show that the automorphism group of affine n-space $A^n$ determines $A^n$ up to isomorphism: If $X$ is a connected affine variety such that $Aut(X)$ is isomorphic to $Aut(A^n)$ as ind-groups, then $X$ is isomorphic to $A^n$ as a variety. 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