{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:MLRNDUTDQ2TTB723FXNTSLUZAY","short_pith_number":"pith:MLRNDUTD","canonical_record":{"source":{"id":"1005.1900","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-11T18:06:31Z","cross_cats_sorted":[],"title_canon_sha256":"7da17dc4bce10dc193a363d71c8c8abcd4cd45e8b357c04f58242d45decff43e","abstract_canon_sha256":"6fadd1531ae5fc9cb8db83a67acd4e9d87ce5f06ca0e2454f4b305b9caf5f192"},"schema_version":"1.0"},"canonical_sha256":"62e2d1d26386a730ff5b2ddb392e9906187c26a90c22476cd58402f6db915411","source":{"kind":"arxiv","id":"1005.1900","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.1900","created_at":"2026-05-18T04:09:53Z"},{"alias_kind":"arxiv_version","alias_value":"1005.1900v6","created_at":"2026-05-18T04:09:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.1900","created_at":"2026-05-18T04:09:53Z"},{"alias_kind":"pith_short_12","alias_value":"MLRNDUTDQ2TT","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MLRNDUTDQ2TTB723","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MLRNDUTD","created_at":"2026-05-18T12:26:10Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:MLRNDUTDQ2TTB723FXNTSLUZAY","target":"record","payload":{"canonical_record":{"source":{"id":"1005.1900","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-11T18:06:31Z","cross_cats_sorted":[],"title_canon_sha256":"7da17dc4bce10dc193a363d71c8c8abcd4cd45e8b357c04f58242d45decff43e","abstract_canon_sha256":"6fadd1531ae5fc9cb8db83a67acd4e9d87ce5f06ca0e2454f4b305b9caf5f192"},"schema_version":"1.0"},"canonical_sha256":"62e2d1d26386a730ff5b2ddb392e9906187c26a90c22476cd58402f6db915411","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:53.803930Z","signature_b64":"MPK2YmjH5Hte8Oaz0erEUt9KTnGDi3JDDwD4mJxN+de/8xOXie1xaqn4cLwrLl7P6ggy7arM7++dqPe4dm/PAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62e2d1d26386a730ff5b2ddb392e9906187c26a90c22476cd58402f6db915411","last_reissued_at":"2026-05-18T04:09:53.803262Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:53.803262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1005.1900","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:09:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZRaYnp6cyJarX/XRpNbJUdBHocgXwZChTckcDLxP/WAd5Cu29tMLmG61Jmtu082Y0MNZIDXFY3WLqWEa2P1HDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T08:47:29.427124Z"},"content_sha256":"3fc9e4ec9a59bf94439cfa1583a316e461b1a9bcb9dc861f8d473f16af594a9d","schema_version":"1.0","event_id":"sha256:3fc9e4ec9a59bf94439cfa1583a316e461b1a9bcb9dc861f8d473f16af594a9d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:MLRNDUTDQ2TTB723FXNTSLUZAY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The graded structure of Leavitt Path algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.RA","authors_text":"R. Hazrat","submitted_at":"2010-05-11T18:06:31Z","abstract_excerpt":"Leavitt path algebras associate to directed graphs a $\\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras $L(1,k)$. In this note, we first study this $\\mathbb Z$-grading and characterize the ($\\mathbb Z$-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, $C_n$-comet and multi-headed graphs. The last two type are examples of graphs whose Leavitt path algebras are strongly graded. We characterize Leavitt path algebras which are strongly graded, along the way obtaining classes of algebras which are group rings or crossed-products. In an "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1900","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:09:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NruC0KfjqyzIVOoGCAjgrFVVg26A0CFZ4Yf45VLjY5ki5TmXumaPhbcpPqO/IDrfTg1Ll/pY1OQuXd3294E4CQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T08:47:29.427477Z"},"content_sha256":"642c8664c15e81733f2c67e7be978429b4191ffb6107dfbaed65ae5e8a5e4a0d","schema_version":"1.0","event_id":"sha256:642c8664c15e81733f2c67e7be978429b4191ffb6107dfbaed65ae5e8a5e4a0d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MLRNDUTDQ2TTB723FXNTSLUZAY/bundle.json","state_url":"https://pith.science/pith/MLRNDUTDQ2TTB723FXNTSLUZAY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MLRNDUTDQ2TTB723FXNTSLUZAY/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-27T08:47:29Z","links":{"resolver":"https://pith.science/pith/MLRNDUTDQ2TTB723FXNTSLUZAY","bundle":"https://pith.science/pith/MLRNDUTDQ2TTB723FXNTSLUZAY/bundle.json","state":"https://pith.science/pith/MLRNDUTDQ2TTB723FXNTSLUZAY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MLRNDUTDQ2TTB723FXNTSLUZAY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:MLRNDUTDQ2TTB723FXNTSLUZAY","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6fadd1531ae5fc9cb8db83a67acd4e9d87ce5f06ca0e2454f4b305b9caf5f192","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-11T18:06:31Z","title_canon_sha256":"7da17dc4bce10dc193a363d71c8c8abcd4cd45e8b357c04f58242d45decff43e"},"schema_version":"1.0","source":{"id":"1005.1900","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1005.1900","created_at":"2026-05-18T04:09:53Z"},{"alias_kind":"arxiv_version","alias_value":"1005.1900v6","created_at":"2026-05-18T04:09:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1005.1900","created_at":"2026-05-18T04:09:53Z"},{"alias_kind":"pith_short_12","alias_value":"MLRNDUTDQ2TT","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_16","alias_value":"MLRNDUTDQ2TTB723","created_at":"2026-05-18T12:26:10Z"},{"alias_kind":"pith_short_8","alias_value":"MLRNDUTD","created_at":"2026-05-18T12:26:10Z"}],"graph_snapshots":[{"event_id":"sha256:642c8664c15e81733f2c67e7be978429b4191ffb6107dfbaed65ae5e8a5e4a0d","target":"graph","created_at":"2026-05-18T04:09:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Leavitt path algebras associate to directed graphs a $\\mathbb Z$-graded algebra and in their simplest form recover the Leavitt algebras $L(1,k)$. In this note, we first study this $\\mathbb Z$-grading and characterize the ($\\mathbb Z$-graded) structure of Leavitt path algebras, associated to finite acyclic graphs, $C_n$-comet and multi-headed graphs. The last two type are examples of graphs whose Leavitt path algebras are strongly graded. We characterize Leavitt path algebras which are strongly graded, along the way obtaining classes of algebras which are group rings or crossed-products. In an ","authors_text":"R. Hazrat","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-11T18:06:31Z","title":"The graded structure of Leavitt Path algebras"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1005.1900","kind":"arxiv","version":6},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3fc9e4ec9a59bf94439cfa1583a316e461b1a9bcb9dc861f8d473f16af594a9d","target":"record","created_at":"2026-05-18T04:09:53Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6fadd1531ae5fc9cb8db83a67acd4e9d87ce5f06ca0e2454f4b305b9caf5f192","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.RA","submitted_at":"2010-05-11T18:06:31Z","title_canon_sha256":"7da17dc4bce10dc193a363d71c8c8abcd4cd45e8b357c04f58242d45decff43e"},"schema_version":"1.0","source":{"id":"1005.1900","kind":"arxiv","version":6}},"canonical_sha256":"62e2d1d26386a730ff5b2ddb392e9906187c26a90c22476cd58402f6db915411","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62e2d1d26386a730ff5b2ddb392e9906187c26a90c22476cd58402f6db915411","first_computed_at":"2026-05-18T04:09:53.803262Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:53.803262Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MPK2YmjH5Hte8Oaz0erEUt9KTnGDi3JDDwD4mJxN+de/8xOXie1xaqn4cLwrLl7P6ggy7arM7++dqPe4dm/PAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:53.803930Z","signed_message":"canonical_sha256_bytes"},"source_id":"1005.1900","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3fc9e4ec9a59bf94439cfa1583a316e461b1a9bcb9dc861f8d473f16af594a9d","sha256:642c8664c15e81733f2c67e7be978429b4191ffb6107dfbaed65ae5e8a5e4a0d"],"state_sha256":"d8b00dc555922b0295e9800d0e0f48983c882a48e44b75795e052bbe319883ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/JwTEjwu8VCjVmnQdvcmEuq+PeNFXRkcgBh6h2tVqr3m/VSRVV019PVMFLS4zUEB7UH1SAop36xuIvIvnMNEDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T08:47:29.429302Z","bundle_sha256":"bbc55e1d2c2d17cb7b2f481e670a42b43afab427d7b7bfe6aeb9399e77576193"}}