{"uid":"cap_1XrC9aSVL5DNQi2g1XkG7","slug":"rational-lie-bracket-universal-enveloping-algebra-pbw-normal-form-7b67240b","name":"Rational Lie Bracket Universal Enveloping Algebra PBW Normal Form Verifier","description":"Verify every ordered-monomial coefficient of a supplied enveloping-algebra expression under an explicit Lie bracket tensor.","url":"https://detect.halowerk.com/v1/F-REST-B20/rational-lie-bracket-universal-enveloping-pbw-normal-form?utm_source=zero.xyz","method":"POST","headers":{},"bodySchema":{"type":"object","properties":{"artifact":{"type":"object","required":["pbw"],"properties":{"pbw":{"type":"object","required":["data","claim"],"properties":{"data":{"type":"object","required":["brackets","terms"],"properties":{"terms":{"type":"array","items":{"type":"object","required":["word","coefficient"],"properties":{"word":{"type":"array","items":{"type":"integer","maximum":255,"minimum":0},"maxItems":6,"minItems":0},"coefficient":{"type":"integer","maximum":100,"minimum":-100}},"additionalProperties":false},"maxItems":40,"minItems":0},"brackets":{"type":"array","items":{"type":"array","items":{"type":"array","items":{"type":"integer","maximum":100,"minimum":-100},"maxItems":8,"minItems":1},"maxItems":8,"minItems":1},"maxItems":3,"minItems":1}},"additionalProperties":false},"claim":{}},"additionalProperties":false}},"additionalProperties":false}}},"responseSchema":null,"example":null,"exampleRequest":null,"tags":["x402"],"displayCostAmount":"0.003","displayCostAsset":"USDC","priceDynamic":false,"priceHint":null,"priceStatus":"priced","priceSource":"probe","requiresHandshake":false,"reviewCount":0,"rating":{"score":"0.00","successRate":"0.00","reviews":0,"stars":null,"state":"unrated"},"availabilityStatus":"unknown","priceObserved":null,"sessionDeposit":null,"pricing":{"kind":"static","summary":"$0.003/call","primary":{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"},"accepted":[{"kind":"static","protocol":"x402","network":"base","amountUsd":"0.003","per":"call","confidence":"exact"}]},"paymentMethods":[{"uid":"pm_wGH5KQxlNA4GumCtJ-DUo","protocol":"x402","methodType":"crypto","chain":"base","mode":"charge","costAmount":"0.003","costPer":"request","priority":0,"asset":"0x833589fCD6eDb6E08f4c7C32D4f71b54bdA02913","unit":"request","depositMicros":null,"planRef":null}],"brandName":null,"brandSlug":null,"brandBaseUrl":null,"brandDocsUrl":null,"whatItDoes":"Verifies the ordered-monomial (PBW basis) coefficients of a universal enveloping algebra expression against an explicitly supplied Lie bracket tensor.","exampleAgentPrompt":"Can you verify that my PBW normal form expression with terms like word [1,2,3] with coefficient 5 and word [0,2] with coefficient -3 is correct given this Lie bracket tensor [[1,0,-1],[0,1,0]]?","exampleUseCases":[{"title":"PBW coefficient correctness check","prompt":"I have a universal enveloping algebra expression with monomials: word [0,1] coefficient 2, word [1,2] coefficient -1, word [2] coefficient 4. My Lie bracket tensor is [[[1,-1],[0,1]]]. Can you verify that all these PBW ordered-monomial coefficients are correct?"},{"title":"Automated Lie algebra homework checker","prompt":"I'm building a tool to check student answers in a Lie theory course. Here are the student's claimed PBW terms — word [0,1,2] with coefficient 3 and word [1] with coefficient -2 — along with the bracket tensor [[[0,1,0],[1,0,0]]]. Please check if the claim is valid."},{"title":"Symbolic algebra system validation","prompt":"My symbolic algebra pipeline produced a PBW normal form for a rank-2 Lie algebra with brackets [[[1,0],[0,-1]]] and terms: word [0,0,1] coefficient 7, word [1,2] coefficient -4. Verify whether these ordered-monomial coefficients are consistent with the given bracket structure."}],"resultDescription":"Returns a verification result indicating whether each ordered-monomial (PBW basis) coefficient in the supplied universal enveloping algebra expression is correct under the given Lie bracket tensor, flagging any coefficient mismatches or ordering violations.","failureModes":["Invalid bracket tensor dimensions (exceeds 3×8×8 shape)","Word indices out of range (must be 0–255)","Coefficients out of supported range (-100 to 100)","Too many terms supplied (max 40)","Malformed or missing 'pbw' artifact object","Claim field inconsistent with data, causing validation rejection"],"whenToPreferThis":"Use this endpoint when you need to programmatically verify the mathematical correctness of a PBW (Poincaré–Birkhoff–Witt) normal form expansion in a universal enveloping algebra given an explicit Lie bracket tensor — particularly useful for automated algebra checkers, symbolic computation pipelines, or educational tools that need ground-truth validation of Lie-theoretic calculations.","instructions":null,"reviewSummary":null,"reviewSummaryHighlights":null,"reviewSummaryConcerns":null,"reviewSummaryGeneratedAt":null,"activationCount":0,"lastUsedAt":null,"lastSuccessfullyRanAt":null,"lastHealthCheckAt":"2026-11-04T06:40:45.775Z","isFirstParty":false,"canonicalSlug":"rational-lie-bracket-universal-enveloping-algebra-pbw-normal-form-7b67240b"}