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We study the quadratic algebras in Artin–Schelter regular algebras of dimension 5 generated in degree 1 under the hypothesis that . All the algebras obtained are proved to be Koszul algebras or piecewise-Koszul algebras. In addition, we find that there don’t exist d-Koszul (d > 2) Artin-Schelter regular algebras of dimension 5.
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