Drinfeld twists of Koszul algebras

arXiv2306.08983MaRDI QIDQ6440382

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Publication date: 15 June 2023

Abstract: Given a Hopf algebra $H$ and a counital $2$-cocycle $mu$ on $H$, Drinfeld introduced a notion of twist which deforms an $H$-module algebra $A$ into a new algebra $A_{m}u$. We show that when $A$ is a quadratic algebra, and $H$ acts on $A$ by degree-preserving endomorphisms, then the twist $A_{m}u$ is also quadratic. Furthermore, if $A$ is a Koszul algebra, then $A_{m}u$ is a Koszul algebra. As an application, we prove that the twist of the $q$-quantum plane by the quasitriangular structure of the quantum enveloping algebra $U_q(mathfrak{sl}_2)$ is a quadratic algebra equal to the $q^{-1}$-quantum plane.

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