Drinfeld twists of Koszul algebras (Q6561437)

Let \(H\) be Hopf algebra over a field \(k\). Given a quadratic \(H\)-module algebra \(A\) with degree-preserving \(H\)-action, and a counital 2-cocycle (or \textit{twist}) \(\mu \in H \otimes H\), the author shows that the corresponding Drinfeld twist \(A_\mu\) is also a quadratic algebra and determines its (quadratic) relations explicitly. He then shows that if \(A\) is a Koszul algebra, then \(A_\mu\) is also Koszul, extending a result of \textit{A. Davies} [Commun. Algebra 45, No. 3, 1347--1363 (2017; Zbl 1386.16013)] for the case when \(H\) is the group algebra of a finite abelian group. As applications, it is shown 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, and new proofs that the quantum symmetric and exterior algebras \(S_{-1}(V)\) and \(\Lambda_{-1}(V)\) are Koszul are given.