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In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra H , which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of H -modules is equivalent to the category H H 𝒬 𝒴 𝒟 of quasi-Yetter–Drinfeld modules. Next, we prove that H H 𝒬 𝒴 𝒟 is equivalent to the category of generalized Hopf bimodules. Finally, we show that H H 𝒬 𝒴 𝒟 is also equivalent to the category of quasi-coactions over some Majid's braided group if H is quasi-triangular. [ABSTRACT FROM AUTHOR]

In this paper, we give a recursive formula for the interesting PBW basis E A of composition subalgebras ℭ ▵ (n) of Ringel–Hall algebras ℌ ▵ (n) of cyclic quivers after [Generic extensions and canonical bases for cyclic quivers, Canad. J. Math. 59(6) (2007) 1260–1283], and another construction of canonical bases of U v + ( 𝔰 𝔩 ̂ n) from the monomial bases m (A) following [Multiplication formulas and canonical basis for quantum affine, 𝔤 𝔩 n , Canad. J. Math. 70(4) (2018) 773–803]. As an application, we will determine all the canonical basis elements of U v + ( 𝔰 𝔩 ̂ 2) associated with modules of Loewy length ≤ 3. Finally, we will discuss the canonical bases between Ringel–Hall algebras and affine quantum Schur algebras. [ABSTRACT FROM AUTHOR]