โ Scribed by Hui-Xiang Chen
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Let R: M ยฎ M ~ M ยฎ M be a solution to the quantum Yang-Baxter equation, where M is a finite-dimensional vector space over a field k. It was shown that R can be derived from a left H-module structure and a right H-comodule structure on M for some bialgebra H over k (see [FRT, Mjl, Y]). The module and comodule structures satisfy a natural compatibility condition. M together with this structure is called a quantum Yang-Baxter H-module, which is also called a Yetter-Drinfeld module over H (see [Mo, RT]). A quantum Yang-Baxter H-module is a (left) crossed H-bimodule [Y].
Let H be a finite-dimensional Hopf algebra over a field k, and let D(H) be the Drinfeld's quantum double derived from H [Dr]. It is well known [K, Mj2] that a vector space M possesses a crossed H-bimodule structure if and only if M possesses a left D(H)-module structure. Hence the Yetter-Drinfeld module category ny~ n is the same as the left D(H)-module category o(n)~'.
Suppose that n _ 1 and that q ~ k is a primitive nth root of unity. Then to = q-1 is also a primitive nth root of unity. Taft constructed an n2-dimensional Hopf algebra An(to) in [T]. The An(to)'s form an interesting class of pointed Hopf algebras from a combinatorial point of view. When n is odd, (D(An(tO)), R) provides an invariant of three-manifolds [H]. Generally, the double of An(tO) is of interest in connection with knot theory.
In the previous paper [C], we constructed an infinite-dimensional noncommutative and noncocommutative Hopf algebra H(p, q) for any p, q 1 Supported by NSF of China (Grant No. 19971073) and Jiangsu education committee.
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