Let H 8 be the Kac-Paljutkin algebra [Trudy Moskov. Mat. Obšč. 15 (1966), 224-261], which is the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra. All simple Yetter-Drinfel'd modules over H 8 are given, and finite-dimensional Nichols algebras over H 8 are determined completely. It turns out that they are all of diagonal type. In fact, they are of Cartan types A
In this paper, the author gives a complete set of simple Yetter-Drinfeld modules over Suzuki algebra A µλ N 2n+1 [Suz98] and investigates the Nichols algebras over those irreducible Yetter-Drinfeld modules. The finite dimensional Nichols algebras of diagonal type are of Cartan type A 1 , A 1 × A 1 , A 2 , Super type A 2 (q; I 2 ) and the Nichols algebra ufo(8). And the involved finite dimensional Nichols algebras of non-diagonal type are 12, 4m and m 2 dimensional. The left three unsolved cases are set as open problems.
Let (X, r) be any set-theoretical non-degenerate solution of the Yang-Baxter equation and (X, r) be the derived solution of (X, r). As for any braided vector space (W X,r , c) associated to (X, r), is it possible to find some braided vector space (W X,r , c) which is t-equivalent to (W X,r , c)? In case that (X, r) is a near-rack solution, we give a sufficient condition to make an affirmative answer to the question. Examples of t-equivalence are constructed, hence finite dimensional Nichols algebras are obtained. In particular, all finite dimensional Nichols algebras associated to involutive near-rack solutions are classified.
Let H 8 be the neither commutative nor cocommutative semisimple eight dimensional Hopf algebra, which is also called Kac-Paljutkin algebra [KP66]. All simple Yetter-Drinfel'd modules over H 8 are given. As for simple objects and direct sums of two simple objects in H 8H 8 YD, we calculated dimensions for the corresponding Nichols algebras, except four semisimple cases which are generally difficult. Under the assumption that the four undetermined Nichols algebras are all infinite dimensional, we determine all the finite dimensional Nichols algebras over H 8 . It turns out that the already known finite dimensional Nichols algebras are all diagonal type. In fact, they are Cartan types A
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