Finite Quantum Groups and Cartan Matrices

On pointed Hopf algebras of dimension p^5

“…Then the same arguments as in [3,Theorem 1.3] give the condition p 3 or p 1 mod 3. Furthermore, let b be such that b 2 b 1 0 mod p (the condition on p is equivalent to the existence of such a b) and take e 1 p, f 1 Lemma 4.1.…”

“…We denote the generators of C p 2 Â C p by fg 1 Y g 2 g and the generators of d C p 2ÂC p by f g 1 Y g 2 g, where g i g j q ij 3Ài . We denote the generator of C p 3 by fgg and the generator of d C p 3 by f gg, where g g q 3 . It is straightforward that the following Yetter±Drinfeld modules give QLS of dimension p 2 : …”

“…that S S 1 . This is the same as saying that S H S. Now, [2, Theorem 3.2] plus Remark 4.6 solve the problem for the cases in which À has order p 4 or p 3 , and [3,Theorem 8.2] and [3, Lemma 8.5] solve the problem for the case in which À has order p or À C p  C p . The following theorem solves the pending case.…”

“…See also the discussion in [3, §9] for the ®rst step, [3, §6] for the second. In particular, a necessary and su cient condition for two YD-modules to give isomorphic algebras after bosonization is given in [3,Proposition 6.3].…”

“…According to [3], if V has a matrix b ij we say that V is of Cartan type if there exists a (generalized) Cartan matrix a ij such that…”