Automorphism groups of representation rings of the weak Sweedler Hopf algebras

<abstract><p>Let $ \mathfrak{w}^{s}_{2, 2}(s = 0, 1) $ be two classes of weak Hopf algebras corresponding to the Sweedler Hopf algebra, and $ r(\mathfrak{w}^{s}_{2, 2}) $ be the representation rings of $ \mathfrak{w}^{s}_{2, 2} $. In this paper, we investigate the automorphism groups $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $ of $ r(\mathfrak{w}^{s}_{2, 2}) $, and discuss some properties of $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{s}_{2, 2})) $. We obtain that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{0}_{2, 2})) $ is isomorphic to $ K_4 $, where $ K_4 $ is the Klein four-group. It is shown that $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is a non-commutative infinite solvable group, but it is not nilpotent. In addition, $ {{{\rm{Aut}}}}(r(\mathfrak{w}^{1}_{2, 2})) $ is isomorphic to $ (\mathbb{Z}\times \mathbb{Z}_{2})\rtimes \mathbb{Z}_{2} $, and its centre is isomorphic to $ \mathbb{Z}_{2} $.</p></abstract>

On the SL(2, C)-representation rings of free abelian groups

In this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.