Splitting comodules over Hopf algebras and application to representation theory of quantum groups of type \(A_{0|0}\) (Q5958853)

Let \(H\) be a Hopf algebra with a left (or right) integral \(\int_l\), i.e., \(H\) is left or right co-Frobenius as a coalgebra. Then the rational space \(H^{*\text{rat}}\) of the dual \(H^*\) of \(H\) is isomorphic to \(H\) as \(H\)-comodules. However \(H^{*\text{rat}}\) is an ideal of \(H^*\), and hence the isomorphism between \(H\) and \(H^{*\text{rat}}\) induces a convolution product in \(H\) such that \(H\) becomes an associative algebra \(\widetilde H\) without identity. By a new correspondence between right \(H\)-comodules and unital left \(\widetilde H\)-modules, the author defines a new \(H\)-comodule \(M^\bullet\) from any \(H\)-comodule \(M\). For any simple right \(H\)-comodules \(M_\alpha,M_\beta\), the author proves that \(\dim_k\Hom^H({\mathcal J}(M_\alpha),M^\bullet_\beta)=\delta^\alpha_\beta d^2_\beta\), where \({\mathcal J}(M)\) denotes the injective envelope of \(M\) and \(d^2_\beta=\dim_k\text{End}^H(M_\beta)\). A simple \(H\)-comodule \(M\) is called splitting if \(M={\mathcal J}(M)\). In this paper, the author proves that a simple \(H\)-comodule \(M\) is splitting if and only if the bilinear form \(b(x,y):=\int_l(xS(y))\) does not identically vanish on the coefficient space of \(M\) (the simple subcoalgebra of \(H\) coacting on \(M\)). Applying the above result, the author gives the classification of simple comodules of the quantum groups of type \(A_{0|0}\). It is shown that simple comodules of these Hopf algebras can be labeled by pairs of integers \((k,l)\).