Invariants of the adjoint coaction and Yetter-Drinfeld categories (Q5939819)

If \(H\) is a finite-dimensional semisimple Hopf algebra over an algebraically closed field of characteristic zero, a key object related to the structure of \(H\) is the character algebra \(C(H^*)\), that is, the algebra spanned by the irreducible characters (see for instance the work of \textit{G. I. Kac} [Funkts. Anal. Prilozh. 6, No. 2, 88-90 (1972; Zbl 0258.16007)] and \textit{Y. Zhu} [Int. Math. Res. Not. 1994, No. 1, 53-59 (1994; Zbl 0822.16036)]). It is well known that in this case \(C(H^*)\) may be characterized as the subalgebra of all cocommutative elements in the dual Hopf algebra \(H^*\).NEWLINENEWLINENEWLINEIn this paper, for a Hopf algebra \(H\) over a field \(k\) of arbitrary characteristic, the authors study the subalgebra \(O(H)\) of coinvariants of \(H\) under the left adjoint coaction of \(H\) on itself. Several characterizations of \(O(H)\) are given. For finite-dimensional \(H\) these characterizations involve integrals and distinguished group-like elements; some characterizations concern also the behaviour of the distinct powers of the antipode.NEWLINENEWLINENEWLINEThe paper compares the subalgebra \(O(H^*)\) with the subalgebra of cocommutative elements \(C(H^*)\). A relationship between \(O(H^*)\) and the center of \(H\) is stablished: namely, \(O(H^*)=Z(H)\rightharpoonup T\) and \(Z(H)=O(H^*)\rightharpoonup t\), where \(T\) and \(t\) are non-zero left integrals in \(H^*\) and \(H\), respectively. Further properties are discussed in the case where \(H\) is finite-dimensional and the square of the antipode is an inner automorphism, in particular, whenever \(H\) is semisimple or quasitriangular.NEWLINENEWLINENEWLINEFinally, the paper establishes connections between the category of left modules over \(O(H)\) and the category of left Yetter-Drinfeld modules over \(H\). A related reference is the work of the first author [in Contemp. Math. 267, 55-65 (2000; Zbl 0977.16016)].