Comparison of two notions of weak crossed product

We compare the restriction to the context of weak Hopf algebras of the notion of crossed product with a Hopf algebroid introduced in [Cleft extensions of Hopf algebroids, Appl. Categor. Struct. 14(5–6) (2006) 431–469] with the notion of crossed product with a weak Hopf algebra introduced in [Crossed products for weak Hopf algebras with coalgebra splitting, J. Algebra 281(2) (2004) 731–752].

Constructing the hereditary crossed product order containing a given weak crossed product order and a criterion for weakness

Consider a weak crossed product order [Formula: see text] in [Formula: see text], where [Formula: see text] is the integral closure of a discrete valuation ring [Formula: see text] in a tamely ramified Galois extension [Formula: see text] of the field of fractions of [Formula: see text]. Assume that [Formula: see text] is local. In this paper, we show that [Formula: see text] is hereditary if and only if it is maximal among the weak crossed product orders in [Formula: see text]. We also give an algorithm that constructs, in terms of the basis elements [Formula: see text] and the cocycle [Formula: see text], the unique hereditary weak crossed product order in [Formula: see text] that contains a given [Formula: see text], and we give a criterion for determining whether that hereditary order will have a cocycle that takes nonunit values in [Formula: see text].

ON THE REPRESENTATIONS OF WEAK CROSSED PRODUCTS

Let H be a finite-dimensional weak Hopf algebra in the sense of [G. Böhm and K. Szlachányi, A coassociative C*-quantum group with nonintegral dimensions, Lett. Math. Phys.35 (1996) 437–456] which is semisimple as well as its dual H*, A be a finite-dimensional algebra measured by H and A#σH be a weak crossed product. Then we first prove that the representation dimension of A#σH equals to that of A. Moreover, we will show that over an algebraically closed field k, the representation type between A and its weak crossed product algebra A#σH is coincident, which extends the result given by Liu [On the structure of tame graded basic Hopf algebras, J. Algebra 299 (2006) 841–853]. Finally, we show that A#σH is a CM-finite n-Gorenstein algebra if and only if so is A and the global Gorenstein projective dimension of A#σH is equal to that of A.