A note on the discriminant of reflection Hopf algebras

We provide a formula for commputing the discriminant of skew Calabi-Yau algebra over a central Calabi-Yau algebra. This method is applied to study the Jacobian and discriminant for reflection Hopf algebras. Introduction Recently, the discriminant has been used to determine the automorphism group of some PI algebras [5, 7], and solve isomorphism problem [6] and Zariski cancellation proplem [2, 23, 25]. Despite the usefulness of the discriminant, the computation of the discriminant turns out to be a rather diffculty problem, see [8] for example. Nguyen, Trampel and Yakimov presented a general method for computing the discriminants of algebras in [30], by associating discriminants of noncommutative algebras with Poisson geometry. In [22], Levitt and Yakimov maked use of techniques from quantum cluster algebras to derive explicit formulas for the discriminants of quantized Weyl algebras at roots of unity. Almost all these algebras are skew Calabi-Yau algebras which are finitely generated modules over central subalgebras. It worth pointing out that these algebras are Frobenius algebras over commutative Calabi-Yau algebras. Lemma 0.1 (Corollary 2.10). Let R be an affine Calabi-Yau algebra of dimension d and A be a module-finite R-algbera. If A is a skew Calabi-Yau algebra of dimension d, then A is a Frobenius R-algebra. The discriminant of a Frobenius algebra A is the norm of the different of A, see Lemma 1.8. Throughout the rest of this paper, k is a base field with char k = 0, and all vector spaces, algebras and Hopf algebras are over k. Unadorned ⊗ means ⊗k and Hom means Homk. H will stand for a Hopf algebra (H,∆, ε) with bijective antipode S. We use the Sweedler notation ∆(h) = ∑ (h) h1 ⊗ h2 for all h ∈ H. We recommend [28] as a basic reference for the theory of Hopf algebras. We are interested in the following algebra which comes from the noncommutative invariant theory. Let H be a finite-dimensional semisimple Hopf algebra, and A be a Noetherian connected graded Artin-Schelter regular domain [1], which are skew Calabi-Yau algebras by [34, Lemma 1.2]. Suppose that H acts homogeneously and inner faithfully on A such that A is a left H-module algebra. Then H is called a reflection Hopf algebra or reflection quantum group [18, Definition 3.2] if the fixed subring A is again Artin-Schelter regular. Kirkman and Zhang [20], introduced some noncommutative version of Jacobian jA,H , reflection arrangement aA,H and discriminant δA,H of the H-action on A, see Definition 3.2. When A is central in A and H is a dual reflection group, then δA,H and the noncommutative discriminant have the same prime radical [20, Theorem 0.7]. Our main result is to relate the noncommutative discriminant to the Jacobian of a general Hopf action. 2020 Mathematics Subject Classification. 11R29 12W22 16G30 .