A general criterion is given for when the center of a Taft algebra smash product is the fixed ring. This is applied to the study of the noncommutative discriminant. Our method relies on the Poisson methods of Nguyen, Trampel, and Yakimov, but also makes use of Poisson Ore extensions. Specifically, we fully determine the inner faithful actions of Taft algebras on quantum planes and quantum Weyl algebras.We compute the discriminant of the corresponding smash product and apply it to compute the Azumaya locus and restricted automorphism group.

Let k be a field and let R be a left noetherian k-algebra. The algebra R satisfies the Dixmier-Moeglin equivalence if the annihilators of irreducible representations are precisely those prime ideals that are locally closed in the Spec(R) and if, moreover, these prime ideals are precisely those whose extended centres are algebraic extensions of the base field. We show that if R and S are two left noetherian k-algebras with dim k (R), dim k (S) < |k| then if R and S have homeomorphic spectra then R satisfies the Dixmier-Moeglin equivalence if and only if S does. In particular, the topology of Spec(R) can detect the Dixmier-Moeglin equivalence in this case. In addition, we show that if k is uncountable and R is affine noetherian and its prime spectrum is a disjoint union of subspaces that are each homeomorphic to the spectrum of an affine commutative ring then R satisfies the Dixmier-Moeglin equivalence. We show that neither of these results need hold if k is countable and R is infinite-dimensional. Finally, we make the remark that satisfying the Dixmier-Moeglin equivalence is a Morita invariant and finally we show that R and S are left noetherian k-algebras that satisfy the Dixmier-Moeglin equivalence then R ⊗ k S does too, provided it is left noetherian and satisfies the Nullstellensatz; and we show that eRe also satisfies the Dixmier-Moeglin equivalence, where e is a nonzero idempotent of R.

We study the congeniality property of algebras, as defined by Bao, He, and Zhang, in order to establish a version of Auslander's theorem for various families of filtered algebras. It is shown that the property is preserved under homomorphic images and tensor products under some mild conditions. Examples of congenial algebras in this paper include enveloping algebras of Lie superalgebras, iterated differential operator rings, quantized Weyl algebras, down-up algebras, and symplectic reflection algebras.An important result of Auslander [4] shows that if V is a finite-dimensional vector space over an algebraically closed field k of characteristic zero, and G is a finite group of automorphisms acting linearly on k[V ] with no nontrivial reflections (i.e., G is a small group), then there is an isomorphism of graded algebras k. There has been much work done in extending this result to the noncommutative setting, either by replacing k[V ] by a suitable noncommutative algebra, replacing G with a Hopf algebra H, or both.Recent work of Bao, He, and Zhang introduces the pertinency invariant as a way to test whether an algebra A and a Hopf algebra H acting on A satisfy the conclusion of Auslander's Theorem [5, 6]. The general theme of their results is that, for a suitable pair, this holds if and only if the pertinency is at least two. In [5] it is shown that a class of filtered algebras known as congenial algebras, along with certain groups of filtered automorphisms, are sufficiently suitable. Furthermore, they prove that the enveloping algebra of a finite-dimensional Lie algebra is congenial. The authors state that there are 'ample examples of congenial algebras' and part of our goal is to better understand what algebras satisfy this condition. We prove the following theorem via various results in this paper.Main Theorem. The following algebras are congenial with respect to some filtration.

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While it was identified that the growth of any connected Hopf algebras is either a positive integer or infinite (see [15]), we have yet to determine the GK-dimension of a given connected Hopf algebra. We use the notion of anti-cocommutative elements introduced in [13] to analyze the structure of connected Hopf algebras generated by anti-cocommutative elements and compute the Gelfand-Kirillov dimension of said algebras. Additionally, we apply these results to compare global dimension of connected Hopf algebras and the dimension of their corresponding Lie algebras of primitive elements.