Rigidity of down-up algebras with respect to finite group coactions

Abstract

Abstract. If G is a nontrivial finite group coacting on a graded noetherian down-up algebra A inner faithfully and homogeneously, then the fixed subring A co G is not isomorphic to A. Therefore graded noetherian down-up algebras are rigid with respect to finite group coactions, in the sense of Alev-Polo. An example is given to show that this rigidity under group coactions does not have all the same consequences as the rigidity under group actions. IntroductionThroughout this paper, let k be a base field that is algebraically closed of characteristic zero, and let all vector spaces, (co)algebras, and morphisms be over k.A remarkable theorem of Alev-Polo [AP, Theorem 1] states: Let g and g ′ be two semisimple Lie algebras. Let G be a finite group of algebra automorphisms of the universal enveloping algebra U (g) such that the fixed subringThen G is trivial and g ∼ = g ′ . Alev-Polo called this result a rigidity theorem for universal enveloping algebras. In addition, they proved a rigidity theorem for the Weyl algebras [AP, Theorem 2]. Kuzmanovich and the second-and third-named authors proved Alev-Polo's rigidity theorems in the graded case in [KKZ1, Theorem 0.2 and Corollary 0.4].(Commutative) polynomial rings are not rigid; indeed, by the classical ShephardTodd-Chevalley Theorem if G is a reflection group acting on a commutative polynomial ring A then A G is isomorphic to A. Artin-Schelter regular algebras [AS] are considered to be a natural analogue of polynomial rings in many respects. This paper concerns a class of noncommutative Artin-Schelter regular algebras. The rigidity of a noncommutative algebra is closely related to the lack of reflections in the noncommutative setting [KKZ1]. Therefore the rigidity of an algebra leads to a trivialization of the Shephard-Todd-Chevalley theorem [ST,KKZ2], which is one of the key results in noncommutative invariant theory [Ki]. The rigidity property is also related to Watanabe's criterion for the Gorenstein property, see [KKZ3, Theorem 4.10]. Some recent work in noncommutative algebraic geometry connects the rigidity property and the lack of reflections to Auslander's theorem [BHZ], which is one of the fundamental ingredients in the McKay correspondence [CKWZ1,CKWZ2]. Further understanding of the rigidity property will have implications for several other research directions.In [KKZ5], rigidity with respect to group coactions is studied. Let A be a connected (N-)graded k-algebra. A G-coaction on A (preserving the N-grading)2010 Mathematics Subject Classification. 16E10, 16W22. Key words and phrases. Down-up algebra, coaction, rigidity, Artin-Schelter regular algebra, homological determinant. is equivalent to a G-grading of A (compatible with the original N-grading), and the fixed subring A coG is A e , the component of the unit element e ∈ A under the G-grading. We recall a definition [KKZ5, Definition 0.8]: we say that a connected graded algebra A is rigid with respect to group coactions if for every nontrivial finite group G coacting on A homogeneously and inner faithfu...

Auslander’s Theorem for Group Coactions on Noetherian Graded Down-Up Algebras

“…of degree one such that Note that the hypothesis of "G being non-cyclic" is needed in the above lemma which was proved in [11]. In the present paper we will also consider cyclic cases.…”

“…The final case is when A = F and the proof is also quite tricky. We start with a result of [11]. (2) We also have the other relations:…”

“…The groups of graded algebra automorphisms of the down-up algebras were computed in [15]. Recently, the invariant theory of graded down-up algebras under finite group actions and coactions has been studied in [17,11,13].…”

The Jacobian, reflection arrangement and discriminant for reflection Hopf algebras

“…(a) The rigidity of (noetherian) Artin-Schelter regular algebras under finite group or semisimple Hopf algebra actions [AP,CKZ1,KKZ1,KKZ4]. (b) The homological determinant and Watanabe's theorem.…”

“…Let H be the Hopf algebra (kG) * where G is the dihedral group of order 8 as in part (2). This is the setting in[CKZ1, Example 2.1]. By[CKZ1, Example 2.1], we have hdet = hdet −1…”