Quantum Groupoids Acting on Semiprime Algebras

Abstract

Following Linchenko and Montgomery's arguments we show that the smash product of an involutive weak Hopf algebra and a semiprime module algebra, satisfying a polynomial identity, is semiprime.

1. Introduction

Group actions, Lie algebras acting as derivations and finite group gradings are typical examples of Hopf algebra actions which have been studied for many years. Several generalizations of Hopf algebras have emerged in recent years, like weak Hopf algebras (or quantum groupoids) introduced by Böhm et al. [1]. The action of such objects on algebras, as given by quantum groupoids acting on 𝐶^∗-algebras, [2] or weak Hopf algebras arising from Jones towers [3] are particularly interesting. New examples of weak Hopf algebras arose from double groupoids [4], which were also used to find new weak Hopf actions (see [2]).

A long-standing open problem in the theory of Hopf action is to show that the smash product 𝐴#𝐻 of a semiprime module algebra 𝐴 and a semisimple Hopf algebra 𝐻 is again semiprime (see [5]) (an algebra 𝐴 is semiprime if it does not contain nonzero nilpotent ideals.). The case of 𝐴 being commutative had been settled in [6]. The most recent partial answer to this problem has been given by Linchenko and Montgomery in [7] where they prove the semiprimness of 𝐴#𝐻 under the condition of 𝐴 satisfying a polynomial identity. The purpose of this note is that their result carries over to actions of weak Hopf algebras. We reach more generality by considering actions of linear operators that satisfy certain intertwining relations with the regular multiplications on the algebra.

Let 𝑘 be a commutative ring and let 𝐴 be an associative unital 𝑘-algebra. For any 𝑎∈𝐴 define two linear operators 𝐿(𝑎) and 𝑅(𝑎) in End_𝑘(𝐴) given by ⟨𝐿(𝑎),𝑥⟩=𝑎𝑥 and ⟨𝑅(𝑎),𝑥⟩=𝑥𝑎 for all 𝑥∈𝐴. We identify 𝐴 with the subalgebra 𝐿(𝐴) of End_𝑘(𝐴) generated by all left multiplications 𝐿(𝑎) and denote the subalgebra generated by all operators 𝐿(𝑎) and 𝑅(𝑎) by 𝑀(𝐴), which is also sometimes referred to as the multiplication algebra of 𝐴. As a left 𝐿(𝐴)-module, 𝐴 is isomorphic to 𝐿(𝐴) since we assume 𝐴 to be unital. We will be interested in certain actions on an algebra 𝐴 that may stem from a bialgebra or more generally a bialgebroid. The situation we will encounter is the one where we have an extension 𝐴⊆𝐵 where 𝐵 acts on 𝐴 through a ring homomorphism 𝜙∶𝐵→End_𝑘(𝐴) such that (𝑎)𝜙=𝐿(𝑎) for all 𝑎∈𝐴. For the intrinsic properties of 𝐴 under this action it is enough to look at the subalgebra 𝜙(𝐵) in End_𝑘(𝐴) generated by this action and we might consider intermediate algebras 𝐿(𝐴)⊆𝐵⊆End_𝑘(𝐴) instead. Hence let 𝐵 be a subalgebra of End_𝑘(𝐴) that contains 𝐿(𝐴). Then 𝐴 becomes a cyclic faithful left 𝐵-module by evaluating endomorphisms, that is, for all 𝑏∈𝐵,𝑎∈𝐴∶𝑏⋅𝑎∶=⟨𝑏,𝑎⟩. Note that for any 𝑎′∈𝐴 we have 𝐿(𝑎′)⋅𝑎=⟨𝐿(𝑎′),𝑎⟩=𝑎′𝑎.

Since we assume 𝐴 to be unital, the map Ψ∶End_𝐵(𝐴)→𝐴 with Ψ(𝑓)=(1)𝑓, for all 𝑓∈End_𝐵(𝐴)—evaluating endomorphisms at 1—is an injective ring homomorphism, since for all 𝑓,𝑔∈End_𝐵(𝐴)∶Ψ(𝑓∘𝑔)=((1)𝑓)𝑔=((1)𝑓⋅1)𝑔=(1)𝑓⋅(1)𝑔=Ψ(𝑓)Ψ(𝑔). Moreover if Ψ(𝑓)=(1)𝑓=0, then (𝑎)𝑓=𝑎(1)𝑓=0 and 𝑓=0. The subalgebra Ψ(End_𝐵(𝐴)) can be described as the set of elements 𝑎∈𝐴 such that for any 𝑏∈𝐵∶𝑏⋅𝑎=(𝑏⋅1)𝑎, which we will denote by 𝐴^𝐵. On one hand if 𝑎=(1)𝑓=Ψ(𝑓) for some 𝑓∈End_𝐵(𝐴), then for any 𝑏∈𝐵∶𝑏⋅𝑎=(𝑏⋅1)𝑓=𝐿((𝑏⋅1))⋅(1)𝑓=(𝑏⋅1)𝑎 and on the other hand if 𝑎∈𝐴^𝐵, then 𝑓=𝑅(𝑎) is left 𝐵-linear since for any 𝑏∈𝐵 and 𝑥∈𝐴: (𝑥)()(𝐿(𝑥))=((𝑥((𝑥))𝑎=((𝐿(𝑥)()()𝑏⋅𝑓=𝑏⋅𝑥𝑎=𝑏⋅⋅𝑎𝑏𝐿))⋅𝑎=𝑏𝐿⋅1𝑏⋅⋅1))𝑎=𝑏⋅𝑥𝑎=𝑏⋅𝑥𝑓. (1.1) Thus End_𝐵(𝐴)≃𝐴^𝐵.

Let 𝑀 be any left 𝐵-module and define 𝑀^𝐵={𝑚∈𝑀∶∀𝑏∈𝐵∶𝑏⋅𝑚=(𝑏⋅1)𝑚}. With the same argument as above one sees that Ψ_𝑀∶Hom_𝐵(𝐴,𝑀)→𝑀^𝐵 with Ψ(𝑓)=(1)𝑓 is an isomorphism of abelian groups, hence yielding a left 𝐴^𝐵-module structure on 𝑀^𝐵. Moreover it is possible to show that Hom_𝐵(𝐴,−) is isomorphic to (−)^𝐵 as functors from 𝐵-Mod to 𝐴^𝐵-Mod.

A subset 𝐼 of 𝐴 is called 𝐵-stable if 𝐵⋅𝐼⊆𝐼. The 𝐵-stable left ideals are precisely the (left) 𝐵-submodules of 𝐴. In particular Hom_𝐵(𝐴,𝐼)≃𝐼^𝐵=𝐼∩𝐴^𝐵, for any 𝐵-stable left ideal of 𝐴.

Examples 1.1. The following list illustrates that our aproach reflects many interesting cases of algebras with actions. (1)Let 𝐵=𝑀(𝐴) be the multiplication algebra of 𝐴, then 𝐴 is a faithful cyclic left 𝐵-module. The left 𝐵-modules are precisely the 𝐴-bimodules, in particular the left ideals of 𝐴 are the two-sided ideals of 𝐴, and Hom_𝐵(𝐴,𝑀)≃𝑀^𝐵=𝑍(𝑀)={𝑚∈𝑀∣∀𝑎∈𝐴∶𝑎𝑚=𝑚𝑎} holds for any 𝐴-bimodule 𝑀. The operator algebra 𝐵 is a quotient of the enveloping algebra 𝐴^𝑒=𝐴⊗𝐴^op through the map 𝑎⊗𝑏↦𝐿(𝑎)∘𝑅(𝑏), for all 𝑎⊗𝑏∈𝐴^𝑒. (2)Let 𝐺 be a group acting as (𝑘-linear) automorphisms on 𝐴, that is, there exists a group homomorphism 𝜂∶𝐺→Aut_𝑘(𝐴). Set ^𝑔𝑎=⟨𝜂(𝑔),𝑎⟩ for any 𝑎∈𝐴,𝑔∈𝐺. Define 𝐵=⟨𝐿(𝐴)∪{𝜂(𝑔)∣𝑔∈𝐺}⟩⊆End_𝑘(𝐴). Then the left 𝐵-submodules 𝐼 of 𝐴 are precisely the 𝐺-stable left ideals of 𝐴 and Hom_𝐵(𝐴,𝐼)≃𝐼^𝐵={𝑥∈𝐼∣^𝑔𝑥=𝑥}. 𝐵 is a quotient of the skew group ring 𝐴#𝐺 whose underlying 𝐴-submodule is the free left 𝐴-module with basis {𝑔∣𝑔∈𝐺} and whose multiplication is given by (𝑎#𝑔)(𝑏#ℎ)=𝑎^𝑔𝑏#𝑔ℎ. Note that for any left 𝐴#𝐺-module 𝑀 we have, Hom_𝐴#𝐺(𝐴,𝑀)=Hom_𝐵(𝐴,𝑀)≃𝑀^𝐺 is the set of fixed elements of 𝑀.(3)Let 𝐴 be an 𝑘-algebra with involution ∗ and let 𝐵 be the subalgebra of End_𝐴() generated by 𝐴 and ∗. Since for any 𝑎∈𝐴∶𝑅(𝑎)=∗∘𝐿(𝑎^∗)∘∗ we got 𝑀(𝐴)⊆𝐵. This means (as it is well-known) that any left ideal of 𝐴 which is stable under ∗ is a two-sided ideal. Note that 𝐵 can be seen as the factor ring of the skew-group ring 𝐴^𝑒#𝐺 where 𝐺={id,∗} is the cyclic group of order two and ∗∈Aut(𝐴^𝑒) is given by (𝑎⊗𝑏)^∗∶=𝑏^∗⊗𝑎^∗.(4)Let 𝛿∈Der_𝑘(𝐴) be an 𝑘-linear derivation of 𝐴 and consider 𝐵=⟨𝐿(𝐴)∪{𝛿}⟩⊆End_𝑘(𝐴). The left 𝐵-submodules of 𝐴 are the left ideals 𝐼 that satisfy 𝛿(𝐼)⊆𝐼. The operator algebra 𝐵 is a factor of the ring of differential operator 𝐴[𝑧;𝛿], which as a left 𝐴-module is equal to 𝐴[𝑧] and its multiplication is given by 𝑧𝑎=𝑎𝑧+𝛿(𝑎). The map 𝐴[𝑧;𝛿]→𝐵 with ∑^𝑛_𝑖=0𝑎_𝑖𝑧^𝑖↦∑^𝑛_𝑖=0𝐿(𝑎_𝑖)∘𝛿^𝑖∈𝐵 is a surjective 𝑘-algebra homomorphism and for any left 𝐴[𝑧;𝛿]-module 𝑀 we have Hom_{𝐴[𝑧;𝛿]}(𝐴,𝑀)=Hom_𝐵(𝐴,𝑀)=𝑀^𝛿={𝑚∈𝑀∣𝑧𝑚=0}. In particular End_{𝐴[𝑧;𝛿]}(𝐴)≃𝐴^𝛿=Ker(𝛿). The subring 𝐴^𝛿 of 𝐴 is called the ring of constants of 𝛿.(5)Let 𝐻 be an 𝑘-Hopf algebra action on 𝐴. Let us denote the action of an element ℎ∈𝐻 on 𝐴 by 𝜆_ℎ∈End_𝑘(𝐴) and define 𝐵=⟨𝐿(𝐴)∪{𝜆_ℎ∣ℎ∈𝐻}⟩⊆End_𝑘(𝐴). The smash product 𝐴#𝐻 is an extension with additional module structure. Define 𝜑∶𝐴#𝐻→End_𝑘(𝐴) by 𝜑(𝑎#ℎ)∶=𝐿(𝑎)∘𝜆_ℎ.

2. Linear Operators Acting on Algebras Satisfying a Polynomial Identity

Let 𝐿(𝐴)⊆𝐵⊆End_𝑘(𝐴) be any intermediate algebra as above.

The first technical lemma generalizes a corresponding result of Linchenko [8, Theorem  3.1] for Hopf actions and Nikshych [9, Theorem  6.1.3] for weak Hopf actions. Recall that an ideal whose elements are nilpotent is called a nil ideal.

Lemma 2.1. Let 𝐿(𝐴)⊆𝐵⊆End_𝑘(𝐴) and suppose that for all 𝜓∈𝐵 there exist 𝑚≥1 and elements 𝜓¹₁,…,𝜓¹_𝑚, 𝜓²₁,…,𝜓²_𝑚∈𝐵 such that 𝐿𝜓,𝑎=^𝑚_𝑖=1𝜓¹_𝑖∘𝐿(𝑎)∘𝜓²_𝑖,^𝑛_𝑖=1𝜓²_𝑖∘𝜓¹_𝑖∈𝐿(𝐴) (2.1) for any 𝑎∈𝐴. If 𝐴 is finite dimensional over a field of characteristic 0 and if 𝐼 is a nil ideal, then 𝐵⋅𝐼 is nil. In particular the Jacobson radical of 𝐴 is 𝐵-stable.

Proof. Denote the trace of a 𝑘-linear endomorphism 𝑓 of 𝐴 by Tr(𝑓). Let 𝜓∈𝐵, 𝑎∈𝐴. Using Tr(𝑓𝑔ℎ)=Tr(ℎ𝑓𝑔) and the hypotheses we get Tr𝐿𝜓,𝑎=Tr^𝑛_𝑖=1𝜓²_𝑖∘𝜓¹_𝑖∘𝐿(𝑎)=Tr𝐿𝑦∘𝐿(𝑎)=Tr𝐿𝑦𝑎 (2.2) for some 𝑦∈𝐴. Suppose that 𝑎∈𝐼 with 𝐼 a nil ideal, then 𝑦𝑎∈𝐼 is nilpotent, hence Tr(𝐿(⟨𝜓,𝑎⟩))=Tr(𝐿(𝑦𝑎))=0. For any 𝑘>0 set 𝑧^𝑘∶=⟨𝜓,𝑎⟩^𝑘. Then 𝑧^𝑘=𝐿𝜓,𝑎,𝑧^𝑘−1=^𝑛_𝑖=1𝜓¹_𝑖,𝑎_𝑖 (2.3) for 𝑎_𝑖=𝑎⟨𝜓²_𝑖,𝑧^𝑘−1⟩. Since 𝐼 is an ideal, 𝑎_𝑖∈𝐼. Hence Tr𝐿(𝑧)^𝑘=Tr𝐿𝑧^𝑘=^𝑛_𝑖=1Tr𝐿𝜓¹_𝑖,𝑎_𝑖=0. (2.4) Since 𝐴 is finite dimensional, char(𝑘)=0 and the trace of all powers of 𝐿(𝑧) is zero, 𝐿(𝑧) is a nilpotent operator, that is, 𝑧=⟨𝜓,𝑎⟩ is nilpotent. Thus 𝐵⋅𝐼 is a nil ideal. Since the Jacobson radical of an Artinian ring is the largest nilpotent ideal, we have 𝐵⋅Jac(𝐴)=Jac(𝐴).

The last lemma, which had been proven first by Linchenko for Hopf actions and then by Nikshych for weak Hopf actions allows us to show the stability of the Jacobson radical of an algebra 𝐴 which satisfies a polynomial identity and on which act some operator algebra 𝐵 which is finitely generated over 𝐴. The hypotheses of the following theorem allow the reduction to finite-dimensional factors.

Theorem 2.2. Let 𝐿(𝐴)⊆𝐵⊆End_𝑘(𝐴) over some field 𝑘 of characteristic 0 with 𝐵 being finitely generated as right 𝐴-module. Suppose that for all 𝜓∈𝐵 there exist 𝑛≥1 and elements 𝜓¹₁,…,𝜓¹_𝑚,𝜓²₁,…,𝜓²_𝑚∈𝐵 satisfying 𝐿𝜓,𝑎=^𝑚_𝑖=1𝜓¹_𝑖∘𝐿(𝑎)∘𝜓²_𝑖,^𝑚_𝑖=1𝜓²_𝑖∘𝜓¹_𝑖∈𝐿(𝐴) (2.5) for any 𝑎∈𝐴. If 𝐴 satisfies a polynomial identity or if 𝑘 is an uncountable algebraically closed field, 𝐴 countably generated and all left primitive factor rings of 𝐴 are Artinian, then 𝐵⋅𝐼⊆Jac(𝐴) for all nil ideals 𝐼 of 𝐴.

Proof. Let 𝐼 be a nil ideal. It is enough to show that (𝐵⋅𝐼)𝑉=0 for all simple left 𝐴-modules 𝑉, then 𝐵⋅𝐼⊆Jac(𝐴). Let 𝑉 be a simple left 𝐴-module and 𝑃=Ann_𝐴(𝑉) be its annihilator. If 𝑘 is an uncountable algebraically closed field and 𝐴 is countably generated, then it satisfies the Nullstellensatz, hence End_𝑘(𝑉)=𝑘 (see [10, 9.1.8]). If primitive factors of 𝐴 are Artinian, then by the Weddeburn-Artin Theorem 𝐴/𝑃≃𝑀_𝑛(𝑘) for some 𝑛, hence 𝑉 is a finite-dimensional simple left 𝐴-module. On the other hand, if 𝐴 satisfies a polynomial identity, then 𝐴/𝑃≃𝑀_𝑛(𝐷) where 𝐷 is a finite-dimensional division algebra over 𝐹=𝑍(𝐴/𝑃) by Kaplansky's theorem [10, 13.3.8]. Tensoring 𝐴 by 𝐹 yields an 𝐹-algebra 𝐴∶=𝐴⊗𝐹 with 𝐹-action on the right. Then 𝐿𝐴=𝐿(𝐴)⊗𝐹⊆𝐵⊗𝐹⊆End_𝑘(𝐴)⊗𝐹⊆End_𝐹(𝐴⊗𝐹)=End_𝐹𝐴. (2.6) Moreover 𝑉 is a finite-dimensional simple left 𝐴-module since Ann_𝐴(𝑉)=𝑃⊗𝐹=∶𝑃 and 𝐴/𝑃≃𝑀_𝑛(𝐷) is finite dimensional over 𝐹. Note also that the nil ideal 𝐼 extends to a nil ideal 𝐼̃∶=𝐼⊗𝐹 since by [11, Theorem  5] 𝐼 is a locally nilpotent algebra and hence any element ∑^𝑚_𝑖=1𝑎_𝑖⊗𝑓_𝑖 lies in a nilpotent finitely generated subalgebra generated by the 𝑎_𝑖's and 𝐹.
To summarize, our hypothesis on 𝐴 allows us to consider 𝑉 to be a finite dimensional simple left 𝐴-module, where 𝐴 and 𝐵 are algebras over some field 𝑘 of characteristic 0. Denote by 𝑊=𝐵⊗_𝐴𝑉 the induced left 𝐵-module. Since 𝐵_𝐴 is finitely generated and 𝑉 is finite dimensional, 𝑊 is finite dimensional. Note that the left 𝐵-action on 𝑊 is given by 𝜓⋅(𝜙⊗𝑣)∶=(𝜓∘𝜙)⊗𝑣. Let 𝑄=Ann_𝐴(𝑊). Then 𝑄 is 𝐵-stable, because if 𝑎∈𝑄 and 𝜓∈𝐵, then by hypothesis there exist elements 𝜓¹₁,…,𝜓¹_𝑚,𝜓²₁,…,𝜓²_𝑚∈𝐵 satisfying (2.5). Thus for any 𝑤=𝜙⊗𝑣∈𝑊 we have ⋅=𝐿=𝜓,𝑎𝜙⊗𝑣𝜓,𝑎∘𝜙⊗𝑣_𝑚𝑖=1𝜓¹_𝑖∘𝐿(𝑎)∘𝜓²_𝑖=∘𝜙⊗𝑣_𝑚𝑖=1𝜓¹_𝑖⋅𝑎⋅𝜓²_𝑖⋅𝑤=0, (2.7) since 𝜓²_𝑖⋅𝑤∈𝑊 and 𝑎⋅𝑊=0. Hence 𝐵⋅𝑄⊆𝑄. Let 𝑄_𝐵=Ann_𝐵(𝐴/𝑄). Then 𝐴𝑄𝐴≃𝐿𝑄⊆𝐵𝑄_𝐵⊆End_𝑘𝐴𝑄. (2.8) Since 𝑊 is finite dimensional, 𝐴/𝑄 is finite dimensional. Note that 𝑉 is a simple left 𝐴/𝑄-module. Any nil ideal 𝐼 of 𝐴 yields a nil ideal (𝐼+𝑄)/𝑄 of 𝐴/𝑄. Moreover every element 𝜓+𝑄_𝐵∈𝐵/𝑄_𝐵 satisfies (2.5). By Lemma 2.1, ((𝐵⋅𝐼)+𝑄)/𝑄=𝐵/𝑄_𝐵⋅(𝐼+𝑄)/𝑄 is included in Jac(𝐴/𝑄). Thus ()𝐵𝐵⋅𝐼𝑉=𝑄_𝐵⋅(𝐼+𝑄)𝑄𝐴⋅𝑉⊆Jac𝑄⋅𝑉=0. (2.9) Hence 𝐵⋅𝐼⊆Jac(𝐴) for any nil ideal 𝐼 of 𝐴.

3. Weak Hopf Actions on Algebras Satisfying a Polynomial Identity

Before we apply the results from the previous section, we recall the definition of weak Hopf algebras (or quantum groupoids) as introduced by Böhm et al. in [1].

Definition 3.1. An associative 𝑘-algebra 𝐻 with multiplication 𝑚 and unit 1 which is also a coassociative coalgebra with comultiplication Δ and counit 𝜖 is called a weak Hopf algebra if it satisfies the following properties: (1)the comultiplication is multiplicative, that is, for all 𝑔,ℎ∈𝐻: Δ𝑔ℎ=Δ𝑔Δ(ℎ), (3.1)(2)the unit and counit satisfy: ()Δ(1)=(Δ(1)(1=((1Δ(1)),𝜖Δ⊗id⊗1)(1⊗Δ))1⊗Δ))(⊗1𝑓𝑔ℎ=𝜖𝑓𝑔₁𝜖𝑔₂ℎ=𝜖𝑓𝑔₂𝜖𝑔₁ℎ, (3.2)(3)there exists a linear map 𝑆∶𝐴→𝐴, called antipode, such that ℎ₁𝑆ℎ₂=(𝜖⊗id)(Δ(1)(ℎ⊗1))=∶𝜖_𝑡(ℎ),𝑆ℎ₁ℎ₂=(id⊗𝜖)((1⊗ℎ)Δ(1))=∶𝜖_𝑠(ℎ),𝑆(ℎ)=𝑆ℎ₁ℎ₂𝑆ℎ₃. (3.3)Note that we will use Sweedler’s notation for the comultiplication with suppressed summation symbol.

The image of 𝜖_𝑡 and 𝜖_𝑠 are subalgebras 𝐻_𝑡 and 𝐻_𝑠 of 𝐻 which are separable over 𝑘 [15, 2.3.4] and their images commute with each other. Those subalgebras are also characterized by 𝐻_𝑡={ℎ∈𝐻∶Δ(ℎ)=1₁ℎ⊗1₂}, respectively, 𝐻_𝑠={ℎ∈𝐻∶Δ(ℎ)=1₁⊗1₂ℎ}.

A left 𝐻-module algebra 𝐴 over a weak Hopf algebra 𝐻 is an associative unital algebra 𝐴 such that 𝐴 is a left 𝐻-module and for all 𝑎,𝑏∈𝐴,ℎ∈𝐻:ℎ⋅(𝑎𝑏)=ℎ₁⋅𝑎ℎ₂⋅𝑏,ℎ⋅1_𝐴=𝜖_𝑡(ℎ)⋅1_𝐴. (3.4)

Let 𝐴 be a left 𝐻-module algebra over a weak Hopf algebra 𝐻 and let 𝜆 be the ring homomorphism from 𝐻 to End_𝑘(𝐴) that defines the left module structure on 𝐴, that is, ⟨𝜆(ℎ),𝑎⟩∶=ℎ⋅𝑎 for all ℎ∈𝐻,𝑎∈𝐴. Property (3.4) of the definition above can be interpreted as an intertwining relation 𝜆(ℎ)∘𝐿(𝑎)=𝐿(ℎ₁⋅𝑎)∘𝜆(ℎ₂) of left multiplications 𝐿(𝑎) and left 𝐻-actions 𝜆(ℎ).

The following properties are now deduced from the axioms.

Lemma 3.2. Let 𝐴 be a left 𝐻-module algebra over a weak Hopf algebra 𝐻. Then (1)for all 𝑧∈𝐻_𝑡∶𝜆(𝑧)=𝐿(𝑧⋅1_𝐴) and for all 𝑧∈𝐻_𝑠∶𝜆(𝑧)=𝑅(𝑧⋅1_𝐴), (2)for all ℎ∈𝐻,𝑎∈𝐴∶𝐿(ℎ⋅𝑎)=𝜆(ℎ₁)∘𝐿(𝑎)∘𝜆(𝑆(ℎ₂)),(3)if 𝑆²=id, then 𝜆(𝑆(ℎ₂))∘𝜆(ℎ₁)∈𝐿(𝐴) for all ℎ∈𝐻.

Proof. (1) Let 𝑧∈𝐻_𝑡. Since Δ(𝑧)=1₁𝑧⊗1₂, we have for all 𝑎∈𝐴: 𝑧⋅𝑎=𝑧⋅1_𝐴𝑎=1₁𝑧⋅1_𝐴1₂⋅𝑎=1_𝐻⋅𝑧⋅1_𝐴𝑎=𝑧⋅1_𝐴𝑎. (3.5) The proof of the second statement is analogous.
(2) For ℎ∈𝐻,𝑎,𝑥∈𝐴 we have (ℎ⋅𝑎)𝑥=ℎ₁⋅𝑎𝜖_𝑡ℎ₂⋅1_𝐴𝑥=ℎ₁⋅𝑎𝜖_𝑡ℎ₂⋅𝑥=ℎ₁⋅𝑎ℎ₂𝑆ℎ₃⋅𝑥=ℎ₁⋅𝑎𝑆ℎ₃⋅𝑥. (3.6)
(3) Suppose 𝑆²=id, then 𝑆(𝜖_𝑠(ℎ))=𝑆(ℎ₂)ℎ₁ and as 𝑆(𝐻_𝑠)⊆𝐻_𝑡, we have using (1): 𝜆𝑆ℎ₂∘𝜆ℎ₁=𝜆𝑆𝜖_𝑠(ℎ)=𝐿𝑆𝜖_𝑠(ℎ)⋅1_𝐴∈𝐿(𝐴). (3.7)

We say that a weak Hopf algebra 𝐻 is involutive if its antipode is an involution. Any groupoid algebra is an involutive weak Hopf algebra. Moreover any semisimple Hopf algebra over a field of characteristic zero is involutive. Say that 𝐻 acts finitely on a left 𝐻-module algebra 𝐴 if the image of 𝜆∶𝐻→End_𝑘(𝐴) is finite dimensional. The following statement follows from the last lemma and Theorem 2.2.

Theorem 3.3. Let 𝐻 be an involutive weak Hopf algebra over a field 𝑘 of characteristic zero acting finitely on a left 𝐻-module algebra 𝐴. If 𝐴 satisfies a polynomial identity or if 𝑘 is an uncountable algebraically closed field, 𝐴 is countably generated and all left primitive factor rings of 𝐴 are Artinian, then the Jacobson radical of 𝐴 is 𝐻-stable.

Proof. Let 𝜆∶𝐻→End_𝑘(𝐴) be the ring homomorphism inducing the left 𝐻-module structure on 𝐴. Denote by 𝐵 the subalgebra of End_𝑘(𝐴) generated by 𝐿(𝐴) and 𝜆(𝐻). Let ℎ₁,…,ℎ_𝑚 be elements of 𝐻 such that {𝜆(ℎ₁),…,𝜆(ℎ_𝑚)} forms a basis of 𝜆(𝐻). We claim that any element of 𝐵 is of the form ∑^𝑚_𝑖=1𝜆(ℎ_𝑖)∘𝐿(𝑎_𝑖) for some 𝑎_𝑖∈𝐴. It is enough to show 𝐿(𝐴)𝜆(𝐻)⊆𝜆(𝐻)𝐿(𝐴). So take elements ℎ∈𝐻 and 𝑎,𝑏∈𝐴. Then using Lemma 3.2(2), 𝑆⁻¹=𝑆 and 𝑆(𝐻_𝑡)=𝐻_𝑠 we have ℎ₂⋅𝑆ℎ₁⋅𝑎𝑏=ℎ₂𝑆ℎ₁⋅𝑎ℎ₃=𝑆𝜖⋅𝑏_𝑡ℎ₁⋅𝑎ℎ₂⋅𝑏=𝑎𝜖_𝑡ℎ₁⋅1_𝐴ℎ₂⋅𝑏=𝑎(ℎ⋅𝑏). (3.8) This shows the intertwining relation 𝐿(𝑎)∘𝜆(ℎ)=𝜆(ℎ₂)∘𝐿(𝑆(ℎ₁)⋅𝑎) in 𝐵 which yields that 𝐵 is finitely generated as a right 𝐴-module. By the definition of module algebras, we also have that 𝜆(ℎ)∘𝐿(𝑎)=𝐿(ℎ₁⋅𝑎)∘𝜆(ℎ₂). Hence 𝜆(𝐻)𝐿(𝐴)=𝐿(𝐴)𝜆(𝐻). For any 𝑎∈𝐴 and 𝜓=∑^𝑚_𝑖=1𝐿(𝑎_𝑖)∘𝜆(ℎ_𝑖)∈𝐵 we have by Lemma 3.2(3) and by (3.8): 𝐿𝜓,𝑎=𝐿𝑎_𝑖∘𝐿ℎ_𝑖𝑎,𝑎=𝐿_𝑖∘𝜆ℎ_𝑖1∘𝐿(𝑎)∘𝜆𝑆ℎ_𝑖2=𝜆ℎ_𝑖2∘𝐿𝑆ℎ_𝑖1⋅𝑎_𝑖∘𝐿(𝑎)∘𝜆𝑆ℎ_𝑖3=_𝑗𝜓¹_𝑗∘𝐿(𝑎)∘𝜓²_𝑗 (3.9) for 𝜓¹_𝑗=𝜆(ℎ_𝑖2)∘𝐿(𝑆(ℎ_𝑖1)⋅𝑎_𝑖), 𝜓²_𝑗=𝜆(𝑆(ℎ_𝑖3), and some appropriate choice of indices 𝑗. Moreover 𝜓²_𝑗∘𝜓¹_𝑗=𝜆𝑆ℎ_𝑖3∘𝜆ℎ_𝑖2∘𝐿𝑆ℎ_𝑖1⋅𝑎_𝑖=𝐿𝑦_ℎ𝑖2∘𝐿𝑆ℎ_𝑖1⋅𝑎_𝑖∈𝐿(𝐴) (3.10) for some elements 𝑦_ℎ𝑖2∈𝐴 that exist by Lemma 3.2(4). Therefore the hypotheses of Theorem 2.2 are fulfilled and the statement follows.

3.1. Smash Products of Weak Hopf Actions

Recall that the smash product 𝐴#𝐻 of a left 𝐻-module algebra 𝐴 and a weak Hopf algebra 𝐻 is defined on the tensor product 𝐴⊗_𝐻𝑡𝐻 where 𝐴 is considered a right 𝐻_𝑡-module by 𝑎⋅𝑧=𝑎(𝑧⋅1_𝐴) for 𝑎∈𝐴,𝑧∈𝐻_𝑡. The (𝑘-linear) dual 𝐻^∗ of 𝐻 becomes also a weak Hopf algebra and acts on 𝐴#𝐻 by 𝜙⋅(𝑎#ℎ)∶=𝑎#𝜙(⇀ℎ), where 𝜙⇀ℎ=ℎ₁⟨𝜙,ℎ₂⟩. Using the Montgomery-Blattner duality theorem for weak Hopf algebras proven by Nikshych we have the following.

Lemma 3.4. Let 𝐻 be a finite-dimensional weak Hopf algebra and 𝐴 a left 𝐻-module algebra. Then 𝐴#𝐻 is a finitely generated projective right 𝐴-module and 𝐴#𝐻#𝐻^∗≃𝑒𝑀_𝑛(𝐴)𝑒 for some idempotent 𝑒∈𝑀_𝑛(𝐴) where 𝑀_𝑛(𝐴) denotes the ring of 𝑛×𝑛-matrixes for some number 𝑛>0.

Proof. By [14, Theorem 3.3] 𝐴#𝐻#𝐻^∗≃End(𝐴#𝐻_𝐴). Since 𝐻_𝑡 is a separable 𝑘-algebra, it is semisimple Artinian. Hence 𝐻 is a (finitely generated) projective right 𝐻_𝑡-module and 𝐻 is a direct summand of 𝐻^𝑛_𝑡 for some 𝑛>0. Moreover it follows from the proof of Lemma 3.2 that 𝐴#𝐻=(1#𝐻)(𝐴#1). Thus 𝐻⊗_𝐻𝑡𝐴≃𝐴#𝐻 as right 𝐴-modules by ℎ⊗𝑎↦(1#ℎ)(𝑎#1). On the other hand 𝐻⊗_𝐻𝑡𝐴 is a direct summand of 𝐻^𝑛_𝑡⊗_𝐻𝑡𝐴≃𝐴^𝑛 as right 𝐴-module. Hence 𝐴#𝐻 is a projective right 𝐴-module of rank 𝑛 and End(𝐴#𝐻)_𝐴≃𝑒𝑀_𝑛(𝐴)𝑒 for some idempotent 𝑒∈𝑀_𝑛(𝐴).

3.2. Semiprime Smash Products for Weak Hopf Actions

We can now transfer Linchenko and Montgomery's result [7, Theorem 3.4] on the semiprimness of smash products to weak Hopf actions.

Theorem 3.5. Let 𝐴 be a left 𝐻-module algebra over a finite dimensional involutive weak Hopf algebra 𝐻 over a field of characteristic zero. If 𝐴 is semiprime and satisfies a polynomial identity, then 𝐴#𝐻 is semiprime.

Proof. Set 𝐵=𝐴#𝐻#𝐻^∗. Note that 𝐻^∗ is also involutive since its antipode is defined by ⟨𝑆^∗(𝜙),ℎ⟩∶=⟨𝜙,𝑆(ℎ)⟩ for all 𝜙∈𝐻^∗,ℎ∈𝐻. By [12, Corollary  6.5] 𝐻^∗ is semisimple and by [1, 3.13] there exists a normalized left integral Λ∈𝐻^∗. This implies that 𝐴#𝐻 is a projective left 𝐵-module as the left 𝐵-linear map 𝐴#𝐻→𝐵 with 𝑎#ℎ↦𝑎#ℎ#Λ splits the projection 𝐵→𝐴#𝐻 given by 𝑎#ℎ#𝜙↦𝑎#ℎ(𝜙⇀1).
First suppose that Jac(𝐴)=0. By Lemma 3.4, Jac(𝐵)≃𝑒𝑀_𝑛(Jac(𝐴))𝑒=0 for some idempotent 𝑒. This implies also that Rad_𝐵(𝐴#𝐻)=0 as well, since 𝐴#𝐻 is supposed to be a projective left 𝐵-module. Recall that the radical Rad(𝑀) of a module 𝑀 is the intersection of all maximal submodules of 𝑀 or equivalently the sum of all small submodules, that is, of those submodules 𝑁 of 𝑀 such that 𝑁+𝐿≠𝑀 for all 𝐿≠𝑀.
Since 𝐴#𝐻 is a finite extension of 𝐴, also 𝐴#𝐻 satisfies a polynomial identity and since 𝐻^∗ is finite dimensional it acts finitely on 𝐴#𝐻. Thus Theorem 3.3 applies and for any nil ideal 𝐼 of 𝐴#𝐻 we have 𝐵⋅𝐼⊆Jac(𝐴#𝐻). On the other hand any 𝐵-submodule 𝑁 of Jac(𝐴#𝐻) is contained in Rad_𝐵(𝐴#𝐻), which is zero. Hence 𝐼=0 and 𝐴#𝐻 is semiprime.
In general, if 𝐴 is semiprime, we can extend the 𝐻-action of 𝐴 to the polynomial ring 𝐴[𝑥] by identifying 𝐴[𝑥] with 𝐴⊗_𝐻𝑡𝐻_𝑡[𝑥], which is a left 𝐻-module algebra, where 𝐻 acts on 𝑥 by ℎ⋅𝑥=(𝜖_𝑡(ℎ)⋅1_𝐴)𝑥. Since 𝐴 is semiprime, satisfying a polynomial identity, Jac(𝐴[𝑥])=0 by [13]. Moreover 𝐴[𝑥] also satisfies a polynomial identity and by the argument above 𝐴[𝑥]#𝐻 is semiprime. As any ideal 𝐼 of 𝐴#𝐻 can be extended to an ideal 𝐼[𝑥] of 𝐴#𝐻[𝑥]=𝐴[𝑥]#𝐻, also 𝐴#𝐻 is semiprime.

Acknowledgment

The first author was supported by grant SFRH/PROTEC/49857/2009. The second author was partially supported by Centro de Matemática da Universidade do Porto (CMUP), financed by FCT (Portugal) through the programs POCTI (Programa Operacional Ciência, Tecnologia, Inovação) and POSI (Programa Operacional Sociedade da Informação), with national and European community structural funds.