Quantum Groupoids Acting on Semiprime Algebras

Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2011, Article ID 546058, 9 pages doi:10.1155/2011/546058 Research Article Quantum Groupoids Acting on Semiprime Algebras Ineˆs Borges1 and Christian Lomp2 <ul style="display: flex;"><li style="flex:1">1 Instituto Superior De Contabilidade e Administra c ¸ao de Coimbra, Quinta Agr ı cola-Bencanta, </li><li style="flex:1">˜</li><li style="flex:1">´</li></ul>3040-316 Coimbra, Portugal <ul style="display: flex;"><li style="flex:1">2 Departamento de Matematica, Faculdade de Ciencias, Universidade do Porto, Rua Campo Alegre 687, </li><li style="flex:1">´</li><li style="flex:1">ˆ</li></ul>4169-007 Porto, Portugal Correspondence should be addressed to Christian Lomp, <a href="mailto:[email protected]" target="_blank">[email protected] </a>Received 30 March 2011; Accepted 5 June 2011 Academic Editor: Olaf Lechtenfeld Copyright q 2011 I. Borges and C. Lomp. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 <ul style="display: flex;"><li style="flex:1">groupoidsꢁ introduced by Bohm et al. ꢂ1ꢃ. The action of such objects on algebras, as given by </li><li style="flex:1">¨</li></ul>quantum groupoids acting on C∗-algebras, ꢂ2ꢃ or weak Hopf algebras arising from Jones towers ꢂ3ꢃ are particularly interesting. New examples of weak Hopf algebras arose from double groupoids ꢂ4ꢃ, which were also used to find new weak Hopf actions ꢀsee ꢂ2ꢃꢁ. 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 ꢂ5ꢃꢁ ꢀan algebra A is semiprime if it does not contain nonzero nilpotent ideals.ꢁ. The case of A being commutative had been settled in ꢂ6ꢃ. The most recent partial answer to this problem has been given by Linchenko and Montgomery in ꢂ7ꢃ 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. <ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">Advances in Mathematical Physics </li></ul>Let k be a commutative ring and let A be an associative unital k-algebra. For any a ∈ A define two linear operators Lꢀaꢁ and Rꢀaꢁ in EndkꢀAꢁ given by ꢀLꢀaꢁ, xꢁ ꢄ ax and ꢀRꢀaꢁ, xꢁ ꢄ xa for all x ∈ A. We identify A with the subalgebra LꢀAꢁ of EndkꢀAꢁ generated by all left multiplications Lꢀaꢁ and denote the subalgebra generated by all operators Lꢀaꢁ and Rꢀaꢁ by MꢀAꢁ, which is also sometimes referred to as the multiplication algebra of A. As a left LꢀAꢁ-module, A is isomorphic to LꢀAꢁ 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 → EndkꢀAꢁ such that ꢀaꢁφ ꢄ Lꢀaꢁ for all a ∈ A. For the intrinsic properties of A under this action it is enough to look at the subalgebra φꢀBꢁ in EndkꢀAꢁ generated by this action and we might consider intermediate algebras LꢀAꢁ ⊆ B ⊆ EndkꢀAꢁ instead. Hence let B be a subalgebra of EndkꢀAꢁ that contains LꢀAꢁ. Then A becomes a cyclic faithful left B-module by evaluating endomorphisms, that is, for all b ∈ B, a ∈ A : b·a :ꢄ ꢀb, aꢁ. Note that for any aꢂ ∈ A we have Lꢀaꢂꢁ·a ꢄ ꢀLꢀaꢂꢁ, aꢁ ꢄ aꢂa. Since we assume A to be unital, the map Ψ : EndBꢀAꢁ → A with Ψꢀfꢁ ꢄ ꢀ1ꢁf, for all f ∈ EndBꢀAꢁ—evaluating endomorphisms at 1—is an injective ring homomorphism, since for all f, g ∈ EndBꢀAꢁ : Ψꢀf ◦ gꢁ ꢄ ꢀꢀ1ꢁfꢁg ꢄ ꢀꢀ1ꢁf · 1ꢁg ꢄ ꢀ1ꢁf · ꢀ1ꢁg ꢄ ΨꢀfꢁΨꢀgꢁ. Moreover if Ψꢀfꢁ ꢄ ꢀ1ꢁf ꢄ 0, then ꢀaꢁf ꢄ aꢀ1ꢁf ꢄ 0 and f ꢄ 0. The subalgebra ΨꢀEndBꢀAꢁꢁ can be described as the set of elements a ∈ A such that for any b ∈ B : b · a ꢄ ꢀb · 1ꢁa, which we will denote by AB. On one hand if a ꢄ ꢀ1ꢁf ꢄ Ψꢀfꢁ for some f ∈ EndBꢀAꢁ, then for any b ∈ B : b·a ꢄ ꢀb·1ꢁf ꢄ Lꢀꢀb·1ꢁꢁ·ꢀ1ꢁf ꢄ ꢀb·1ꢁa and on the other hand if a ∈ AB, then f ꢄ Rꢀaꢁ is left B-linear since for any b ∈ B and x ∈ A: b · ꢀxꢁf ꢄ b · ꢀxaꢁ ꢄ b · ꢀLꢀxꢁ · aꢁ ꢄ ꢀbLꢀxꢁꢁ · a ꢄ ꢀbLꢀxꢁ · 1ꢁa ꢀ1.1ꢁ ꢄ ꢀb · ꢀLꢀxꢁ · 1ꢁꢁa ꢄ ꢀb · xꢁa ꢄ ꢀb · xꢁf. Thus EndBꢀAꢁ ꢃ AB. Let M be any left B-module and define MB ꢄ {m ∈ M : ∀b ∈ B : b · m ꢄ ꢀb · 1ꢁm}. With the same argument as above one sees that ΨM : HomBꢀA, Mꢁ → MB with Ψꢀfꢁ ꢄ ꢀ1ꢁf is an isomorphism of abelian groups, hence yielding a left AB-module structure on MB. Moreover it is possible to show that HomBꢀA, −ꢁ is isomorphic to ꢀ−ꢁB as functors from B-Mod to AB- Mod. A subset I of A is called B-stable if B · I ⊆ I. The B-stable left ideals are precisely the ꢀleftꢁ B-submodules of A. In particular HomBꢀA, Iꢁ ꢃ IB ꢄ I ∩ AB, 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. ꢀ1ꢁ Let B ꢄ MꢀAꢁ 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 HomBꢀA, Mꢁ ꢃ MB ꢄ ZꢀMꢁ ꢄ {m ∈ M | ∀a ∈ A : am ꢄ ma} holds for any A-bimodule M. The operator algebra B is a quotient of the enveloping algebra Ae ꢄ A⊗Aop through the map a⊗b → Lꢀaꢁ◦Rꢀbꢁ, for all a ⊗ b ∈ Ae. ꢀ2ꢁ Let G be a group acting as ꢀk-linearꢁ automorphisms on A, that is, there exists a group homomorphism η : G → AutkꢀAꢁ. Set ga ꢄ ꢀηꢀgꢁ, aꢁ for any a ∈ A, g ∈ G. <ul style="display: flex;"><li style="flex:1">Advances in Mathematical Physics </li><li style="flex:1">3</li></ul>Define B ꢄ ꢀLꢀAꢁ ∪ {ηꢀgꢁ | g ∈ G}ꢁ ⊆ EndkꢀAꢁ. Then the left B-submodules I of A gx <ul style="display: flex;"><li style="flex:1">are precisely the G-stable left ideals of A and HomBꢀA, Iꢁ ꢃ IB ꢄ {x ∈ I | </li><li style="flex:1">ꢄ x}. </li></ul>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#gꢁꢀb#hꢁ ꢄ agb#gh. Note that for any left A#G-module M we have, HomA#GꢀA, Mꢁ ꢄ HomBꢀA, Mꢁ ꢃ MG is the set of fixed elements of M. ꢀ3ꢁ Let A be an k-algebra with involution ∗ and let B be the subalgebra of EndAꢀ ꢁ generated by A and ∗. Since for any a ∈ A : Rꢀaꢁ ꢄ ∗ ◦ Lꢀa∗ꢁ ◦ ∗ we got MꢀAꢁ ⊆ B. This means ꢀas it is well-knownꢁ 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 Ae#G where G ꢄ {id, ∗} is the cyclic group of order two and ∗ ∈ AutꢀAeꢁ is given by ꢀa ⊗ bꢁ∗ :ꢄ b∗ ⊗ a∗. ꢀ4ꢁ Let δ ∈ DerkꢀAꢁ be an k-linear derivation of A and consider B ꢄ ꢀLꢀAꢁ ∪ {δ}ꢁ ⊆ EndkꢀAꢁ. The left B-submodules of A are the left ideals I that satisfy δꢀIꢁ ⊆ I. The operator algebra B is a factor of the ring of differential operator Aꢂz; δꢃ, which as a left A-module is equal to Aꢂzꢃ and its multiplication is given by za ꢄ az δꢀaꢁ. <ul style="display: flex;"><li style="flex:1">ꢀ</li><li style="flex:1">ꢀ</li></ul><ul style="display: flex;"><li style="flex:1">n</li><li style="flex:1">n</li></ul>The map Aꢂz; δꢃ → B with iꢄ0 aizi → iꢄ0 Lꢀaiꢁ ◦ δi ∈ B is a surjective k-algebra homomorphism and for any left Aꢂz; δꢃ-module M we have HomAꢂz;δꢃꢀA, Mꢁ ꢄ HomBꢀA, Mꢁ ꢄ Mδ ꢄ {m ∈ M | zm ꢄ 0}. In particular EndAꢂz;δꢃꢀAꢁ ꢃ Aδ ꢄ Kerꢀδꢁ. The subring Aδ of A is called the ring of constants of δ. ꢀ5ꢁ Let H be an k-Hopf algebra action on A. Let us denote the action of an element h ∈ H on A by λh ∈ EndkꢀAꢁ and define B ꢄ ꢀLꢀAꢁ ∪ {λh | h ∈ H}ꢁ ⊆ EndkꢀAꢁ. The smash product A#H is an extension with additional module structure. Define ϕ : A#H → EndkꢀAꢁ by ϕꢀa#hꢁ :ꢄ Lꢀaꢁ ◦ λh. 2. Linear Operators Acting on Algebras Satisfying a Polynomial Identity Let LꢀAꢁ ⊆ B ⊆ EndkꢀAꢁ be any intermediate algebra as above. The first technical lemma generalizes a corresponding result of Linchenko ꢂ8, Theorem 3.1ꢃ for Hopf actions and Nikshych ꢂ9, Theorem 6.1.3ꢃ for weak Hopf actions. Recall that an ideal whose elements are nilpotent is called a nil ideal. Lemma 2.1. Let LꢀAꢁ ⊆ B ⊆ EndkꢀAꢁ and suppose that for all ψ ∈ B there exist m ≥ 1 and elements ψ11, . . . , ψm1 , ψ12, . . . , ψm2 ∈ B such that <ul style="display: flex;"><li style="flex:1">m</li><li style="flex:1">n</li></ul><ul style="display: flex;"><li style="flex:1">ꢅ</li><li style="flex:1">ꢅ</li></ul><ul style="display: flex;"><li style="flex:1">ꢁꢂ </li><li style="flex:1">ꢃꢄ </li></ul><ul style="display: flex;"><li style="flex:1">L</li><li style="flex:1">ψ, a </li></ul>ꢄψi1 ◦ Lꢀaꢁ ◦ ψi2, ψi2 ◦ ψi1 ∈ LꢀAꢁ ꢀ2.1ꢁ <ul style="display: flex;"><li style="flex:1">iꢄ1 </li><li style="flex:1">iꢄ1 </li></ul>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. <ul style="display: flex;"><li style="flex:1">4</li><li style="flex:1">Advances in Mathematical Physics </li></ul>Proof. Denote the trace of a k-linear endomorphism f of A by Trꢀfꢁ. Let ψ ∈ B, a ∈ A. Using Trꢀf ghꢁ ꢄ Trꢀh f gꢁ and the hypotheses we get ꢆꢆ nꢅ<ul style="display: flex;"><li style="flex:1">ꢁ ꢁꢂ </li><li style="flex:1">ꢃꢄꢄ </li><li style="flex:1">ꢁ ꢁ ꢄ </li><li style="flex:1">ꢄ</li><li style="flex:1">ꢁ ꢁ ꢄꢄ </li></ul>Tr L ψ, a ꢄ Tr ψi2 ◦ ψi1 ◦ Lꢀaꢁ ꢄ Tr L y ◦ Lꢀaꢁ ꢄ Tr L ya ꢀ2.2ꢁ iꢄ1 for some y ∈ A. Suppose that a ∈ I with I a nil ideal, then ya ∈ I is nilpotent, hence kTrꢀLꢀꢀψ, aꢁꢁꢁ ꢄ TrꢀLꢀyaꢁꢁ ꢄ 0. For any k > 0 set zk :ꢄ ꢀψ, aꢁ . Then n<ul style="display: flex;"><li style="flex:1">ꢇ</li><li style="flex:1">ꢈ</li><li style="flex:1">ꢇ</li><li style="flex:1">ꢈ</li></ul>ꢅ<ul style="display: flex;"><li style="flex:1">ꢁꢂ </li><li style="flex:1">ꢃꢄ </li></ul>zk ꢄ <ul style="display: flex;"><li style="flex:1">L</li><li style="flex:1">ψ, a , zk−1 </li></ul>ꢄψi1, ai ꢀ2.3ꢁ iꢄ1 for ai ꢄ aꢀψi2, zk−1ꢁ. Since I is an ideal, ai ∈ I. Hence n<ul style="display: flex;"><li style="flex:1">ꢉ</li><li style="flex:1">ꢊ</li><li style="flex:1">ꢉ ꢉ ꢊꢊ </li><li style="flex:1">ꢉ ꢉꢇ </li><li style="flex:1">ꢈꢊꢊ </li></ul>ꢅTr Lꢀzꢁk ꢄ Tr L zk ꢄ<ul style="display: flex;"><li style="flex:1">Tr L ψi1, ai </li><li style="flex:1">ꢄ 0. </li></ul>ꢀ2.4ꢁ iꢄ1 Since A is finite dimensional, charꢀkꢁ ꢄ 0 and the trace of all powers of Lꢀzꢁ is zero, Lꢀzꢁ is a nilpotent operator, that is, z ꢄ ꢀψ, aꢁ 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 · JacꢀAꢁ ꢄ JacꢀAꢁ. 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 LꢀAꢁ ⊆ B ⊆ EndkꢀAꢁ 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, . . . , ψm1 , ψ12, . . . , ψm2 ∈ B satisfying <ul style="display: flex;"><li style="flex:1">m</li><li style="flex:1">m</li></ul><ul style="display: flex;"><li style="flex:1">ꢅ</li><li style="flex:1">ꢅ</li></ul><ul style="display: flex;"><li style="flex:1">ꢁꢂ </li><li style="flex:1">ꢃꢄ </li></ul><ul style="display: flex;"><li style="flex:1">L</li><li style="flex:1">ψ, a </li></ul>ꢄψi1 ◦ Lꢀaꢁ ◦ ψi2, ψi2 ◦ ψi1 ∈ LꢀAꢁ

Quantum Groupoids Acting on Semiprime Algebras

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