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Quantum Groupoids Acting on Semiprime Algebras
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.
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. . The action of such objects on algebras, as given by quantum groupoids acting on -algebras,  or weak Hopf algebras arising from Jones towers  are particularly interesting. New examples of weak Hopf algebras arose from double groupoids , which were also used to find new weak Hopf actions (see ).
A long-standing open problem in the theory of Hopf action is to show that the smash product of a semiprime module algebra and a semisimple Hopf algebra is again semiprime (see ) (an algebra is semiprime if it does not contain nonzero nilpotent ideals.). The case of being commutative had been settled in . The most recent partial answer to this problem has been given by Linchenko and Montgomery in  where they prove the semiprimness of under the condition of 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 be a commutative ring and let be an associative unital -algebra. For any define two linear operators and in given by and for all . We identify with the subalgebra of generated by all left multiplications and denote the subalgebra generated by all operators and by , which is also sometimes referred to as the multiplication algebra of . As a left -module, is isomorphic to since we assume to be unital. We will be interested in certain actions on an algebra that may stem from a bialgebra or more generally a bialgebroid. The situation we will encounter is the one where we have an extension where acts on through a ring homomorphism such that for all . For the intrinsic properties of under this action it is enough to look at the subalgebra in generated by this action and we might consider intermediate algebras instead. Hence let be a subalgebra of that contains . Then becomes a cyclic faithful left -module by evaluating endomorphisms, that is, for all . Note that for any we have .
Since we assume to be unital, the map with , for all —evaluating endomorphisms at 1—is an injective ring homomorphism, since for all . Moreover if , then and . The subalgebra can be described as the set of elements such that for any , which we will denote by . On one hand if for some , then for any and on the other hand if , then is left -linear since for any and : Thus .
Let be any left -module and define . With the same argument as above one sees that with is an isomorphism of abelian groups, hence yielding a left -module structure on . Moreover it is possible to show that is isomorphic to as functors from -Mod to -Mod.
A subset of is called -stable if . The -stable left ideals are precisely the (left) -submodules of . In particular , for any -stable left ideal of .
Examples 1.1. The following list illustrates that our aproach reflects many interesting cases of algebras with actions. (1)Let be the multiplication algebra of , then is a faithful cyclic left -module. The left -modules are precisely the -bimodules, in particular the left ideals of are the two-sided ideals of , and holds for any -bimodule . The operator algebra is a quotient of the enveloping algebra through the map , for all . (2)Let be a group acting as (-linear) automorphisms on , that is, there exists a group homomorphism . Set for any . Define . Then the left -submodules of are precisely the -stable left ideals of and . is a quotient of the skew group ring whose underlying -submodule is the free left -module with basis and whose multiplication is given by . Note that for any left -module we have, is the set of fixed elements of .(3)Let be an -algebra with involution and let be the subalgebra of generated by and . Since for any we got . This means (as it is well-known) that any left ideal of which is stable under is a two-sided ideal. Note that can be seen as the factor ring of the skew-group ring where is the cyclic group of order two and is given by .(4)Let be an -linear derivation of and consider . The left -submodules of are the left ideals that satisfy . The operator algebra is a factor of the ring of differential operator , which as a left -module is equal to and its multiplication is given by . The map with is a surjective -algebra homomorphism and for any left -module we have . In particular . The subring of is called the ring of constants of .(5)Let be an -Hopf algebra action on . Let us denote the action of an element on by and define . The smash product is an extension with additional module structure. Define by .
2. Linear Operators Acting on Algebras Satisfying a Polynomial Identity
Let 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 and suppose that for all there exist and elements , such that for any . If is finite dimensional over a field of characteristic 0 and if is a nil ideal, then is nil. In particular the Jacobson radical of is -stable.
Proof. Denote the trace of a -linear endomorphism of by . Let , . Using and the hypotheses we get for some . Suppose that with a nil ideal, then is nilpotent, hence . For any set . Then for . Since is an ideal, . Hence Since is finite dimensional, and the trace of all powers of is zero, is a nilpotent operator, that is, is nilpotent. Thus is a nil ideal. Since the Jacobson radical of an Artinian ring is the largest nilpotent ideal, we have .
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 which satisfies a polynomial identity and on which act some operator algebra which is finitely generated over . The hypotheses of the following theorem allow the reduction to finite-dimensional factors.
Theorem 2.2. Let over some field of characteristic 0 with being finitely generated as right -module. Suppose that for all there exist and elements satisfying for any . If satisfies a polynomial identity or if is an uncountable algebraically closed field, countably generated and all left primitive factor rings of are Artinian, then for all nil ideals of .
Proof. Let be a nil ideal. It is enough to show that for all simple left -modules , then . Let be a simple left -module and be its annihilator. If is an uncountable algebraically closed field and is countably generated, then it satisfies the Nullstellensatz, hence (see [10, ]). If primitive factors of are Artinian, then by the Weddeburn-Artin Theorem for some , hence is a finite-dimensional simple left -module. On the other hand, if satisfies a polynomial identity, then where is a finite-dimensional division algebra over by Kaplansky's theorem [10, ]. Tensoring by yields an -algebra with -action on the right. Then
Moreover is a finite-dimensional simple left -module since and is finite dimensional over . Note also that the nil ideal extends to a nil ideal since by [11, Theorem 5] is a locally nilpotent algebra and hence any element lies in a nilpotent finitely generated subalgebra generated by the 's and .
To summarize, our hypothesis on allows us to consider to be a finite dimensional simple left -module, where and are algebras over some field of characteristic 0. Denote by the induced left -module. Since is finitely generated and is finite dimensional, is finite dimensional. Note that the left -action on is given by . Let . Then is -stable, because if and , then by hypothesis there exist elements satisfying (2.5). Thus for any we have since and . Hence . Let . Then Since is finite dimensional, is finite dimensional. Note that is a simple left -module. Any nil ideal of yields a nil ideal of . Moreover every element satisfies (2.5). By Lemma 2.1, is included in . Thus Hence for any nil ideal of .
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 .
Definition 3.1. An associative -algebra with multiplication and unit 1 which is also a coassociative coalgebra with comultiplication and counit is called a weak Hopf algebra if it satisfies the following properties: (1)the comultiplication is multiplicative, that is, for all : (2)the unit and counit satisfy: (3)there exists a linear map , called antipode, such that Note that we will use Sweedler’s notation for the comultiplication with suppressed summation symbol.
The image of and are subalgebras and of which are separable over [15, ] and their images commute with each other. Those subalgebras are also characterized by , respectively, .
A left -module algebra over a weak Hopf algebra is an associative unital algebra such that is a left -module and for all :
Let be a left -module algebra over a weak Hopf algebra and let be the ring homomorphism from to that defines the left module structure on , that is, for all . Property (3.4) of the definition above can be interpreted as an intertwining relation of left multiplications and left -actions .
The following properties are now deduced from the axioms.
Lemma 3.2. Let be a left -module algebra over a weak Hopf algebra . Then (1)for all and for all , (2)for all ,(3)if , then for all .
Proof. (1) Let . Since , we have for all :
The proof of the second statement is analogous.
(2) For we have
(3) Suppose , then and as , we have using :
We say that a weak Hopf algebra 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 acts finitely on a left -module algebra if the image of is finite dimensional. The following statement follows from the last lemma and Theorem 2.2.
Theorem 3.3. Let be an involutive weak Hopf algebra over a field of characteristic zero acting finitely on a left -module algebra . If satisfies a polynomial identity or if is an uncountable algebraically closed field, is countably generated and all left primitive factor rings of are Artinian, then the Jacobson radical of is -stable.
Proof. Let be the ring homomorphism inducing the left -module structure on . Denote by the subalgebra of generated by and . Let be elements of such that forms a basis of . We claim that any element of is of the form for some . It is enough to show . So take elements and . Then using Lemma 3.2(2), and we have This shows the intertwining relation in which yields that is finitely generated as a right -module. By the definition of module algebras, we also have that . Hence . For any and we have by Lemma 3.2(3) and by (3.8): for , , and some appropriate choice of indices . Moreover for some elements 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 of a left -module algebra and a weak Hopf algebra is defined on the tensor product where is considered a right -module by for . The (-linear) dual of becomes also a weak Hopf algebra and acts on by , where . Using the Montgomery-Blattner duality theorem for weak Hopf algebras proven by Nikshych we have the following.
Lemma 3.4. Let be a finite-dimensional weak Hopf algebra and a left -module algebra. Then is a finitely generated projective right -module and for some idempotent where denotes the ring of -matrixes for some number .
Proof. By [14, Theorem 3.3] . Since is a separable -algebra, it is semisimple Artinian. Hence is a (finitely generated) projective right -module and is a direct summand of for some . Moreover it follows from the proof of Lemma 3.2 that . Thus as right -modules by . On the other hand is a direct summand of as right -module. Hence is a projective right -module of rank and for some idempotent .
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 be a left -module algebra over a finite dimensional involutive weak Hopf algebra over a field of characteristic zero. If is semiprime and satisfies a polynomial identity, then is semiprime.
Proof. Set . Note that is also involutive since its antipode is defined by for all . By [12, Corollary 6.5] is semisimple and by [1, 3.13] there exists a normalized left integral . This implies that is a projective left -module as the left -linear map with splits the projection given by .
First suppose that . By Lemma 3.4, for some idempotent . This implies also that as well, since is supposed to be a projective left -module. Recall that the radical of a module is the intersection of all maximal submodules of or equivalently the sum of all small submodules, that is, of those submodules of such that for all .
Since is a finite extension of , also satisfies a polynomial identity and since is finite dimensional it acts finitely on . Thus Theorem 3.3 applies and for any nil ideal of we have . On the other hand any -submodule of is contained in , which is zero. Hence and is semiprime.
In general, if is semiprime, we can extend the -action of to the polynomial ring by identifying with , which is a left -module algebra, where acts on by . Since is semiprime, satisfying a polynomial identity, by . Moreover also satisfies a polynomial identity and by the argument above is semiprime. As any ideal of can be extended to an ideal of , also is semiprime.
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.
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Copyright © 2011 Inês Borges and Christian 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.