On the notion of retractable modules' in the context of algebras.

Palestine Journal of Mathematics Vol. 3(Spec 1) (2014) , 343–355 © Palestine Polytechnic University-PPU 2014 ON THE NOTION OF ’RETRACTABLE MODULES’ IN THE CONTEXT OF ALGEBRAS Christian Lomp Dedicated to Patrick Smith and John Clark on the occasion of their 70th birthdays. Communicated by Jawad Abuhlail MSC 2010 Classifications: Primary 16N60, 16D10; Secondary 16S40, 16W30. Keywords and phrases: retractable modules, algebras with large centre, fix rings, Hopf algebras, endomorphism rings. This research was funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT - Fundação para a Ciência e a Tecnologia under the project PEst-C/MAT/UI0144/2011. Abstract. This is a survey on the usage of the module theoretic notion of a “retractable module" in the study of algebras with actions. We explain how classical results can be interpreted using module theory and end the paper with some open questions. 1 Introduction This note is written for module theorists and intends to show where the module theoretic notion of a retractable module plays a role in the context of algebras with certain additional structure. These "additional structures" include group actions, involutions, Lie algebra actions or more generally Hopf algebra actions as well as the bimodule structure of the algebra (and combinations of all these). Such algebra Ais usually a subalgebra of a larger algebra Band has the structure of a cyclic left B-module, while its endomorphism ring EndB(A)is isomorphic to a subalgebra ABof A. In this intrinsic situation the condition on Ato be a retractable B-module means that the subring ABhas non-zero intersection with all non-zero left ideals of Athat are stable under the module action of B. We will first recall the module theoretical notion of a retractable module and set it in a categorical and lattice theoretical context. In the second section we will examine various situations of algebras Awith additional structures and recall manytheorem classical theorems that can be expressed in terms of module theory. The last section deals with open problems around retractable modules in the context of algebras. Note that all rings/algebras are considered to be associative and unital. Modules are usually meant to be left modules and homomorphisms are acting from the right. 2 Module Theory A retractable module is a (left) A-module M, over some ring A, such that there exist non-zero homomorphisms from Minto each of its non-zero submodules. The notion of a retractable module appeared first in the work of Khuri [17] and had since then been used in connection with primness conditions and the nonsingularity of a module and its endomorphism ring (see [12, 14–16, 33]). One of Khuri’s result is the establishment of a bijective correspondence between closed submodules of a module Mand closed left ideals of its endomorphism ring S=EndA(M)in case M is non-degenerated (see [13, 33, 34]). A module is non-degenerated if its standard Morita context is non-degenerated. Recall that a Morita context between two rings Aand Sis a quadruple (A, M, N, S)where AMSand SNAare bimodules with bimodule maps (−,−):M⊗SN→A and [−,−]:N⊗AM→Ssatisfying m[n, m0] = (m, n)m0and n(m, n0)=[n, m]n0for all m, m0∈Mnon-degenerated and n, n0∈N. The context is called non-degenerated if MSis faithful and for all 0 6=m∈Malso [N, m]6=0. The standard Morita context of a left A-module Mis the context (A, M, M∗, S)with S=EndA(M)and M∗=HomA(M, A)and the maps (−,−):M⊗SM∗→A(m, f):= (m)f∀m∈M, f ∈M∗(2.1) [−,−]:M∗⊗AM→S[f, m]:= [n7→ (n)fm]∀m∈M, f ∈M∗(2.2) 344 Christian Lomp MSis obviously always faithful and Mis non-degenerated if and only if [M∗, m]6=0 for all 0 6=m∈M. In this case there exist for any non-zero submodule Nof Mand non-zero element m∈Na homomorphism f:M→Asuch that the map ˜ f:M→Nwith (n)˜ f= (n)fm ∈Am ⊆N, for n∈M, is non-zero. Hence any non-degenerated module is in particular retractable. Retractable modules have gained recently further attention in [8–10, 18,19,26–28], but have previously also played a major role in the context of algebras. The theorems of Bergman-Isaacs or Rowen say that in certain situations an algebra Awith a group action Gor considered as bimodule is a retractable module considered over the skew group ring A∗Gor over the enveloping algebra Ae=A⊗Aop. In case ∂is a derivation on an algebra A, then Ais a retractable A[x;∂]- module if ∂is locally nilpotent. Furthermore Cohen’s question raises the problem as to whether a semiprime algebra Awith an action of a semisimple Hopf algebra His a retractable A#Hmodule. With this in mind, the survey was written to illustrate the use of the module theoretic notion of retractability in the context of algebras. 2.1 Categorical notions A retractable module is clearly a generalisation of a self-generator. Here we shortly review this notion in the context of category theory. Definition 2.1. Let Cbe a category. An object Xof Cis generated by an object Gof Cif for every pair of distinct morphisms f, g :X→Yin Cthere exists a morphism h:G→Xwith hf 6=hg. In particular if Cis an Abelian category and Xis not the zero object, then Mor(G, X)6={0}, because for the identity f=idXand the zero morphism g=0, there exist a morphism h:G→ Xsuch that h6=0. Having this in mind the definition of a retractable object seems to be a direct generalisation of a generator. Definition 2.2. An object Mof an Abelian category Cis called retractable if Mor(M, N)6={0} for all subobjects Nof M, different from the zero object. Ler Cbe an Abelian category with arbitrary coproducts. Let Mbe any object in Cand Na subobject of it. Then there exists a unique subobject Tr(M, N)of Nsuch that every morphism f:M→Nfactors through Tr(M, N). Suppose that Mis a retractable object, then Tr(M, N) is essential in Nfor each non-zero subobject N∈ L in the sense that for all non-zero subobjects Kof Nthe meet K∩Tr(M, N)is non-zero (as any f:M→Kcan be considered a morphism f:M→Nand hence factored through Tr(M, N)). This means in the case of a module category C, that a module Mis retractable if and only if for all submodules Nof M, the trace Tr(M, N), which is the sums of images of all homomorphisms f:M→N, is essential in N. Loosely speaking every submodule of a retractable module Mcan be "approximated" by an M-generated submodule. 2.2 Lattice theoretical meaning Let Rbe ring and Ma left R-module with endomorphism ring S=End(M). To link module theoretical properties of Mwith properties of Sone can use the following map from the lattice L(M)of left R-submodules of Mto the lattice L(S)of left ideals of S: Hom(M, −):L(M)→ L(S)N7→ Hom(M, N) = {f∈S|(M)f⊆N}. This map Hom(M, −)is always a homomorphism of semilattices between (L(M),∩)and (L(S),∩)since Hom(M, N ∩L) = Hom(M, N)∩Hom(M, L)holds for all N, L ∈ L(M). Call a homomorphism ϕ:L → L0of semilattice with least element 0 faithful if ϕ(x) = 0⇒x=0. The following Lemma can be proven easily: Lemma 2.3. Let Mbe a left A-module. (i) Hom(M, −)is injective if and only if Mis a self-generator. (ii) Hom(M, −)is faithful if and only if Mis a retractable module. While the injectivity or faithfulness of Hom(M, −)has to do with Mbeing a generator or retractable, the surjectivity of Hom(M, −)deals with the projectivity of M(we refer the reader to [30] for all undefined notion.): RETRACTABLE MODULES 345 Lemma 2.4. Let Mbe a left A-module and S=EndA(M). (i) All cyclic left ideals of Sbelong to the range of Hom(M, −)if and only if Mis semiprojective. (ii) All finitely generated left ideals of Sbelong to the range of Hom(M, −)if and only if Mis intrinsically-projective. (iii) Hom(M, −)is surjective if Mis Σ-self-projective, i.e. projective in the Wisbauer category σ[M]. Considering semi-projective retractable modules combines the light self-generator and selfprojectivity condition on M. Such modules were studied for example in [9]. 3 Algebras with additional structures An (associative, unital) algebra Aover a commutative ring Ris an R-module Asuch that there exist R-linear maps µ:A⊗RA→Aand η:R→A, called the multiplication of Aand unit of Arespectively, such that Awith µas multiplication, defined as ab =µ(a⊗b)for a, b ∈A, and 1A=η(1)as unit element forms an associative and unital ring. It is easy to see that by taking R=Z, any (associative, unital) ring is an (associative, unital) algebra over Z. Thus for those that do not like the idea of R-algebras, they might for the beginning just ignore Rand think of Abeing and ordinary ring. Clearly ηdoes not need to be injective, just think of A=Zn[x], for some n > 1, which is a Z-algebra and η:Z→Zn⊆Zn[x]is the canonical map, where Zn=Z/nZ. Moreover the image of η lies always in the centre of Aand in particular Ais an R0-algebra for any subring R0of the centre Z(A) = {a∈A|ab =ba ∀b∈A}.In the following let Rbe always a commutative ring and Aan R-algebra. 3.1 Algebras that are retractable as bimodule The endomorphism ring EndR(A)of Aas R-module is itself an R-algebra whose R-module structure is given as follows: for all r∈R, f ∈EndR(A)set r·f:A→Aby (r·f)(x):=rf(x) for all x∈A. The multiplication of EndR(A)is given by the composition of functions and the unit map is given by η:R→EndR(A)sending r7→ r·idA. For each element a∈Athere are two R-linear maps of Awhich are the left and the right multiplication by a: la:A→A la(x):=ax ∀x∈A, ra:A→A ra(x):=xa ∀x∈A. Note that since the multiplication of Ais supposed to be associative, laand rbcommute, i.e. la◦rb=rb◦lain EndR(A), for any a, b ∈A. The subalgebra of EndR(A)generated by all maps laand rbfor a, b ∈A. Is called the multiplication algebra of Aand denoted by M(A). Left M(A)-modules Mcan be considered as bimodules over A, where one defines for all a, b ∈Aand m∈M: am :=la•mand mb :=rb•m. The bimodule compatibility condition (am)b=a(mb)for all m∈Mholds, because of (rb◦ la−la◦rb)•M=0. Analogously any A-bimodule has a natural structure as left M(A)- module given by la•m=am and rb•m=mb, for a, b ∈Aand m∈M. The enveloping algebra Ae=A⊗RAop is also an R-algebra, where Aop denotes the opposite ring of A. The multiplication of Aeis defined as (a⊗x)(b⊗y):= (ab)⊗(yx)∀a, b, x, y ∈A. Moreover the map ψ:Ae→EndR(A)given by a⊗b7→ la◦rb∀a, b ∈A is a surjective algebra map from Aeto M(A)whose kernel is the annihilator of A, where Ais naturally considered a left Ae-module by (a⊗b)•x=axb for all a, b, x ∈A. Hence Ker(ψ) = (n X i=1 ai⊗bi∈Ae| n X i=1 aixbi=0∀x∈A)=AnnAe(A). 346 Christian Lomp For a bimodule M∈Ae-Mod one defines its centre as Z(M) = {m∈M|am =ma ∀a∈A}. The canonical map ψM: HomAe(A, M)−→ Z(M)given by f7→ (1)f∀f∈HomAe(A, M).(3.1) is a bijection, where the left Ae-module homomorphism is applied from the right. The inverse of this map is ψ−1 M:Z(M)−→ HomAe(A, M)given by m7→ [a7→ a·m]∀m∈Z(M).(3.2) In particular ψA: EndAe(A)≃Z(A)is an isomorphism of R-algebras and the bijections ψM are left Z(A)-linear maps. Lemma 3.1. Ais a retractable left Ae-module if and only if Ahas a large centre Z(A), i.e. every non-zero two-sided ideal of Acontains a non-zero central element. Proof. This follows from the the fact that the Ae-submodules of Aare precisely the two-sided ideals Iand from the bijection ψI: HomAe(A, I)≃Z(I) = I∩Z(A). There are at least two important results that have to be mentioned in this context. The first is due to Rowen and says the following (see [25]): Theorem 3.2 (Rowen, 1972). Any semiprime PI algebra has a large centre. Recall that an R-algebra Ais a PI-algebra if it there exists an element f(x1, . . . , xn)in the free algebra Rhx1, . . . , xniover Rsuch that f(a1, . . . , an) = 0 for any substitution a1, . . . , an∈A and such that one of the coefficients of a monomial of highest degree of fis 1. Examples of semiprime PI-algebras are matrix algebras over division algebras that are finite dimensional over their centre or more generally any semiprime algebra that is a finitely generated when considered a module over its centre. Thus Rowen’s theorem says that any semiprime PI-algebra is a retractable left Ae-module. For a non-trivial example one might consider the quantum plane at root of unity. Let q∈C\ {0}. The quantum plane over Cwith parameter qis the algebra A=Cq[x, y] = Chx, yi/hyx −qxyi. Elements of Acan be uniquely written as linear combinations of monomials of the form xiyj for i, j ≥0. The relation yx =qxy makes Cq[x, y]a non-commutative algebra if q6=1. An elementary calculation shows that the centre of Cq[x, y]is Z(A) = Cif qis not a root of unity and that it is Z(A) = C[xn, yn]if qis a primitive n-th root of unity. In the later case Ais generated by all monomials of the form xiyjwith 0 ≤i, j < n as a module over Z(A). Hence Ais a PI-algebra. Since Ais also a domain the centre is large, i.e. any non-zero ideal of Cq[x, y] contains a non-zero polynomial of the form f(xn, yn). The second result in this context is Puczyłowski and Smoktunowicz’ description of the Brown-McCoy radical of an algebra Afrom [23]. Recall that the Brown-McCoy radical BMc(A) of Ais the intersection of all maximal two-sided ideals. This means that the Brown-McCoy radical is the module theoretic radical of Aas bimodule, i.e. BMc(A) = Rad(AeA). Puczyłowski and Smoktunowicz described the Brown-McCoy radical of A[x]using the following ideal: PS(A) = \{P⊆A|Pis a prime ideal and A/P has a large centre}. Theorem 3.3 (Puczyłowski-Smoktunowicz, 1998). BMc(A[x]) = P S(A)[x]. Their result relies on the following (surprising) Lemma from [23]: Lemma 3.4. Let Mbe a maximal ideal of A[x]. If A∩M=0, then Ahas a large centre. In the case of the Lemma, if such maximal ideal Mof A[x]exist with M∩A=0, then A will also be a prime ring. Recall that a module Mover some ring Ais called compressible if Membeds into any non-zero submodule of it, i.e. for any 0 6=N⊆Mthere exists an injective A-linear map f:M→N. Prime algebras with large centre are precisely the algebras that are compressible as bimodule. Lemma 3.5 (see [30, 35.10]). An algebra Ais a compressible Ae-module if and only if it is prime and has a large centre. Hence, in module theoretic terms, P S(A)is the intersection of all those Ae-submodules P of Asuch that A/P is a compressible Ae-module. RETRACTABLE MODULES 347 3.2 Derivations A (R-linear) derivation of an R-algebra Ais an R-linear map ∂:A→Asuch that ∂(ab) = ∂(a)b+a∂(b)for all a, b ∈A. Examples are ordinary partial derivations ∂xion the polynomial ring R[x1, . . . , xn]over R. For any a∈Aof an R-algebra A, its commutator ∂(a)=[a, −], with [a, b] = ab −ba for b∈A, is a derivation, called an inner derivation of A. Given a derivation ∂one constructs the differential operator ring B=A[x;∂]as the Ralgebra generated by Aand xsubject to xa =ax +∂(a)∀a∈A. The algebra A[x;∂]can be constructed as a subalgebra of EndA(A[x]) such that A[x;∂]is a free left A-module with basis {xi|i∈N}. Hence the elements of Bcan be uniquely written as (left) polynomials Pn i=0aixiwith ai∈A. Moreover Abecomes a left A[x;∂]-module with respect to the action x·a=∂(a)or more generally n X i=0 aixi!·b= n X i=0 aiδi(b)∀b∈A, ∀ n X i=0 aixi∈B. The left A[x;∂]-submodules of Aare precisely those left ideals Iof Athat are stable under the derivation, i.e. ∂(I)⊆I. For any left A[x;∂]-module Mone defines its submodule of constants as M∂={m∈M|x·m=0}=AnnM(x). For M=Aone has A∂=Ker(∂)which is easily seen to be a subalgebra of A. Analogously to the bimodule situation we have the following R-linear isomorphisms for any left A[x;∂]-module M: ψM: HomA[x;∂](A, M)−→ M∂given by f7→ (1)f∀f∈HomA[x;∂](A, M). (3.3) Its inverse map is ψ−1 M:M∂−→ HomA[x;∂](A, M)given by m7→ [a7→ a·m]∀m∈M∂.(3.4) In particular ψA: EndA[x;∂](A)≃A∂is an isomorphism of R-algebras and the bijections ψM are left A∂-linear maps. Using these isomorphisms the following Lemma is obvious: Lemma 3.6. Ais a retractable A[x;∂]-module if and only if A∂is large in A, i.e. A∂intersects all non-trivial ∂-stable left ideals of Anon-trivially. A sufficient condition for this to happen is the local nilpotency of ∂, i.e. if for every a∈A, there exists n∈Nsuch that ∂n(a) = 0. Proposition 3.7. If ∂is locally nilpotent, then Ais a retractable A[x;∂]-module. Proof. The proof of this fact is obvious, because if 0 6=a∈Iis a non-zero element of an ∂- stable left ideal Iof A, then by hypothesis there exists n∈Nsuch that ∂n(a) = 0. Take the least n∈Nsuch that ∂n(a) = 0, then ∂n−1(a)is a non-zero element of I∩A∂, which proves that A∂ is large in Aand hence Ais a retractable A[x;∂]-module. For example the partial derivatives ∂ ∂xiof A=R[x1, . . . , xn]for any i=1, . . . , n are locally nilpotent. However it is unknown when Ais a retractable A[x;∂]-module for an arbitrary derivation ∂. Question 3.8. What are necessary and sufficient conditions for Ato be a retractable A[x;∂]- module? in other words, find conditions on Aand ∂such that any non-zero ∂-stable left ideal contains a non-zero constant. Zelmanowitz called a left R-module fully retractable if for any non-zero submodule Nand non-zero g:N→Mthere exists h:M→Nsuch that hg 6=0. It is not clear when Ais fully retractable as A[x;∂]-module. The next Proposition can be found in [6]. Proposition 3.9 (Borges-Lomp, 2011). Let ∂be a locally nilpotent derivation of A. (i) Ais a compressible left A[x;∂]-module, provided A∂is a domain. (ii) A∂is a left Ore domain if and only if Ais a uniform left A[x;∂]-module and A∂is a domain. 348 Christian Lomp (iii) Ais critically compressible as left A[x;∂]-module if and only if A∂is a left Ore domain and Ais fully retractable as left A[x;∂]-module. Let R=kbe a field of characteristic zero and ∂a locally nilpotent derivation on Asuch that there exists an element a∈Awith ∂(a) = 1. Then by [6, Proposition 3.10] Ais a self-projective A[x;∂]-module and module theory yields the following result (see [6, Proposition 3.12]) Proposition 3.10 (Borges-Lomp, 2011). Let Abe an algebra over a field kof characteristic zero and ∂a locally nilpotent derivation of Asuch that ∂(a) = 1, for some a∈A. Then the following statements are equivalent: (a) A∂is a left Ore domain; (b) Ais a left Ore domain; (c) Ais a critically compressible A[x;∂]-module. Example 3.11 (Goodearl, 1980). Let A=k[[t]] be the power series ring over a field kof characteristic 0 and let ∂be the derivation with ∂(tn) = ntnfor all n≥0. Certainly ∂is not locally nilpotent. Any ideal of Ais ∂-stable, because the proper ideals are of the form I=Atnfor n≥0. Since for any a=P∞ n=0antn∈Aone has ∂(a) = P∞ n=0nantn6=1 we have that there does not exist any a∈Awith ∂(a) = 1. Nevertheless Ais a self-projective left A[x;∂]-module. To see this note that by the correspondence (3.3) it is enough to show that (A/I)∂= (A∂+I)/I for any ∂-stable left ideal Iof A. Let I=Atmbe any ideal of Aand a=P∞ n=0antnwith ∂(a)∈I. Then there exists b∈Asuch that ∂(a) = P∞ n=0nantn=btm∈I, which implies that ai=0 for all 1 ≤i<m. Thus a=a0+b0tmfor some b0∈Aand a+I=a0+Iin A/I. Since a0∈A∂this shows (A/I)∂⊆(A∂+I)/I while the reversed inclusion is obvious. Since Ais a Noetherian integral domain, A[x;∂]is a left Noetherian Ore domain. However as A∂=k is the base field and Ais not simple as left A[x;∂]-module, Ais not retractable and hence not compressible as A[x;∂]-module. Note that the set DerR(A)of derivations on the R-algebra Aforms a Lie algebra with the ordinary Lie product induced by the product(=composition) of EndR(A), i.e. if ∂, ∂ ∈DerR(A), then their commutator [∂, δ] = ∂◦δ−δ◦∂∈DerR(A) is again a derivation of A. An action of an arbitrary abstract Lie algebra gover Rby derivations is given by a map of Lie algebras d:g→DerR(A). We shall write the image of x∈gunder d as dx. An analogous construction to the construction of the differential operator ring is given by a new product on the tensor product of Aand the universal enveloping algebra U(g)of g. The new algebra is called the smash product of Aand U(g)and is denoted by A#U(g). The product is determined by (1#x)(a#1) = a#x+dx(a)#1 ∀x∈g, a ∈A. Later we will shortly mention Hopf algebras and their action on rings and U(g)is one of the examples. Again Abecomes a left A#U(g)-module where the module action is given by (a#x)· b=a dx(b)for all a, b ∈Aand x∈g. Again one can consider the subset of all those elements a∈Asuch that dx(a) = 0 for all x∈g, i.e. Ag=\ x∈g Ker(dx). For an arbitrary left A#U(g)-module Mone sets Mg=Tx∈gAnnM(1#x).As before there are functorial Rlinear isomorphisms Mg≃HomA#U(g)(A, M)and in particular Ag≃EndA#U(g)(A). Retractability for Aas a left A#U(g)-module also means here that Agis large in Awith respect to all those left ideals of Athat are stable under alll derivations dxwith x∈g. In the case of a single derivation ∂∈DerR(A)one considers the trivial Lie algebra g=R with zero bracket. The map d:R→DerR(A)is then given by r7→ r∂ for all r∈R. Note that the enveloping algebra of the trivial Lie algebra is the polynomial ring R[x]in one variable. Moreover A#U(g) = A⊗RR[x] = A[x] is determined by the product: (1#x)(a#1) = a#x+∂(a)#1 or better xa =ax +∂(a)∀a∈A, showing that A#R[x]≃A[x;∂]. RETRACTABLE MODULES 349 3.3 Group Actions A group Gact on an R-algebra Aby automorphism, which means that there is a homomorphism of groups ρ:G→AutR(A)from Gto the group of R-linear automorphisms of A. We denote the image of g∈Gunder ρby ρg, although we sometimes write gainstead of ρg(a)for a∈Aand g∈G. As in the last section, a new algebra can be attached to Gand A, which is the skew-group ring A∗Gdefined on A⊗RR[G], where R[G]is the group ring of Gover R. Alternatively one might consider A∗Gas the free left A-module with basis {g|g∈G}such that the multiplication is defined as ag ·bh =aρg(b)gh =a(gb)gh ∀a, b ∈A, ∀g, h ∈G. If Gis cyclic infinite, i.e. G=hσi, then A∗Gis equal to the Laurent skew-polynomial ring A[x, x−1;σ]whose underlying space are the Laurent polynomials with coefficients in Aand whose multiplication is determined by xna=σn(a)xn∀a∈A, n ∈Z. If G=hσiis cyclic of order n, then A∗Gis equal to the factor A[x;σ]/hxn−1iof the skewpolynomial ring A[x;σ]. Let Gbe any group acting on Aas automorphism. Then Ahas a left A∗G-module structure defined by ag ·b=aρg(b)∀a, b ∈A, g ∈G. The left A∗G-submodules of Aare precisely the G-stable left ideals of A. Let Mbe any left A∗G-module. Then the submodule of fixed elements of Mis MG={m∈M| ∀g∈G:g·m=m.} Moreover one has again R-linear isomorphisms for any left A∗G-module M: ψM: HomA∗G(A, M)−→ MGgiven by f7→ (1)f∀f∈HomA∗G(A, M). (3.5) with inverse map ψ−1 M:MG−→ HomA∗G(A, M)given by m7→ [a7→ a·m]∀m∈MG.(3.6) In particular ψA: EndA∗G(A)≃AGis an isomorphism of R-algebras and the bijections ψMare left AG-linear maps. Lemma 3.12. Ais a retractable A∗G-module if and only if AGis large in A, i.e. AGintersects all non-trivial G-stable left ideals of Anon-trivially. The existence of non-trivial fixed elements in G-stable left ideals reduces the study of the structure of G-stable left ideals of Ato the study of left ideals of AG. The following result is a classical theorem in the study of group action: Theorem 3.13 (Bergman-Isaacs, 1973; Kharchenko, 1974). Let Gbe a finite group of order n acting on an R-algebra A. Assume that one of the following conditions is verified: (i) Ais n-torsionfree and does not contain any non-zero nilpotent G-stable ideal or (ii) Ais reduced, i.e. does not contain any nilpotent element. Then Ais retractable as left A∗G-module. For the proof of (i) see [3,22] for the proof of (ii) see [11]. 3.4 Involutions Let Abe an R-algebra. An involution of Ais an R-linear map ∗:A→Awith a7→ a∗that is an anti-algebra homomorphism and has order 2, i.e. (ab)∗=b∗a∗and (a∗)∗=afor all a, b ∈A. An element a∈Ais called symmetric (respectively anti-symmetric) with respect to ∗if a∗=a(respectively. a∗=−a). Ideals that are stable under the involution ∗are called ∗-ideals. Consider the subalgebra Bof EndR(A)generated by ∗and all left multiplications la for a∈A, i.e. B=h{∗} ∪ {la|a∈A}i ⊆ EndR(A). 350 Christian Lomp Note that for any a, b ∈Aone has ra(b) = ba = (a∗b∗)∗= (∗ ◦ la∗◦ ∗)(b). Hence ra=∗ ◦ la∗◦ ∗ ∈ B. It is clear that Abecomes a left B-module by simply applying f∈B⊆EndR(A), i.e. for any f∈Band a∈Aset f·a:=f(a). The left B-submodules of Aare stable under left and right multiplications laand rafor any a∈Aand hence are twosided ideals of A. Moreover they are stable under ∗. On the other hand any ∗-ideal is also a left B-submodule. The algebra Bcan be seen as a factor of a skew group algebra. Let Ae=A⊗RAop be the enveloping algebra of Aand consider the map α:Ae→Aedefined by α(a⊗b) = b∗⊗a∗for all a, b ∈A. The map αis an automorphism of Ae, because for any a, b, c, d ∈A: α((a⊗b)(c⊗d))=α(ac ⊗db)= (db)∗⊗(ac)∗=b∗d∗⊗c∗a∗=(b∗⊗a∗) (d∗⊗c∗)=α(a⊗b)α(c⊗d). As αis its own inverse it is an automorphism of order 2. Let G=hαi={id, α}and consider the (surjective) map ψ:Ae∗G→Bgiven by (a⊗b)⊗id + (c⊗d)⊗α7→ la◦rb+lc◦rd◦ ∗ for all a, b, c, d ∈A. Then ψis an algebra homomorphism. The calculations are easy but tedious and will be illustrated on the example of the product of (1⊗1)⊗αand (a⊗b)⊗id. Note first that for any x∈A: lb∗◦ra∗◦ ∗(x) = b∗x∗a∗= (axb)∗=∗ ◦ la◦rb(x). Hence ψ((1⊗1⊗α)(a⊗b⊗id))=ψ(b∗⊗a∗⊗α)=lb∗◦ra∗◦∗ =∗◦la◦rb=ψ(1⊗1⊗α)(a⊗b⊗id)). Thus Bis a factor algebra of Ae∗G. For any left Ae∗G-module Mone defines the submodule of central symmetric elements as Z(M;∗) = Z(M)∩MG={m∈Z(M)|α·m=m}. For M=Aone obtains the central symmetric elements of A, i.e. Z(A;∗) = {a∈Z(A)|a∗= a}. Again one has R-isomorphisms for any left Ae∗G-module M: ψM: HomAe∗G(A, M)−→ Z(M;∗)given by f7→ (1)f∀f∈HomAe∗G(A, M). (3.7) with inverse map ψ−1 M:Z(M;∗)−→ HomAe∗G(A, M)given by m7→ [a7→ a·m]∀m∈Z(M;∗). (3.8) In particular ψA: EndAe∗G(A)≃Z(A;∗)is an isomorphism of R-algebras and the bijections ψMare left Z(A;∗)-linear maps. Lemma 3.14. Ais a retractable Ae∗G-module if and only if every non-zero ∗-ideal contains a non-zero central symmetric element. Using Rowen’s theorem we have the following: Corollary 3.15. Let ∗be an involution of an R-algebra A. If Ais a semiprime PI-algebra, then Ais a retractable Ae∗G-module. Proof. Let Ibe a non-zero ∗-ideal. By Rowen’s Theorem 3.2, Icontains a non-zero central element, say a∈I. Since Iis ∗-stable, a∗∈I. Thus a+a∗is a central symmetric element of I. If a+a∗=0, then a∗=−aand a2is a central symmetric element as (a2)∗= (−a)2=a2.Note that a26=0 as ais non-zero and central and Ais semiprime. 4 Open Problems If a group Gacts on an algebra Aby automorphisms, then Abecomes a module over the skew group ring A∗Gas well as over the group algebra R[G]. If a Lie algebra gacts on Aby derivations, then Abecomes a module over A#U(g)as well as over the universal enveloping algebra U(g). Both algebras R[G]and U(g)are examples of Hopf algebras and the constructions A∗Grespectively A#U(g)are so-called smash products. A Hopf algebra His an algebra such RETRACTABLE MODULES 351 that there exist algebra maps ∆:H→H⊗H(called the comultiplication of H) and :H→R (called the count of H) such that the following diagrams commute: H ∆// ∆  H⊗H ∆⊗1  H⊗H1⊗∆//H⊗H⊗H R⊗H H ⊗H ⊗1 oo1⊗//H⊗R H ≃ ee ≃ 99 ∆ OO For an element h∈Hits comultiplication ∆(h)is an element of H⊗H. It is common to use the so-called Sweedler’s notation enumerating symbolically the first and second tensorand by writing ∆(h) = P(h)h1⊗h2∈H⊗H. The ring of R-linear endomorphisms of EndR(H)of a Hopf algebra Hhas another ring structure as the usual given by the convolution product which associates to two endomorphisms f, g of Hthe endomorphisms f∗g=µ◦(f⊗g)◦∆where µdenotes the multiplication of H. To obtain a Hopf algebra one also requires that the identity map has an inverse in EndR(H)with respect to the convolution product. This inverse is usually denoted by Sand called the antipode of H. Equivalently there should exist an endomorphism Ssatisfying µ◦(id ⊗S)◦∆=η◦=µ◦(S⊗id)◦∆ where η:R→Hdenotes the map r7→ r1Hfor all r∈R. A Hopf algebra Hacts on an R-algebra Aif Ais a left H-module and an algebra in the category of left H-modules. The later means that the multiplication m:A⊗RA→Aand the unit map R→Awith r7→ r1Aare maps of left H-modules. Note that due to the comultiplication the category of left H-modules is closed under tensor products, i.e. it is a tensor category. For left H-modules Nand M, elements x∈Nand y∈Mand h∈Hwith ∆(h) = P(h)h1⊗h2∈H⊗Hone sets h·(x⊗y) = P(h)(h1·x)⊗(h2·y).The base ring Rbecomes a left H-module by h·r=(h)r for all h∈H, r ∈R. Hence for Hto act on A,Ahas to be a left H-module and the following conditions have to be fulfilled for all a, b ∈Aand h∈H. h·(ab) = X (h) (h1·a)(h2·b)and h·1A=(h)1A. The smash product of Aand His denoted by A#Hand defined on the tensor product A⊗RH with multiplication given by (a⊗h)·(b⊗g) = X (h) a(h1·b)⊗h2g∀a, b ∈A, h, g ∈H. Then Abecomes a cyclic left A#H-module by the action (a⊗h)•b=a(h·b)for all a, b ∈A, h∈H. For any left A#H-module Mone defines the submodule of H-invariants of Mas MH={m∈M|h·m=(h)m∀h∈H}. As in the previous sections one has a for any left A#H-module Mcanonical maps: ψM: HomA#H(A, M)−→ MHgiven by f7→ (1)f∀f∈HomA#H(A, M). (4.1) with inverse map ψ−1 M:MH−→ HomA#H(A, M)given by m7→ [a7→ a·m]∀m∈MH.(4.2) In particular ψA: EndA#H(A)≃AHis an isomorphism of R-algebras and the bijections ψMare left AH-linear maps. For a group Gand its group algebra H=R[G], the Hopf algebra structure of His given by the comultiplication ∆(g) = g⊗g, the counit (g) = 1 and the antipode S(g) = g−1, for all g∈G. The group algebra Hacts on Aif there is a module action H⊗A→A, say by h⊗a7→ h·a, for all h∈H, a ∈Aand the two conditions from above are satisfied, i.e. g·(ab)=(g·a)(g·b)and g·1A=(g)1A=1A∀g∈G. Hence if one denotes by αgthe map a7→ αg(a) = g·a, then one sees from this two conditions that αgis an endomorphism of rings. Since Gis a group and Ais supposed to be a left H-module, αg−1=α−1 gfor all g∈G. It is easy to see that A∗Gequals the smash product A#R[G].