Injective Hulls of Simple Modules Over Nilpotent Lie Color Algebras
aa r X i v : . [ m a t h . R A ] M a r INJECTIVE HULLS OF SIMPLE MODULES OVER NILPOTENT LIE COLORALGEBRAS
CAN HAT˙IPO ˘GLU
Abstract.
Using cocycle twists for associative graded algebras, we characterize finite dimensionalnilpotent Lie color algebras L graded by arbitrary abelian groups whose enveloping algebras U ( L ) havethe property that the injective hulls of simple right U ( L )-modules are locally Artinian. We also collecttogether results on gradings on Lie algebras arising from this characterization. Introduction
When Jategaonkar proved in [10] that fully bounded Noetherian rings satisfy Jacobson’s conjecture,a key step in his proof was to observe that over such rings finitely generated essential extensions ofsimple modules were Artinian. This created an interest in this finiteness property, and a number ofNoetherian rings have been studied in relation with it. The obvious question whether this propertywas shared by every Noetherian ring was answered negatively by Musson in [12, 13].Recall that a module M is said to be locally Artinian if every finitely generated submodule of it isArtinian. For a Noetherian ring A , finitely generated essential extensions of simple right modules areArtinian if and only if the property( ⋄ ) Injective hulls of simple right A -modules are locally Artinianis satisfied. We should stress that property ( ⋄ ) is sufficient for a Noetherian ring to satisfy Jacobson’sconjecture.Interest in property ( ⋄ ) have increased in recent years, with Noetherian down-up algebras havingthis property have been characterized in a series of papers by Carvalho et al. [5], Carvalho andMusson [6] and Musson [14]. Later, finite dimensional solvable Lie superalgebras over algebraicallyclosed fields whose enveloping algebras have property ( ⋄ ) have been characterized in [9] and then acharacterization of Ore extensions k [ x ][ y ; α, d ] of k [ x ] having property ( ⋄ ) has been obtained in [4].Most recently, a characterization involving torsion theories and some sufficient conditions have beengiven by the author in [8].Lie color algebras are natural generalizations of Lie superalgebras. In this note we show using atechnique which is sometimes called “Scheunert’s trick”, that the result on nilpotent Lie superalgebrascould easily be generalized to nilpotent Lie color algebras. The characterization of nilpotent Lie super-algebras L with property ( ⋄ ) has shown that it only depends on the even part of the Lie superalgebra.We will see that the same holds in this more general setting of Lie color algebras.Some word on notation. In the rest of the paper, we assume that k is an algebraically closed fieldof characteristic zero, G is an additive abelian group and all the graded rings and modules are gradedby G . The group of nonzero elements of k will be denoted by k ∗ .2. Cocycle Twists of Associative Graded Algebras and Property ( ⋄ )Let A be a graded associative algebra over k . That is, A is a graded k -vector space and there is afamily { A g | g ∈ G } of subspaces of A such that A = ⊕ g ∈ G A g and A g A h ⊆ A g + h for every g, h ∈ G .The set of all g ∈ G such that A g = 0 is called the support of the grading. A nonzero element a of A g is said to be a homogeneous element of degree g , and in this case we will write | a | = g to indicate thedegree of a .A graded right A -module is a right A -module M such that there exists a family { M g | g ∈ G } of k -subspaces of M such that M = ⊕ g ∈ G M g and for every g, h ∈ G we have M g A h ⊆ M g + h . For agraded ring A , we will denote the category of graded right A -modules by Gr- A and Mod- A will denotethe category of right A -modules. Date : September 24, 2018.2010
Mathematics Subject Classification.
Primary .
Key words and phrases.
Simple modules, injective modules, graded algebras, cocycle twists, Lie color algebras.
By a on G we mean a map σ : G × G −→ k ∗ which satisfies(2.1) σ ( a, b ) σ ( a + b, c ) = σ ( b, c ) σ ( a, b + c )for all a, b, and c ∈ G . For our purposes, we will also assume that a 2-cocycle σ satisfies the normal-ization condition σ (0 ,
0) = 1. It follows from this assumption that σ ( a,
0) = σ (0 , a ) = σ (0 ,
0) = 1.Note that if σ is a 2-cocycle, then the map σ − : G × G −→ k ∗ defined by σ − ( a, b ) = 1 σ ( a, b )is also a 2-cocycle.Let σ be a 2-cocycle on G . The cocycle twist of the algebra A is the algebra A σ whose underlying G -graded vector space is the same as A , and whose multiplication is defined as(2.2) a · σ b = σ ( | a | , | b | ) ab for every homogeneous elements a, b ∈ A . Observe that the defining relation (2.1) of a 2-cocycle is ex-actly what is needed for the twisted algebra A σ to be an associative algebra, because, for homogeneouselements a, b, c ∈ A we have a · σ ( b · σ c ) = a · σ ( σ ( | b | , | c | ) bc ) = σ ( | a | , | b | + | c | ) σ ( | b | , | c | ) abc = σ ( | a | , | b | ) σ ( | a | + | b | , | c | ) abc = ( a · σ b ) · σ c There is a corresponding relation between the categories of graded modules over A and its cocycletwist A σ . Let M be a graded right A -module. We define a right A σ -module structure on M as follows.For homogeneous elements a ∈ A σ and m ∈ M , we define the action of a on m by m ∗ σ a = σ ( | a | , | m | ) ma. where the action on the right hand side is the right A -action on M . We will shortly see that thisdefines an equivalence of categories between the categories of graded right modules Gr- A and Gr- A σ but first we need to introduce more terminology.For a graded right module M over A and g ∈ G , we define the g - suspension of M to be thegraded right A -module M ( g ) = ⊕ h ∈ G M ( g ) h where M ( g ) h = M g + h . This defines a functor T g : Gr- A −→ Gr- A by letting T g ( M ) = M ( g ). For G -graded rings A and B , Gordon and Green [7] calla functor F : Gr- A −→ Gr- B a graded functor if it commutes (in the appropriate categories) withthe g -suspension functor for every g ∈ G . A graded functor U : Gr- A −→ Gr- B is called a gradedequivalence if there is a graded functor V : Gr- B −→ Gr- A such that V U ≃ Gr- A and U V ≃ Gr- B .If this is the case, the categories Gr- A and Gr- B are said to be graded equivalent . We are ready tostate the following lemma. Lemma 2.1.
Let A be a graded ring and σ be a 2-cocycle on G . Then Gr- A is graded equivalent toGr- A σ .Proof. Let ( − ) σ : Gr- A −→ Gr- A σ denote the functor which assigns to each graded right A -module M the twisted module M σ , leaving homomorphisms unchanged. Since the functor does not changethe underlying G -grading on M , it is clear that it commutes with the suspension functor T g for every g ∈ G . Moreover, if we let σ − : G × G −→ k ∗ be the 2-cocycle defined by σ − ( a, b ) = σ ( a,b ) , thenwe have ( M σ ) σ − = M and hence ( − ) σ − ◦ ( − ) σ ≃ Gr- A and similarly ( − ) σ ◦ ( − ) σ − ≃ Gr- A σ . So itfollows that the categories Gr- A and Gr- A σ are graded equivalent. (cid:3) We would like to connect ( ⋄ ) property of A to that of A σ . We already know that there is anequivalence of categories between Gr- A and Gr- A σ , so that the ( ⋄ ) property for graded injective hullspasses from Gr- A to Gr- A σ . The question is whether the same is true for the categories Mod- A andMod- A σ . It turns out that this is indeed the case, as the following result shows that graded equivalencesbetween graded module categories give rise to Morita equivalences between module categories. Theorem 2.2 ([11], Theorem 1.3. See also [7] Theorem 5.4.) . Let A and B be G -graded rings. Let Φ A (resp. Φ B ) denote the forgetful functor Gr- A −→ Mod- A (resp. Gr- B −→ Mod- B ) . Then thefollowing statements are equivalent. (i) The categories Gr- A and Gr- B are graded equivalent; (ii) There is a Morita equivalence L : Mod- A −→ Mod- B and a graded functor F : Gr- A −→ Gr- B such that Φ B ◦ F = L ◦ Φ A ; NJECTIVE HULLS OF SIMPLE MODULES OVER NILPOTENT LIE COLOR ALGEBRAS 3 (iii)
There exists an object P ∈ Gr- A such that Φ A ( P ) is a finitely generated projective generatorin Mod- A and the graded ring End A ( P ) is isomorphic to B as graded rings. Since ( ⋄ ) is a Morita invariant property, the next result follows from the above theorem. Corollary 2.3.
Let A be a G -graded associative k -algebra and let σ be a 2-cocycle on G . Then A hasproperty ( ⋄ ) if and only if A σ does. Cocycle twists of Lie color algebras A commutation factor ε on G with values in k ∗ is a mapping ε : G × G −→ k ∗ such that ε ( a, b ) ε ( b, a ) = 1 , ε ( a, b + c ) = ε ( a, b ) ε ( a, c ) , ε ( a + b, c ) = ε ( a, c ) ε ( b, c )for all a, b, c ∈ G . It is easy to see from the definition that ε ( a, a ) = ± a ∈ G and accordinglywe define two sets G + = { a ∈ G | ε ( a, a ) = 1 } and G − = { a ∈ G | ε ( a, a ) = − } . Here, G + is asubgroup of G of index at most two and G − is the other coset when the index of G + is two.A G -graded Lie color algebra L with a commutation factor ε is a G -graded vector space L = ⊕ g ∈ G L g with a bracket operation [ , ] : L × L −→ L which satisfies [ L g , L h ] ⊆ L g + h for all g, h ∈ G , along withcolor skew symmetry and color Jacobi identities:(3.1) [ x, y ] = − ε ( | x | , | y | )[ y, x ] , (3.2) [[ x, y ] , z ] = [ x, [ y, z ]] − ε ( | x | , | y | )[ y, [ x, z ]]for all homogeneous elements x, y, z of L . For example, if ε is chosen to be the trivial commutationfactor defined by ε ( a, b ) = 1 for all a, b ∈ G , then L is nothing but a G -graded ordinary Lie algebra.If G = Z and ε is defined as ε ( a, b ) = ( − ab for all a, b ∈ Z , then L is a Lie superalgebra.Like the superalgebra case, we have the notion of even and odd elements for Lie color algebras. Fora Lie color algebra L , we define two sets L + = ⊕ g ∈ G + L g and L − = ⊕ g ∈ G − L g . The elements of L + arecalled even while the elements of L − are called the odd elements of L .The universal enveloping algebra U ( L ) of a Lie color algebra L is defined as U ( L ) = T ( L ) /J ( L )where T ( L ) is the tensor algebra of L and J ( L ) is the ideal of T ( L ) which is generated by the elements x ⊗ y − ε ( | x | , | y | ) y ⊗ x − [ x, y ] for every homogeneous elements x, y of L . The enveloping algebra U ( L )of a Lie color algebra is a G -graded associative algebra and it is well-known that when L is finitedimensional U ( L ) is Noetherian.Let L be a G -graded Lie color algebra with a commutation factor ε . For a 2-cocycle σ on G , if wedefine a new multiplication on L by(3.3) [ x, y ] σ = σ ( | x | , | y | )[ x, y ]for every homogeneous elements x and y of L , then with this multiplication L σ becomes a G -gradedLie color algebra with commutation factor ε ′ = εδ where ε ′ ( a, b ) = ε ( a, b ) δ ( a, b ) and δ ( a, b ) = σ ( a, b ) /σ ( b, a ) for all a, b ∈ G .The descending central sequence of a Lie color algebra L is defined as C ( L ) = L, C n +1 ( L ) = [ C n ( L ) , L ] , ∀ n ≥ .L is called nilpotent if C n ( L ) = 0 for some n . Before we proceed any further, we note the followingeasy fact that being nilpotent is preserved under cocycle twists. Lemma 3.1.
Let L be a G -graded Lie color algebra and let σ be a 2-cocycle on G . Then, L is nilpotentif and only if L σ is nilpotent.Proof. Obviously, C ( L σ ) = L = C ( L ). It is also clear that as k -spaces C i +1 ( L σ ) = C i +1 ( L ) for any i ≥
0. Hence, C n ( L ) = 0 if and only if C n ( L σ ) = 0, i.e. L is nilpotent if and only if L σ is nilpotent. (cid:3) Cocycle twists can be applied to pass from a Lie color algebra to a Lie superalgebra and we brieflyexplain this process now. Let L be a G -graded Lie color algebra with a commutation factor ε . If σ isa 2-cocycle on G , we know that L σ is a G -graded Lie color algebra with a commutation factor ε ′ = εδ , CAN HAT˙IPO ˘GLU where δ is as in the lemma. Let ε : G × G −→ k ∗ denote the superalgebra commutation factor, whichis defined by the rule ε ( a, b ) = 1if a ∈ G + or b ∈ G + and ε ( a, b ) = − a, b ∈ G − . The following result is the crucial step in the process of passing from a Lie coloralgebra to a Lie superalgebra. Theorem 3.2. [2, Theorem 2.3]
Let G be an arbitrary abelian group, and k an arbitrary commutativering with and with group of units k ∗ . Then for any commutation factor ε : G × G −→ k ∗ there existsa 2-cocycle σ : G × G −→ k ∗ such that if we set δ ( g, h ) = σ ( g, h ) /σ ( h, g ) , then εδ = ε . That is, for any G -graded Lie color algebra L with a commutation factor ε , we can find a 2-cocycle σ such that the twisted Lie color algebra L σ is a Lie superalgebra L σ = L σ ⊕ L σ where L σ = L + and L σ = L − .Hence, the discussions of the previous section apply to Lie color algebras as well and there is acorresponding equivalence of categories between the graded representations of a Lie color algebra L and the graded representations of its cocycle twist L σ . While this correspondence only exists betweenthe graded modules over Lie color algebras and Lie superalgebras, we are interested in modules overthe enveloping algebras of such algebras. Fortunately, the cocycle twists of Lie color algebras areconnected to their enveloping algebras in the following way. Lemma 3.3. [15, Theorem 2]
Let L be a G -graded Lie color algebra and let σ be a 2-cocycle on G .Then there is an algebra isomorphism U ( L σ ) ∼ = U ( L ) σ where U ( L ) is considered as a G -graded associative algebra and U ( L ) σ is its cocycle twist. Together with Corollary 2.3, we have:
Lemma 3.4.
Let L be a G -graded Lie color algebra and let σ be a 2-cocycle on G . Then U ( L ) hasproperty ( ⋄ ) if and only if U ( L σ ) does. The previous lemma and the Morita equivalence between U ( L ) and its cocycle twists allow us toreduce our study to that of the enveloping algebra of a Lie superalgebra. Recall that a central abeliandirect factor of a Lie algebra L is an abelian Lie subalgebra A of L such that L = A × B for some Liesubalgebra B of L . We can now state the main result of this note. Theorem 3.5.
Let L be a finite dimensional nilpotent G -graded Lie color algebra with a commutationfactor ε . Let σ be the 2-cocycle such that the twisted algebra L σ is a G -graded Lie superalgebra with L σ = L + and L σ = L − . The following statements are equivalent. (i) U ( L ) -Mod has property ( ⋄ ) ; (ii) U ( L σ ) -Mod has property ( ⋄ ) ; (iii) U ( L σ ) -Mod has property ( ⋄ ) ; (iv) Up to a central abelian direct factor, L σ is isomorphic to either (a) a nilpotent Lie algebra with an abelian ideal of codimension one, (b) the five dimensional Lie algebra L with basis { e , . . . , e } and nonzero brackets [ e , e ] = e , [ e , e ] = e , [ e , e ] = e , (c) the six dimensional Lie algebra L with basis { e , . . . , e } and nonzero brackets [ e , e ] = e , [ e , e ] = e , [ e , e ] = e .Proof. The equivalence of ( i ) and ( ii ) follows from Lemma 3.4. ( ii ) ⇔ ( iii ) ⇔ ( iv ) from Lemma 3.1and [9, Theorem 1.1]. (cid:3) Nilpotent Lie algebras of almost maximal index
A Lie algebra g is said to have almost maximal index if ind( g ) = dim( g ) −
2. According to [9,Proposition 5.3], a finite dimensional nilpotent Lie algebra has almost maximal index if and only if itis one of the Lie algebras appearing in Theorem 3.5(iv).Finite dimensional Lie algebras with an abelian ideal of codimension one are in one-to-one cor-respondence with finite dimensional vector spaces V and nilpotent endomorphisms f : V −→ V . NJECTIVE HULLS OF SIMPLE MODULES OVER NILPOTENT LIE COLOR ALGEBRAS 5
Given such a vector space and a nilpotent endomorphism, one defines a Lie algebra structure on thevector space L = kx ⊕ V by defining [ x, v ] = f ( v ). A typical example of this construction is the n -dimensional standard filiform Lie algebra, which is the Lie algebra L n with a basis { e , e , . . . , e n } andwhose nonzero brackets are [ e , e i ] = e i +1 for i = 2 , , . . . , n −
1. For instance the three dimensionalHeisenberg Lie algebra is the three dimensional standard filiform Lie algebra.The n -dimensional standard filiform Lie algebra L n is realized in the above sense as a vector space L n = kx ⊕ V where V is the span of the elements x , x , . . . , x n with a nilpotent endomorphism f : V −→ V given by f ( x i ) = x i +1 for all i = 2 , , . . . , n − f on a finite dimensional vector space V , there exists a basis { v , v , . . . , v n } of V suchthat f ( v i ) is either zero or equal to v i +1 for all i = 1 , , . . . , n −
1. Now, let L = kx ⊕ V be a finitedimensional nilpotent Lie algebra with an abelian ideal V . Then, V has a basis { v , v , . . . , v n } suchthat [ x, v i ] is either zero or v i +1 , for all i = 1 , , . . . , n −
1. Without loss of generality, we may assumethat [ x, v ] = 0, applying a change of basis if necessary. If [ x, v i ] is nonzero for each i , then L isstandard filiform. If [ x, v i ] = 0 for some i >
1, then L can be written as the direct sum of a standardfiliform Lie algebra L = h x, v , v , . . . , v i i and an abelian Lie ideal L = h v i +1 , v i +2 , . . . , v n i , thereforeproving our claim.The above discussion shows that if L is a finite dimensional nilpotent Lie algebra, then U ( L ) satisfiesproperty ( ⋄ ) if and only if L is isomorphic, up to a central abelian direct factor, to L , L or to astandard filiform Lie algebra.5. Gradings on nilpotent Lie algebras of almost maximal index
In this section we collect together results on the classification of group gradings on finite dimensionalnilpotent Lie algebras of almost maximal index. Two gradings Γ : L = ⊕ L g and Γ ′ : ⊕ g L ′ g on aLie algebra L by an abelian group G are called isomorphic if there is a Lie algebra automorphism ϕ : L −→ L such that ϕ ( L g ) = L ′ g .5.1. Gradings on L and L . Let us first consider the gradings on L and L . These are freenilpotent Lie algebras, in particular, L is the free Lie algebra with two generators of step threeand L is the free nilpotent Lie algebra with three generators of step two. Finite dimensional freenilpotent Lie algebras are members of the larger class of free algebras of finite rank in nilpotent varietiesof algebras . The gradings on these algebras have been classified by Bahturin in [1]. If F n is such analgebra with a generating set { x , x , . . . , x n } then there is a standard Z n -grading on F n obtainedin the following way. We let α = ( d , d , . . . , d n ) ∈ Z n and F αn be the span of all the monomialswhose degree with respect to each generator x i is d i , i = 1 , , . . . , n . Then the subspaces F αn form a Z n -grading on F n as F n = ⊕ α ∈ Z n F αn (see [1]).Then the corresponding standard gradings on L and L are as follows.(1) If L = L then L = ⊕ ( a,b ) ∈ Z L ( a,b ) where L (0 , = h e i , L (0 , = h e i , L (1 , = h e i , L (2 , = h e i , L (1 , = h e i and all other summands are zero.(2) If L = L , then L = ⊕ ( a,b,c ) ∈ Z where L (1 , , = h e i , L (0 , , = h e i , L (0 , , = h e i , L (1 , , = h e i , L (0 , , = h e i , and L (1 , , = h e i and all the other summands are zero.Essentially, the above standard gradings are the only gradings on these Lie algebras in the sensethat any grading of these algebras are induced from the standard gradings. More precisely, if G and H are abelian groups and α : G −→ H is a group homomorphism, then a grading Γ ′ : V = ⊕ h ∈ H V ′ h issaid to be induced from the grading Γ : V = ⊕ g ∈ G V g if V ′ h ∈ H = M g ∈ G : α ( g )= h V g . Then, it follows from [1, Theorem 1] that any G -grading Γ on L n is induced from the standard Z n -grading by a homomorphism α : Z n −→ H , n = 2 , ⊕ g ∈ G A g and Γ ′ : ⊕ h ∈ H A ′ h we say that Γ ′ is a refinement of Γ (or that Γ is a coarsening of Γ ′ ) if for each h ∈ H there exists g ∈ G such that A ′ h ⊂ A g . A grading Γ which doesnot have a proper refinement is called fine . It turns out that the standard grading is also the only fineabelian group grading of the Lie algebras L and L , up to equivalence, see [1, Corollary 3]. CAN HAT˙IPO ˘GLU
Gradings on Standard Filiform Lie Algebras.
Gradings on standard filiform Lie algebrashave been classified by Bahturin et al. in [3] and they are parallel to gradings on free nilpotent Liealgebras. Let L = L n denote the n -dimensional standard filiform Lie algebra. Then L has the standardgrading defined by L = ⊕ ( a,b ) ∈ Z L ( a,b ) where L (1 , = h e i , L ( s − , = h e s i for all s = 2 , , . . . , n andall other summands are zero. In this case, any G -grading of L n is isomorphic to a coarsening of thestandard Z n -grading [3, Theorem 12]. References [1] Bahturin, Yuri. Group Gradings on Free Algebras of Nilpotent Varieties of Algebras.
Serdica Math. J. , 38, (2012),1 - 6.[2] Bahturin, Yuri; Montgomery, Susan. PI-envelopes of Lie superalgebras.
Proc. Amer. Math. Soc . 127 (1999), no. 10,2829-2839.[3] Bahturin, Yuri; Goze, Michel; Remm, Elisabeth. Group Gradings on Filiform Lie Algebras.
Comm. Algebra , 44: 40- 62, 2016.[4] Carvalho, Paula A. A. B.; Hatipo˘glu, Can; Lomp, Christian; Injective Hulls of Simple Modules over DifferentialOperator Rings.
Comm. Algebra
43 (2015), no. 10, 4221-4230.[5] Carvalho, Paula A. A. B.; Lomp, Christian; Pusat-Yilmaz, Dilek. Injective modules over down-up algebras.
Glasg.Math. J . 52 (2010), no. A, 53-59.[6] Carvalho, Paula A. A. B.; Musson, Ian M. Monolithic modules over Noetherian rings.
Glasg. Math. J . 53 (2011),no. 3, 683-692.[7] Gordon, Robert; Green, Edward L. Graded Artin algebras.
J. Algebra
76 (1982), no. 1, 111-137.[8] Hatipoglu, C. Stable Torsion Theories and the Injective Hulls of Simple Modules.
Int. Electron. J. Algebra , 16, 2014,89-98.[9] Hatipo˘glu, Can; Lomp, Christian. Injective hulls of simple modules over finite dimensional nilpotent complex Liesuperalgebras.
J. Algebra
361 (2012), 79-91.[10] Jategaonkar, Arun Vinayak. Jacobson’s conjecture and modules over fully bounded Noetherian rings.
J. Algebra
J. Pure Appl. Algebra
51 (1988),no. 3, 277-291.[12] Musson, I. M. Injective modules for group rings of polycyclic groups. I, II.
Quart. J. Math. Oxford Ser . (2) 31 (1980),no. 124, 429-448, 449-466.[13] Musson, I. M. Some examples of modules over Noetherian rings.
Glasgow Math. J . 23 (1982), no. 1, 9-13.[14] Musson, Ian M. Finitely generated, non-Artinian monolithic modules.
New trends in noncommutative algebra , 211-220, Contemp. Math., 562, Amer. Math. Soc., Providence, RI, 2012.[15] Scheunert, M. Generalized Lie algebras.
J. Math. Phys . 20 (1979), no. 4, 712-720.
Research Center for Theoretical Sciences, Mathematics Division, National Cheng Kung University,Tainan, Taiwan .Currently at the American University of the Middle East, Department of Mathematics and Statistics,Kuwait
E-mail address ::