Indecomposable generalized weight modules over the algebra of polynomial integro-differential operators

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Bavula, V., Bekkert, V. and Futorny, V. (2018) Indecomposable generalized weight

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arXiv:1612.09391v1 [math.RT] 30 Dec 2016

ALGEBRA OF POLYNOMIAL INTEGRO-DIFFERENTIAL OPERATORS

V. V. BAVULA, V. BEKKERT AND V. FUTORNY

Abstract. For the algebraI₁₌Khx,dxd,

R

iof polynomial integro-differential operators over a fieldKof characteristic zero, a classification of indecomposable, generalized weightI_1-modules of finite length is given. Each such module is an infinite dimensional uniserial module. Ext-groups are found between indecomposable generalized weight modules, it is proven that they are finite dimensional vector spaces.

Key Words: the algebra of polynomial integro-differential operators, generalized weight mod-ule, indecomposable modmod-ule, simple module.

Mathematics subject classification 2000: 16D60, 16D70, 16P50, 16U20.

1. Introduction

Throughout, ring means an associative ring with 1; module means a left module;N_:={0,1, . . .}

is the set of natural numbers;N_{+:={1,2, . . .} and}Z_≤0:=−N_{;K is a field of characteristic zero}

andK∗_{is its group of units;P}

1:=K[x] is a polynomial algebra in one variablexoverK;∂:= _dxd; EndK(P1) is the algebra of all K-linear maps fromP1 to P1, and AutK(P1) is its group of units (i.e. the group of all the invertible linear maps from P1 to P1); the subalgebras A1 :=Khx, ∂i andI_{1:=Khx, ∂,}R

iof EndK(P1) are called the (first)Weyl algebraand thealgebra of polynomial integro-differential operators respectively whereR

:P1 →P1, p7→R p dx, is theintegration, i.e.

R

:xn_7→ xn+1

n+1 for alln∈N. The algebraI1is neither left nor right Noetherian and not a domain. Moreover, it contains infinite direct sums of nonzero left and right ideals, [2].

In Section 2, a classification of indecomposable, generalized weightI_{1-modules of finite length is}

given (Theorem 2.5). A similar classification is given in [1] for the generalized Weyl algebras where a completely different approach was taken. Properties of the algebras I_{n :=} I⊗₁n _{of polynomial}

integro-differential operators in arbitrary many variables are studied in [2] and [5]. The groups AutK−alg(In) are found in [3]. The simpleI1-modules are classified in [4].

AcknowledgmentThe first author is grateful to the University of S˜ao Paulo for hospitality during his visit and to Fapesp for financial support (processo 2013/24392-5). The third first author is supported in part by the CNPq (301320/2013-6), by the Fapesp (2014/09310-5).

2. Classification of indecomposable, generalized weight I₁-modules of finite length

In this section, a classification of indecomposable, generalized weightI_{1-modules of finite length}

is given (Theorem 2.5).

As an abstract algebra, the algebra I_{1 is generated by the elements ∂, H :=∂x and}R

(since

x=R

H) that satisfy the defining relations, [2, Proposition 2.2] (where [a, b] :=ab−ba):

∂

Z

= 1, [H,

Z

] =

Z

, [H, ∂] =−∂, H(1−

Z

∂) = (1−

Z

∂)H = 1−

Z

∂.

The elements of the algebraI_1,

eij :=

Z i

∂j−

Z i+1

∂j+1, i, j∈N_{, (1)}

2 V. V. BAVULA, V. BEKKERT AND V. FUTORNY

satisfy the relationseijekl =δjkeil where δjk is the Kronecker delta function. Notice thateij =

Ri

e00∂j. The matrices of the linear maps eij ∈ EndK(K[x]) with respect to the basis {x[s] := xs

s!}s∈N of the polynomial algebraK[x] are the elementary matrices, i.e.

eij∗x[s] =

(

x[i] ifj=s,

0 ifj6=s.

LetEij ∈EndK(K[x]) be the usual matrix units, i.e. Eij∗xs=δjsxi for alli, j, s∈N. Then

eij = j!

i!Eij, (2)

Keij = KEij, and F := Li,j≥0Keij = Li,j≥0KEij ≃ M∞(K), the algebra (without 1) of

infinite dimensional matrices.

Z-grading on the algebra I₁ and the canonical form of an integro-differential op-erator, [2]. The algebraI_{1 =} L

i∈ZI1,i is a Z-graded algebra (I1,iI1,j ⊆ I1,i+j for alli, j ∈ Z)

where

I_1,i=

  

 

D1Ri=RiD1 ifi >0,

D1 ifi= 0,

∂|i|_D

1=D1∂|i| ifi <0,

(3)

the algebraD1:=K[H]L Li∈NKeii is a commutative non-Noetherian subalgebra ofI1,Heii=

eiiH = (i+ 1)eii for i ∈ N(notice that Li∈NKeii is the direct sum of non-zero ideals of D1);

(Ri

D1)D1 ≃D1,

Ri

d 7→d; D1(D1∂

i_{)≃D1, d∂}i _{7→ d, for all i ≥0 since∂}iRi

= 1. Notice that the maps ·Ri

: D1 → D1Ri, d 7→ dRi, and ∂i· : D1 → ∂iD1, d 7→ ∂id, have the same kernel

Li−1

j=0Kejj.

Each elementaof the algebraI_{1 is the unique finite sum}

a=X

i>0

a−i∂i+a0+

X

i>0

Z i

ai+

X

i,j∈N

λijeij (4)

where ak ∈K[H] andλij ∈K. This is the canonical formof the polynomial integro-differential operator [2].

Letvi:=

     Ri

ifi >0,

1 ifi= 0, ∂|i| _{ifi <0.}

ThenI_{1,i=D1vi =viD1 and an elementa∈}I_{1is the unique finite sum}

a=X

i∈Z

bivi+

X

i,j∈N

λijeij (5)

wherebi∈K[H] andλij ∈K. So, the set{Hj∂i, Hj,R i

Hj_{, e}

st|i≥1;j, s, t≥0}is aK-basis for the algebraI_{1. The multiplication in the algebra}I_{1 is given by the rule:}

Z

H = (H−1)

Z

, H∂=∂(H−1),

Z

eij =ei+1,j,

eij

Z

=ei,j−1, ∂eij=ei−1,j, eij∂=∂ei,j+1,

Heii =eiiH = (i+ 1)eii, i∈N, wheree−1,j:= 0 andei,−1:= 0.

The algebra I_{1 has the only proper ideal F =} L

i,j∈NKeij ≃ M∞(K) and F

2 _{= F. The}

factor algebra I_{1/F is canonically isomorphic to the skew Laurent polynomial algebra B1 :=}

K[H][∂, ∂−1_{;τ], τ(H) =H + 1, via∂7→∂,}R

7→∂−1_{, H 7→H (where∂}±1_α=τ±1_(α)∂±1 _{for all} elementsα∈K[H]). The algebraB1is canonically isomorphic to the (left and right) localization

AnI_{1-moduleMis called aweightmodule ifM =}⊕λ∈KMλwhereMλ:={m∈M|Hm=λm}. An I_{1-module M is called a generalized weight module if M = ⊕λ∈KM}λ _{where M}λ _{:= {m ∈}

M|(H −λ)n_{m= 0 for some n=n(m)}. The set Supp(M) :={λ∈K|M}λ _{6= 0} is called the} support of the generalized weight moduleM. For allλ∈K andn≥1,

∂nMλ⊆Mλ−n and

Z n

Mλ⊆Mλ+n.

Let 0 →N → M → L →0 be a short exact sequence of I_{1-modules. Then M is a generalized}

weight module iff so are the modules N andL, and in this case

Supp(M) = Supp(N)∪Supp(L).

For eachI_{1-moduleM, there is a short exact sequence of}I_1-modules

0→F M →M →M :=M/F M →0 (6)

where

(i)F·F M =F M, and (ii)F·M = 0,

and the properties (i) and (ii) determine the short exact sequence (6) uniquely, i.e. if 0 →

M1→M →M2→0 is a short exact sequence ofI1-modules such thatF M1=M1andF M2= 0 thenM1∼=F M andM2≃M.

Notice that

F M ≃K[x]I, (7)

i.e. theI_{1-moduleF M is isomorphic to the direct sum ofI copies of the simple weight}I_1-module

K[x]. Clearly,M is a B1-module.

The indecomposableI₁-modulesM(n, λ). Forλ∈Kand a natural numbern≥1, consider theB1-module

M(n, λ) :=B1⊗K[H]K[H)/(H−λ)n. (8)

Clearly,

M(n, λ)≃B1/B1(H−λ)n≃I1/(F+I1(H−λ)n). (9) TheI_{1-module/B1-moduleM(n, λ) is a generalized weight module with SuppM(n, λ) =λ+}Z_,

M(n, λ) =M i∈Z

M(n, λ)λ+i _{and dimM(n, λ)}λ+i_{=n for all i∈Z. (10)}

For an algebraA, we denote byA−Mod its module category. The next proposition describes the set of indecomposable, generalized weightI_{1-modules of finite lengthM withF M= 0.}

Proposition 2.1. (1) M(n, λ) is an indecomposable, generalized weight I_{1-module of finite}

lengthn.

(2) M(n, λ)∼=M(m, µ)if and only ifn=m andλ−µ∈Z_.

(3) LetM be a generalized weightB1-module of lengthn(i.e. letM be a generalized weightI1

-module such thatF M = 0, by (6)). ThenM is indecomposable if and only ifM ≃M(n, λ)

for someλ∈K.

Proof. 1. Since (B1)K[H]=⊕i∈Z∂iK[H] is a free rightK[H]-module, the functor

B1⊗K[H]− :K[H]−Mod→B1−Mod, N7→B1⊗K[H]N,

is an exact functor. TheK[H]-moduleK[H]/(H−λ)n_{is an indecomposable, hence theB1-module}

M(n, λ) is indecomposable and generalized weight of lengthn.

2. (⇒) Suppose thatI_{1-modulesM(n, λ) andM(m, µ) are isomorphic. Then Supp(M(n, λ)) =}

Supp(M(n, λ)), i.e. λ+Z_=µ+Z_{, i.e. λ−µ∈}Z_{. Thenn=m, by (10).}

(⇐) Suppose thatk:=λ−µ∈Z_{andn=m. We may assume thatk≥1. Using the equality}

(H−λ)n_∂k _=∂k_(H−λ−k)n _=∂k_(H−µ)n_{, we see that the B1-homomorphism}

M(n, λ) =B1/B1(H−λ)n→M(n, µ) =B1/B1(H−µ)n, 1 +B1(H−λ)n7→∂k+B1(H−µ)n,

4 V. V. BAVULA, V. BEKKERT AND V. FUTORNY

(⇒) Each indecomposable, generalized weightB1-moduleM is of the typeB1⊗K[H]N for an indecomposableK[H]-moduleN of lengthn. Notice thatN ≃K[H]/(H−λ)n _{for someλ∈K.} Therefore,M ≃M(n, λ).

Lemma 2.2. LetM be an indecomposable, generalized weightI_{1-module. ThenSupp(M)⊆λ+}Z

for someλ∈K.

Proof. LetM =⊕µ∈Supp(M)Mµbe a generalized weight I1-module. Then

M = M

µ+Z_∈Supp(M)/Z

Mµ+Z

is a direct sum of I_{1-submodules Mµ+}Z := ⊕i∈ZMµ+i where Supp(M)/Z is the image of the

support Supp(M) under the abelian group epimorphismK →K/Z_{, γ7→γ+}Z_{. The}I_1-module

M is indecomposable, henceM =Mλ+Z for someλ∈K, i.e. Supp(M)⊆λ+Z.

The next lemma describes the set of indecomposable, generalized weightI_{1-modulesM with}

F M =M.

Lemma 2.3. LetM be an indecomposable, generalized weightI_{1-modulesM. Then the following}

statements are equivalent.

(1) F M =M.

(2) M ≃K[x].

(3) Supp(M)⊆N_.

Proof. (1)⇒(2) : IfF M =M thenM ≃K[x](I) _{for some setI necessarily with|I|= 1 since}

M is indecomposable, i.e. M ≃K[x]. (2)⇒(3) : Supp(K[x]) ={1,2, . . .} ⊆N_.

(3) ⇒ (1) : Suppose that Supp(M) ⊆ N_{. Using the short exact sequence of} I_1-modules

0 → F M → M → M := M/F M → 0 we see that Supp(M) = Supp(F M)∪Supp(M). Since Supp(F M) = Supp(K[x](I)_{) ={1,2, . . .} and Supp(M) is an abelian group, we must haveM = 0} (since Supp(M)⊆N_{), i.e. M =F M.}

The following result is a key step in obtaining a classification of indecomposable, generalized weightI_{1-modules of finite length.}

Theorem 2.4. Let M be a generalized weight I_{1-module of finite length. Then the short exact}

sequence (6) splits.

Proof. We can assume that F M 6= 0 and M 6= 0. It is obvious that F M ≃K[x]s _{for some}

s≥1 and theB1-moduleM ≃Lt

i=1M(ni, λi) for someni≥1,λi ∈K andt ≥1. It suffices to show that

Ext1I₁(M(n, λ), K[x]) = 0 (11)

for alln≥1 andλ∈K. Ifλ∈Z_{we can assume that λ= 0, by Proposition 2.1.(2).}

(i)F(H−λ)n_{=F: The equality follows from the equalitiese}

ij(H−λ)n=eij(j+ 1−λ) and the choice ofλ.

(ii)M(n, λ) =I_1/I_1(H−λ)n_{: By (i),}I_1(H−λ)n_⊇F(H−λ)n_{=F. Hence,}

M(n, λ) =I_1/(F+I_1(H−λ)n) =I_1/I_1(H−λ)n.

(iii)The equality (11) holds: LetM =M(n, λ). By (ii), the short exact sequence ofI_1-modules

0→I_1(H−λ)n_→I_{1→M →0 (12)}

is a projective resolution of theI_{1-moduleM since the map}

·(H−λ)n :I₁→I_1(H−λ)n, a7→a(H−λ)n,

is an isomorphism ofI_{1-modules, by the choice ofλ. Then}

Ext1I1(M, K[x])≃Z

1_/B1

whereZ1_{= Hom}

I₁(I1(H−λ)n, K[x])≃K[x] andB1≃(H−λ)nK[x] =K[x], by the choice ofλ.

The next theorem is a classification of the set of indecomposable, generalized weightI_1-modules

of finite length.

Theorem 2.5. Each indecomposable, generalized weight I_{1-module of finite length is isomorphic}

to one of the modules below:

(1) K[x],

(2) M(n, λ) where n ≥ 1 and λ ∈ Λ where Λ is any fixed subset of K such that the map

Λ→(K/Z_),λ7→λ+Z_{, is a bijection.}

The I_{1-modules above are pairwise non-isomorphic, indecomposable, generalized weight and of}

finite length.

Proof. The theorem follows from Theorem 2.4, Proposition 2.1 and Lemma 2.3.

Corollary 2.6. Every indecomposable, generalized weightI_{1-module is an uniserial module.}

Proof. The statement follows from Theorem 2.5.

Homomorphisms and Ext-groups between indecomposables.

Proposition 2.7. (1) Let M andN be generalized weight I_{1-modules such that Supp(M)∩}

Supp(N) =∅. ThenHomI₁(M, N) = 0.

(2) HomI₁(M(n, λ), K[x]) = 0.

(3) HomI₁(K[x], M(n, λ)) = 0.

(4) HomI₁(M(n, λ), M(m, λ))≃Hom_K[H](K[H]/((H−λ)n),(K[H]/((H−λ)m))≃

K[H]/((H−λ)min(n,m)_).

Proof. 1. Statement 1 is obvious.

2. Statement 2 follows from the fact thatF M(n, λ) = 0 andF p=K[x] for all nonzero elements

p∈K[x] (sinceK[x] is a simpleI_{1-module,F is an ideal of the algebra}I_{1such thatF K[x] =K[x]).}

3. Statement 3 follows from the fact that F K[x] = K[x] and F M(n, λ) = 0: f(K[x]) =

f(F K[x]) =F f(K[x]) = 0 for anyf ∈HomI1(K[x], M(n, λ)).

4. The first isomorphism is obvious. Then the second isomorphism follows.

Proposition 2.8. (1) Ext1I₁(K[x], K[x]) = 0.

(2) Ext1I₁(M(n, λ), K[x]) = 0.

(3) Ext1I1(K[x], M(n, λ)) = 0.

(4) Ext1I1(M(n, λ), M(m, µ)) =

(

K if λ−µ∈Z_,

0 if λ−µ6∈Z_.

Proof. 1. Let 0→ K[x] →N → K[x] →0 be a s.e.s. of I_{1-modules. Then F N =N (since}

F K[x] = K[x]), and so N is an epimorphic image of the semisimple I_{1-moduleF ⊕F. Hence,}

N ≃K[x]⊕K[x] (sinceI₁F ≃K[x]( N₎

). 2. See (11).

3. Let 0 → M = M(n, λ) → L → K[x] → 0 be a s.e.s. of I_{1-modules. Since F M = 0,}

we have F L = F K[x] ≃ K[x] is a submodule of L such that F L∩M = 0 (since otherwise

F L⊆M by simplicity of theI_{1-moduleF L}≃K[x], and so 0=6 K[x]≃F L=F2L⊆F M = 0, a contradiction). ThenF L⊕M ⊆L. Furthermore,F L⊕M =LsincelI1(F L⊕M) =lI1(L). This

means that the s.e.s. splits.

4. Let 0→M1 →M →M1 →0 be a s.e.s. of generalized weightI1-modules. If Supp(M1)∩ Supp(M2) =∅, it splits. In particular, Ext1I1(M(n, λ), M(m, µ)) = 0 if λ−µ6∈Z. Ifλ−µ∈ Z

we can assume thatλ=µ(sineM(m, λ)≃M(m, µ)). Using (12), where we assume thatλ= 0 if

λ∈Z_{, we see that Ext}1_I

1(M(n, λ), M(m, λ))≃M(m, λ)/(H−λ)M(m, λ)≃K.

Since the left global dimension of the algebraI_{1 is 1, [6], Proposition 2.7 and Proposition 2.8}

describe all the Ext-groups between indecomposable, generalized weightI_{1-modules. This is also}

6 V. V. BAVULA, V. BEKKERT AND V. FUTORNY

References

[1] V. V. Bavula and V. Bekkert, Indecomposable representations of generalized Weyl algebras,Comm. Algebra, 28_{(2000), no. 11, 5067-5100.}

[2] V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra,J. Lond. Math. Soc. (2), 83_{(2011), no. 2, 517-543.}

[3] V. V. Bavula, The group of automorphisms of the algebra of polynomial integro-differential operators, J. Algebra,348_{(2011), 233-263.}

[4] V. V. Bavula, The algebra of integro-differential operators on an affine line and its modules, J. Pure Appl. Algebra217_{(2013), no. 3, 495-529.}

[5] V. V. Bavula, The algebra of polynomial integro-differential operators is a holonomic bimodule over the subalgebra of polynomial differential operators.Algebr. Represent. Theory17_{(2014), no. 1, 275-288.} [6] V. V. Bavula, The global dimension of the algebra of integro-differential operators, submitted.

Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, UK

E-mail address:[email protected]

Departamento de Matem´atica, ICEx, Universidade Federal de Minas Gerais, Av. Antˆonio Carlos, 6627, CP 702, CEP 30123-970, Belo Horizonte-MG, Brasil

E-mail address:[email protected]

Instituto de Matem´atica e Estat´ıstica, Universidade de S˜ao Paulo, Caixa Postal 66281, S˜ao Paulo, CEP 05315-970, Brasil