1. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Indecomposable injective modules over Down-Up
algebras
Christian Lomp
jointly with Paula Carvalho and Dilek Pusat-Yilmaz
Universidade do Porto
21. May 2010
Christian Lomp Indecomposable injective modules over Down-Up algebras

2. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Pr¨ufer Groups
Prime number p: the Pr¨ufer group Zp∞ is the union of the chain
Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂
∞
n=1
Zpn = Zp∞ .
Christian Lomp Indecomposable injective modules over Down-Up algebras

3. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Pr¨ufer Groups
Prime number p: the Pr¨ufer group Zp∞ is the union of the chain
Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂
∞
n=1
Zpn = Zp∞ .
Every proper subgroup of Zp∞ is finite.
Christian Lomp Indecomposable injective modules over Down-Up algebras

4. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Pr¨ufer Groups
Prime number p: the Pr¨ufer group Zp∞ is the union of the chain
Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂
∞
n=1
Zpn = Zp∞ .
Every proper subgroup of Zp∞ is finite.
The Pr¨ufer groups are the injective hulls of Zp.
Christian Lomp Indecomposable injective modules over Down-Up algebras

5. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Pr¨ufer Groups
Prime number p: the Pr¨ufer group Zp∞ is the union of the chain
Zp ⊂ Zp2 ⊂ Zp3 ⊂ . . . ⊂
∞
n=1
Zpn = Zp∞ .
Every proper subgroup of Zp∞ is finite.
The Pr¨ufer groups are the injective hulls of Zp.
Definition (Injective Hull)
M ⊆ E(M) :
E(M) is injective;
M ⊆ E(M) is essential, i.e.
∀U ⊆ E(M) : U ∩ M = 0 ⇒ U = 0.
Christian Lomp Indecomposable injective modules over Down-Up algebras

6. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Matlis Theory
Theorem (Matlis, 1960)
Injective hulls of simples over Noetherian commutative rings are
Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

7. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Matlis Theory
Theorem (Matlis, 1960)
Injective hulls of simples over Noetherian commutative rings are
Artinian.
Theorem (Vamos, 1968)
For a commutative ring R the following are equivalent:
Injective hulls of simples are Artinian;
Rm is Noetherian, ∀m ∈ MaxSpec(R).
Christian Lomp Indecomposable injective modules over Down-Up algebras

8. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Weyl algebras
Theorem (Hirano, 2002)
Injective hulls of simples over A1(Z) = Z[x, y | yx = xy + 1]
are Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

9. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Weyl algebras
Theorem (Hirano, 2002)
Injective hulls of simples over A1(Z) = Z[x, y | yx = xy + 1]
are Artinian.
The injective hull of Q[x] as A1(Q)-module is not Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

10. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Weyl algebras
Theorem (Hirano, 2002)
Injective hulls of simples over A1(Z) = Z[x, y | yx = xy + 1]
are Artinian.
The injective hull of Q[x] as A1(Q)-module is not Artinian.
Observation
Injective hulls of simples over A1(Q) are locally Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

11. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Weyl algebras
Theorem (Hirano, 2002)
Injective hulls of simples over A1(Z) = Z[x, y | yx = xy + 1]
are Artinian.
The injective hull of Q[x] as A1(Q)-module is not Artinian.
Observation
Injective hulls of simples over A1(Q) are locally Artinian.
Theorem (Stafford, 1984)
There exist simple modules over An(C) (n ≥ 2) whose injective
hull is not locally Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

12. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Quantum Plane
Injective hulls of simple modules over K[x, y] are Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

13. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Quantum Plane
Injective hulls of simple modules over K[x, y] are Artinian.
Theorem (Carvalho,Musson 2010)
Injective hulls of simples over Kq[x, y] = K[x, y | yx = qxy] are
locally Artinian if and only if q is a root of unity.
Christian Lomp Indecomposable injective modules over Down-Up algebras

14. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Quantum Plane
Injective hulls of simple modules over K[x, y] are Artinian.
Theorem (Carvalho,Musson 2010)
Injective hulls of simples over Kq[x, y] = K[x, y | yx = qxy] are
locally Artinian if and only if q is a root of unity.
Question
Over which non-commutative Noetherian rings are injective hulls
of simples locally Artinian?
Christian Lomp Indecomposable injective modules over Down-Up algebras

15. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be a countably generated Noetherian C-algebra with
Noetherian centre.
1 Injective hulls of simples over A are locally Artinian
2 Injective hulls of simples over A/mA are locally Artinian,
∀m ∈ MaxSpec(Z(A)).
Christian Lomp Indecomposable injective modules over Down-Up algebras

16. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be a countably generated Noetherian C-algebra with
Noetherian centre.
1 Injective hulls of simples over A are locally Artinian
2 Injective hulls of simples over A/mA are locally Artinian,
∀m ∈ MaxSpec(Z(A)).
(3-dimensional complex Heisenberg Lie algebra)
h generated by x, y, z subject to [x, y] = z, [x, z] = 0 = [y, z].
Christian Lomp Indecomposable injective modules over Down-Up algebras

17. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be a countably generated Noetherian C-algebra with
Noetherian centre.
1 Injective hulls of simples over A are locally Artinian
2 Injective hulls of simples over A/mA are locally Artinian,
∀m ∈ MaxSpec(Z(A)).
(3-dimensional complex Heisenberg Lie algebra)
h generated by x, y, z subject to [x, y] = z, [x, z] = 0 = [y, z].
Let A = U(h), then Z(A) = C[z] and
Christian Lomp Indecomposable injective modules over Down-Up algebras

18. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be a countably generated Noetherian C-algebra with
Noetherian centre.
1 Injective hulls of simples over A are locally Artinian
2 Injective hulls of simples over A/mA are locally Artinian,
∀m ∈ MaxSpec(Z(A)).
(3-dimensional complex Heisenberg Lie algebra)
h generated by x, y, z subject to [x, y] = z, [x, z] = 0 = [y, z].
Let A = U(h), then Z(A) = C[z] and
A/mA C[x, y] or A/mA A1(C).
Christian Lomp Indecomposable injective modules over Down-Up algebras

19. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be a countably generated Noetherian C-algebra with
Noetherian centre.
1 Injective hulls of simples over A are locally Artinian
2 Injective hulls of simples over A/mA are locally Artinian,
∀m ∈ MaxSpec(Z(A)).
(3-dimensional complex Heisenberg Lie algebra)
h generated by x, y, z subject to [x, y] = z, [x, z] = 0 = [y, z].
Let A = U(h), then Z(A) = C[z] and
A/mA C[x, y] or A/mA A1(C).
Hence injective hulls of simples over U(h) are locally Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

20. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
(2n + 1-dimensional complex Heisenberg Lie algebra)
h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi , yi ] = z for
1 ≤ i ≤ n and zero for all other combinations of generators.
Christian Lomp Indecomposable injective modules over Down-Up algebras

21. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
(2n + 1-dimensional complex Heisenberg Lie algebra)
h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi , yi ] = z for
1 ≤ i ≤ n and zero for all other combinations of generators.
Then A = U(h) admits a non-locally Artinian injective hull of a
simple if n ≥ 2, because A/ z − 1 An(C).
Christian Lomp Indecomposable injective modules over Down-Up algebras

22. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
(2n + 1-dimensional complex Heisenberg Lie algebra)
h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi , yi ] = z for
1 ≤ i ≤ n and zero for all other combinations of generators.
Then A = U(h) admits a non-locally Artinian injective hull of a
simple if n ≥ 2, because A/ z − 1 An(C).
Theorem (Musson, 1982)
∀ non-nilpotent soluble finite dimensional complex Lie algebras g
∃ a non-locally Artinian injective hulls of a simple U(g)-module.
Christian Lomp Indecomposable injective modules over Down-Up algebras

23. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
(2n + 1-dimensional complex Heisenberg Lie algebra)
h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi , yi ] = z for
1 ≤ i ≤ n and zero for all other combinations of generators.
Then A = U(h) admits a non-locally Artinian injective hull of a
simple if n ≥ 2, because A/ z − 1 An(C).
Theorem (Musson, 1982)
∀ non-nilpotent soluble finite dimensional complex Lie algebras g
∃ a non-locally Artinian injective hulls of a simple U(g)-module.
Any such algebra has C[x, y | yx = xy + x] as a factor algebra.
Christian Lomp Indecomposable injective modules over Down-Up algebras

24. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Down-Up algebras
Theorem (Dahlberg, 1988)
Injective hulls of simples over U(sl2) are locally Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

25. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Down-Up algebras
Theorem (Dahlberg, 1988)
Injective hulls of simples over U(sl2) are locally Artinian.
U(sl2) = A(2, −1, 1) and U(h) = A(2, −1, 0) are Noetherian
Down-up Algebras
Christian Lomp Indecomposable injective modules over Down-Up algebras

26. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Down-Up algebras
Theorem (Dahlberg, 1988)
Injective hulls of simples over U(sl2) are locally Artinian.
U(sl2) = A(2, −1, 1) and U(h) = A(2, −1, 0) are Noetherian
Down-up Algebras
Theorem (Benkart, Roby 1999)
For (α, β, γ) ∈ C3, the Down-Up algebra A(α, β, γ) is generated
by two elements u and d subject to the relations
d2
u = αdud + βud2
+ γd
du2
= αudu + βu2
d + γu
Christian Lomp Indecomposable injective modules over Down-Up algebras

27. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Fully bounded Noetherian
Question (Smith - X.Antalya Algebra Days 2008)
For which Noetherian Down-Up algebras are all injective hulls of
simple modules locally Artinian?
Christian Lomp Indecomposable injective modules over Down-Up algebras

28. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Fully bounded Noetherian
Question (Smith - X.Antalya Algebra Days 2008)
For which Noetherian Down-Up algebras are all injective hulls of
simple modules locally Artinian?
Theorem (Jategaonkar, 1974)
Injective hulls of simples over an FBN ring are locally Artinian.
Christian Lomp Indecomposable injective modules over Down-Up algebras

29. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Fully bounded Noetherian
Question (Smith - X.Antalya Algebra Days 2008)
For which Noetherian Down-Up algebras are all injective hulls of
simple modules locally Artinian?
Theorem (Jategaonkar, 1974)
Injective hulls of simples over an FBN ring are locally Artinian.
Definition (Fully Bounded Noetherian)
R is Fully bounded Noetherian if it is Noetherian and any essential
left/right ideal of a prime factor of R contains a non-zero ideal.
Christian Lomp Indecomposable injective modules over Down-Up algebras

30. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Fully bounded Noetherian
Question (Smith - X.Antalya Algebra Days 2008)
For which Noetherian Down-Up algebras are all injective hulls of
simple modules locally Artinian?
Theorem (Jategaonkar, 1974)
Injective hulls of simples over an FBN ring are locally Artinian.
Definition (Fully Bounded Noetherian)
R is Fully bounded Noetherian if it is Noetherian and any essential
left/right ideal of a prime factor of R contains a non-zero ideal.
(Jacobson’s conjecture)
If J is the Jacobson radical of a Noetherian ring, then ∞
n=1 Jn = 0.
Christian Lomp Indecomposable injective modules over Down-Up algebras

31. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
FBN Down-Up algebras
Question
Which Noetherian Down-Up algebras are FBN ?
Christian Lomp Indecomposable injective modules over Down-Up algebras

32. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
FBN Down-Up algebras
Question
Which Noetherian Down-Up algebras are FBN ?
Theorem (Carvalho, Pusat-Yilmaz,L.)
The following statements are equivalent for a Noetherian Down-up
algebra A = A(α, β, γ):
1 A is module-finite over a central subalgebra;
2 A satisfies a polynomial identity;
3 A is fully bounded Noetherian;
4 The roots of the polynomial X2 − αX − β are distinct roots of
unity such that both are also different from 1 if γ = 0.
Christian Lomp Indecomposable injective modules over Down-Up algebras

33. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Generalized Weyl algebras
Definition (Bavula 1992, Rosenberg 1995)
The generalized Weyl algebra R(σ, a) is the R-algebra generated
by X+ and X− subject to
X+r = σ(r)X+ and X−r = σ−1(r)X− ∀r ∈ R;
X+X− = a and X−X+ = σ−1(a).
Christian Lomp Indecomposable injective modules over Down-Up algebras

34. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Generalized Weyl algebras
Definition (Bavula 1992, Rosenberg 1995)
The generalized Weyl algebra R(σ, a) is the R-algebra generated
by X+ and X− subject to
X+r = σ(r)X+ and X−r = σ−1(r)X− ∀r ∈ R;
X+X− = a and X−X+ = σ−1(a).
Observation (Kirkman-Musson-Passman, 2000)
A is isomorphic to a generalized Weyl algebra, where R = C[x, y]
and σ(x) = β−1(y −αx −γ) and σ(y) = x and ϕ : A → R(σ, x) by
u → X+
; d → X−
; ud → x; du → y
Christian Lomp Indecomposable injective modules over Down-Up algebras

35. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Generalized Weyl algebras
Definition (Bavula 1992, Rosenberg 1995)
The generalized Weyl algebra R(σ, a) is the R-algebra generated
by X+ and X− subject to
X+r = σ(r)X+ and X−r = σ−1(r)X− ∀r ∈ R;
X+X− = a and X−X+ = σ−1(a).
Observation (Kirkman-Musson-Passman, 2000)
A is isomorphic to a generalized Weyl algebra, where R = C[x, y]
and σ(x) = β−1(y −αx −γ) and σ(y) = x and ϕ : A → R(σ, x) by
u → X+
; d → X−
; ud → x; du → y
Observation (Kulkarni, 2001)
R(σ, x) is f.g. over its centre if and only if σ has finite order.
Christian Lomp Indecomposable injective modules over Down-Up algebras

36. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L.)
Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if
η is a root of unity.
Christian Lomp Indecomposable injective modules over Down-Up algebras

37. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L.)
Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if
η is a root of unity.
(Bavula+Lenagan, 2001): Krull dimension of
Aη := A(1 − η, η, 1) is 2.
Christian Lomp Indecomposable injective modules over Down-Up algebras

38. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L.)
Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if
η is a root of unity.
(Bavula+Lenagan, 2001): Krull dimension of
Aη := A(1 − η, η, 1) is 2.
(Praton 2004): Z(Aη) = C[w]
Christian Lomp Indecomposable injective modules over Down-Up algebras

39. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L.)
Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if
η is a root of unity.
(Bavula+Lenagan, 2001): Krull dimension of
Aη := A(1 − η, η, 1) is 2.
(Praton 2004): Z(Aη) = C[w]
⇒ Aη/mAη has Krull dimension 1 for m = w .
Christian Lomp Indecomposable injective modules over Down-Up algebras

40. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L.)
Injective hulls of simples over A(1 − η, η, 1) are locally Artinian, if
η is a root of unity.
(Bavula+Lenagan, 2001): Krull dimension of
Aη := A(1 − η, η, 1) is 2.
(Praton 2004): Z(Aη) = C[w]
⇒ Aη/mAη has Krull dimension 1 for m = w .
Corollary
Injective hulls of simples over A(α, β, γ) are locally Artinian, if the
roots of X2 − αX − β are distinct roots of unity or both 1.
Christian Lomp Indecomposable injective modules over Down-Up algebras

41. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Progress ...
Theorem (Carvalho,Musson 2010)
Injective hulls of simples over A(α, β, γ) are locally Artinian, if
(and only if ?) the roots of X2 − αX − β are roots of unity.
Christian Lomp Indecomposable injective modules over Down-Up algebras

42. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Progress ...
Theorem (Carvalho,Musson 2010)
Injective hulls of simples over A(α, β, γ) are locally Artinian, if
(and only if ?) the roots of X2 − αX − β are roots of unity.
P.Carvalho, C.L., D.Pusat-Yilmaz, ”Injective Modules over
Down-Up algebras”, arxiv:0906.2930 to appear in Glasgow Math. J.
P.Carvalho, I.Musson, ”Monolithic Modules over Noetherian Rings”,
arxiv:1001.1466
Christian Lomp Indecomposable injective modules over Down-Up algebras

43. Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Progress ...
Theorem (Carvalho,Musson 2010)
Injective hulls of simples over A(α, β, γ) are locally Artinian, if
(and only if ?) the roots of X2 − αX − β are roots of unity.
P.Carvalho, C.L., D.Pusat-Yilmaz, ”Injective Modules over
Down-Up algebras”, arxiv:0906.2930 to appear in Glasgow Math. J.
P.Carvalho, I.Musson, ”Monolithic Modules over Noetherian Rings”,
arxiv:1001.1466
Te¸sekk¨ur Ederim !
Christian Lomp Indecomposable injective modules over Down-Up algebras