Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Indecomposable injective modules over Down-U...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Pr¨ufer Groups
Prime number p: the Pr¨ufer g...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Pr¨ufer Groups
Prime number p: the Pr¨ufer g...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Pr¨ufer Groups
Prime number p: the Pr¨ufer g...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Pr¨ufer Groups
Prime number p: the Pr¨ufer g...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Matlis Theory
Theorem (Matlis, 1960)
Injecti...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Matlis Theory
Theorem (Matlis, 1960)
Injecti...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Weyl algebras
Theorem (Hirano, 2002)
Injecti...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Weyl algebras
Theorem (Hirano, 2002)
Injecti...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Weyl algebras
Theorem (Hirano, 2002)
Injecti...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Weyl algebras
Theorem (Hirano, 2002)
Injecti...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Quantum Plane
Injective hulls of simple modu...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Quantum Plane
Injective hulls of simple modu...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Quantum Plane
Injective hulls of simple modu...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Theorem (Carvalho, Pusat-Yilmaz,L.)
Let A be...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
(2n + 1-dimensional complex Heisenberg Lie a...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
(2n + 1-dimensional complex Heisenberg Lie a...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
(2n + 1-dimensional complex Heisenberg Lie a...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
(2n + 1-dimensional complex Heisenberg Lie a...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Down-Up algebras
Theorem (Dahlberg, 1988)
In...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Down-Up algebras
Theorem (Dahlberg, 1988)
In...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Down-Up algebras
Theorem (Dahlberg, 1988)
In...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Fully bounded Noetherian
Question (Smith - X...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Fully bounded Noetherian
Question (Smith - X...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Fully bounded Noetherian
Question (Smith - X...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Fully bounded Noetherian
Question (Smith - X...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
FBN Down-Up algebras
Question
Which Noetheri...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
FBN Down-Up algebras
Question
Which Noetheri...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Generalized Weyl algebras
Definition (Bavula ...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Generalized Weyl algebras
Definition (Bavula ...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Generalized Weyl algebras
Definition (Bavula ...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Conclusion
Theorem (Carvalho, Pusat-Yilmaz,L...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Progress ...
Theorem (Carvalho,Musson 2010)
...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Progress ...
Theorem (Carvalho,Musson 2010)
...
Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras
Progress ...
Theorem (Carvalho,Musson 2010)
...
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Injective hulls of simple modules over Noetherian rings

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Injective hulls of simple modules over Noetherian rings

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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