Indecomposable injective modules over Down-Up algebras

8/8/2019 Indecomposable injective modules over Down-Up algebras

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Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down-Up algebras

Indecomposable injective modules over Down-Up

algebras

Christian Lompjointly with Paula Carvalho and Dilek Pusat-Yilmaz

Universidade do Porto

21. May 2010

Christian Lomp Indecomposable injective modules over Down-Up algebras
http://find/http://goback/


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Prufer Groups

Prime number p: the Prufer group Zp is the union of the chain

Zp Zp2 Zp3 . . .

n=1

Zpn = Zp.



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Prufer Groups


Zp Zp2 Zp3 . . .

n=1

Zpn = Zp.

Every proper subgroup ofZp is finite.



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Prufer Groups


Zp Zp2 Zp3 . . .

n=1

Zpn = Zp.

Every proper subgroup ofZp is finite.The Prufer groups are the injective hulls ofZp.



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M tli Th W l Al b Q t Pl H i b l b D U l b


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Matlis Theory

Theorem (Matlis, 1960)

Injective hulls of simples over Noetherian commutative rings areArtinian.


Matlis Theory Weyl Algebras Quantum Plane Heisenberg algebra Down Up algebras


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Matlis Theory

Theorem (Matlis, 1960)

Injective hulls of simples over Noetherian commutative rings areArtinian.

Theorem (Vamos, 1968)

For a commutative ring R the following are equivalent:

Injective hulls of simples are Artinian;

Rm is Noetherian, m MaxSpec(R).




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Weyl algebras

Theorem (Hirano, 2002)

Injective hulls of simples over A1(Z) = Z[x, y | yx = xy + 1]are Artinian.




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Weyl algebras



The injective hull ofQ[x] as A1(Q)-module is not Artinian.




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Weyl algebras




Observation

Injective hulls of simples over A1(Q) are locally Artinian.




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Matlis Theory eyl lgebras Quantum Plane Heisenberg algebra Do n Up algebras

Weyl algebras




Observation

Injective hulls of simples over A1(Q) are locally Artinian.

Theorem (Stafford, 1984)

There exist simple modules over An(C) (n 2) whose injectivehull is not locally Artinian.




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y y g g g p g

Quantum Plane

Injective hulls of simple modules over K[x, y] are Artinian.




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Quantum Plane


Theorem (Carvalho,Musson 2010)

Injective hulls of simples over Kq[x, y] = K[x, y | yx = qxy] arelocally Artinian if and only if q is a root of unity.




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Quantum Plane



Injective hulls of simples over Kq[x, y] = K[x, y | yx = qxy] arelocally Artinian if and only if q is a root of unity.

Question

Over which non-commutative Noetherian rings are injective hullsof simples locally Artinian?




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Theorem (Carvalho, Pusat-Yilmaz,L.)

Let A be a countably generated Noetherian C-algebra withNoetherian 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)).




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(3-dimensional complex Heisenberg Lie algebra)

h generated by x, y, z subject to [x, y] = z, [x, z] = 0 = [y, z].




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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




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A/mA C[x, y] or A/mA A1(C).




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A/mA C[x, y] or A/mA A1(C).

Hence injective hulls of simples over U(h) are locally Artinian.




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(2n + 1-dimensional complex Heisenberg Lie algebra)h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi, yi] = z for1 i n and zero for all other combinations of generators.




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(2n + 1-dimensional complex Heisenberg Lie algebra)h generated by x1 . . . , xn, y1, . . . , yn, z subject to [xi, yi] = z for1 i n and zero for all other combinations of generators.Then A = U(h) admits a non-locally Artinian injective hull of asimple if n 2, because A/z 1 An(C).




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Theorem (Musson, 1982)

non-nilpotent soluble finite dimensional complex Lie algebrasg a non-locally Artinian injective hulls of a simple U(g)-module.




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Theorem (Musson, 1982)

non-nilpotent soluble finite dimensional complex Lie algebrasg a non-locally Artinian injective hulls of a simple U(g)-module.

Any such algebra hasC[x, y | yx = xy + x] as a factor algebra.




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Down-Up algebras

Theorem (Dahlberg, 1988)

Injective hulls of simples over U(sl2) are locally Artinian.




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Down-Up algebras



U(sl2) = A(2,1, 1) and U(h) = A(2,1, 0) are Noetherian

Down-up Algebras




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Down-Up algebras



U(sl2) = A(2,1, 1) and U(h) = A(2,1, 0) are Noetherian

Down-up AlgebrasTheorem (Benkart, Roby 1999)

For (,,) C3, the Down-Up algebra A(,,) is generatedby two elements u and d subject to the relations

d2u = dud + ud2 + d

du2 = udu+ u2d + u




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Fully bounded Noetherian

Question (Smith - X.Antalya Algebra Days 2008)

For which Noetherian Down-Up algebras are all injective hulls ofsimple modules locally Artinian?




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Theorem (Jategaonkar, 1974)

Injective hulls of simples over an FBN ring are locally Artinian.




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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.




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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.

(Jacobsons conjecture)

If J is the Jacobson radical of a Noetherian ring, then

n=1 Jn = 0.




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FBN Down-Up algebras

QuestionWhich Noetherian Down-Up algebras are FBN ?




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FBN Down-Up algebras

QuestionWhich Noetherian Down-Up algebras are FBN ?


The following statements are equivalent for a Noetherian Down-upalgebra A = A(,,):

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 X are distinct roots of

unity such that both are also different from 1 if= 0.



G


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Generalized Weyl algebras

Definition (Bavula 1992, Rosenberg 1995)

The generalized Weyl algebra R(,a) is the R-algebra generatedby X+ and X subject to

X+r = (r)X+ and Xr = 1(r)X r R;

X+X = a and XX+ = 1(a).



G li d W l l b


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X+X = a and XX+ = 1(a).

Observation (Kirkman-Musson-Passman, 2000)

A is isomorphic to a generalized Weyl algebra, where R = C[x, y]and(x) = 1(yx) and(y) = x and : A R(, x) by

u X+; d X; ud x; du y



G li d W l l b


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X+X = a and XX+ = 1(a).

Observation (Kirkman-Musson-Passman, 2000)

A is isomorphic to a generalized Weyl algebra, where R = C[x, y]and(x) = 1(yx) and(y) = x and : A R(, x) by

u X+; d X; ud x; du y

Observation (Kulkarni, 2001)

R(, x) is f.g. over its centre if and only if has finite order.



C l i


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Conclusion

Theorem (Carvalho, Pusat-Yilmaz,L.)Injective hulls of simples over A(1 ,, 1) are locally Artinian, if is a root of unity.



C l i


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Conclusion


(Bavula+Lenagan, 2001): Krull dimension ofA := A(1 ,, 1) is 2.



C l i


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Conclusion



(Praton 2004): Z(A) = C[w]



Conclusion


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Conclusion



(Praton 2004): Z(A) = C[w]

A/mA has Krull dimension 1 for m = w.



Conclusion


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Conclusion



(Praton 2004): Z(A) = C[w]

A/mA has Krull dimension 1 for m = w.

Corollary

Injective hulls of simples over A(,,) are locally Artinian, if theroots of X2 X are distinct roots of unity or both 1.



Progress


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Progress ...


Injective hulls of simples over A(,,) are locally Artinian, if(and only if ?) the roots of X2 X are roots of unity.



Progress


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Progress ...



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



Progress


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Progress ...



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

Tesekkur Ederim !


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Indecomposable injective modules over Down-Up algebras