Projective modules over universal enveloping...

Projective modules over universal enveloping algebras

September 27, 2011

() Projective modules over universal enveloping algebras September 27, 2011 1 / 14

R is an associative ring with unit

R-module means a right module over R

A module P is said to be projective if it is isomorphic to a direct

summand of R(�)R

Problem: For a (speci�ed) ring give a classi�cation of projective modules.By a theorem of Kaplansky every projective module is a direct sum ofcountably generated ones.

R is an associative ring with unit

R-module means a right module over R

A module P is said to be projective if it is isomorphic to a direct

summand of R(�)R

Problem: For a (speci�ed) ring give a classi�cation of projective modules.By a theorem of Kaplansky every projective module is a direct sum ofcountably generated ones.

why not?

Projective modules are relevant in homological algebra, geometry anddirect sum decompositions problems in module theory.

Theorem

(Swan) Let X be a compact topological space. The category of real vector

bundles on X is equivalent to the category of �nitely generated projective

modules over C (X ).

Add(M) consists of direct summands of M(�) for some �.

Theorem

(Dress) Let M be a �nitely generated module over R. The category

Add(M) is equivalent to the category of projective modules over

EndR(M).

why not?

Theorem

EndR(M).

why not?

Theorem

EndR(M).

Noetherian setting

Typical situation over noetherian rings is that �nitely generated projectivemodules are harder to classify than non�nitely generated ones.

Evolution of so called Serre's problem (1955):

In 1957 Serre proved that �nitely generated projectives overk[x1; : : : ; xn] are stably free.

In 1963 Bass proved that non�nitely generated projectives overk[x1; : : : ; xn] are free.

In 1976 Suslin and Quillen independently proved that all projectivemodules over k[x1; : : : ; xn] are free.

Noetherian setting

Typical situation over noetherian rings is that �nitely generated projectivemodules are harder to classify than non�nitely generated ones.Evolution of so called Serre's problem (1955):

In 1957 Serre proved that �nitely generated projectives overk[x1; : : : ; xn] are stably free.

In 1963 Bass proved that non�nitely generated projectives overk[x1; : : : ; xn] are free.

In 1976 Suslin and Quillen independently proved that all projectivemodules over k[x1; : : : ; xn] are free.

Noetherian setting cont.

On the other hand sometimes there is a nice connection between propertiesof the ring R and the structure theory for non�nitely generated projectivemodules. Let us discuss the case of an integral group ring of a �nite group.

Theorem

(Swan, Akasaki, Linnell) If G is a �nite group then G is solvable if and

only if every non�nitely generated projective module over ZG is free.

If P is a projective module Tr(P) =P

f : P!R f (P). This is an idempotentideal in R (trace ideal of P) and PTr(P) = P

Theorem

(Bass) If R is a connected commutative noetherian ring then every

non�nitely generated projective module over R is free.

Theorem

Fair-sized modulesWe say that a ring R satis�es (*) if every chain of two-sided idealsI1 � I2 � I3 � � � � such that Ik+1Ik = Ik+1 stabilizes.If R is a (left and right) noetherian ring satisfying (*) then countablygenerated projective modules can be classi�ed by pairs (I ;P), where I isan idempotent ideal of R and P is a �nitely generated projective moduleover R=I .

countably generated projective module P corresponds to (I ;P=PI ),where I is the smallest ideal of P such that P=PI is �nitely generated.

(0;P) corresponds to a �nitely generated projective R-module P

(R; 0) corresponds to R(!)R

(I ;P) are decomposable whenever 0 6= I .

Theorem

(Puninski) Let n 6= 0 be a square-free integer. Then every torsion-free

module over Z[pn] is a direct sum of �nitely generated modules if and

only if n 6� 1 mod 8.

Theorem

Lie Algebras

Let k be a �eld, an algebra (L; [�;�]) is called a Lie algebra if

[�;�] is bilinear[x ; x ] = 0 for every x 2 L

[[a; b]; c] + [[b; c]; a] + [[c ; a]; b] = 0 for every a; b; c

A representation of L is a Lie algebra homomorphism of L to gln(k)Any associative algebra give a Lie structure via [a; b] := ab � ba.De�ne L(1) = L, L(2) = [L; L],. . . ,L(i+1) = [L(i); L(i)],. . .A Lie algebra L is said to be solvable if there exists i 2 N such thatL(i) = 0.A radical of a Lie algebra L is the greatest solvable ideal in L.A Lie algebra L is called semisimple if has zero radical.A Lie algebra L is called simple if [L; L] 6= 0 and there are no nontrivialideals of L.

Basic structure theorems for Lie algebras

Theorem

(Cartan) If L is a �nite dimensional semisimple Lie algebra over a �eld of

characteristic zero, then L = L1 � � � � � Ln, where every Li is an ideal of L

which is a simple Lie algebra.

Theorem

(Levi) Let B be a �nite-dimensional Lie algebra over a �eld of

characteristic zero and let R be the radical of B. Then B contains a

�nite-dimensional semisimple subalgebra L such that B = L� R.

Universal enveloping algebras

Let L be a Lie algebra with basis fx1; : : : ; xng over k. Let

[xi ; xj ] =nX

ci ;j ;kxk :

The algebra U(L) = khx1; : : : ; xni=hxixj � xjxi �Pn

k=1 ci ;j ;kxki is calledthe universal enveloping algebra of L

Monomials xk11 xk2

2 � � � xknn form a basis of U(L), so L is contained in

If A is an associative k-algebra and ' : L! A a linear map such that'([a; b]) = '(a)'(b)� '(b)'(a) then there is a unique extension of' to a homomorphism �: U(L)! A.

The category of representations of L is equivalent to the category of�nite-dimensional right U(L)-modules.

U(L) is a noetherian domain.

Krull dimension of U(L) is at most dimk(L).

[xi ; xj ] =nX

ci ;j ;kxk :

Monomials xk11 xk2

[xi ; xj ] =nX

ci ;j ;kxk :

Monomials xk11 xk2

[xi ; xj ] =nX

ci ;j ;kxk :

Monomials xk11 xk2

[xi ; xj ] =nX

ci ;j ;kxk :

Monomials xk11 xk2

Some results on �nitely generated projectives

Theorem

Let L be a Lie algebra of �nite dimension. Then

(Quillen) Every �nitely generated projective module over U(L) isstably free.

(Sta�ord) If P is a �nitely generated projective module over U(L) ofrank bigger than dimk(L), then P is free. In particular, Tr(P) = U(L)for every nonzero �nitely generated projective U(L)-module.

(Sta�ord) If [L; L] 6= 0 then U(L) contains a right ideal which is stably

free but not free.

Theorem

free but not free.

Theorem

free but not free.

Solvable case

Theorem

Let k be a �eld of characteristic 0 and let L be a solvable Lie algebra of

�nite dimension over k. If I1; I2; : : : is a sequence of ideals in U(L) suchthat Ik+1Ik = Ik+1; k 2 N. Then either Ik = U(L); k 2 N or there exists l

such that Il = 0.

Corollary

If L is a solvable Lie algebra of �nite dimension over a �eld of characteristic

zero then every non-�nitely generated projective U(L)-module is free.

If [L; L] = 0 we get the Bass' result for k[x1; : : : ; xn].

Solvable case

Theorem

Let k be a �eld of characteristic 0 and let L be a solvable Lie algebra of

�nite dimension over k. If I1; I2; : : : is a sequence of ideals in U(L) suchthat Ik+1Ik = Ik+1; k 2 N. Then either Ik = U(L); k 2 N or there exists l

such that Il = 0.

Corollary

If L is a solvable Lie algebra of �nite dimension over a �eld of characteristic

zero then every non-�nitely generated projective U(L)-module is free.

If [L; L] = 0 we get the Bass' result for k[x1; : : : ; xn].

Simple case

Theorem

(Weyl) Let L be a semisimple Lie algebra of �nite dimension over a �eld of

characteristic zero. Then every U(L)-module of �nite dimension is

completely reducible. In particular, if I is an ideal of �nite codimension

over U(L), then U(L)=I is semisimple artinian.

Corollary

(Kraft, Small, Wallach) If L is as above then every ideal in U(L) of �nitecodimension is idempotent.

Put R = U(sl2(C)). For every k there is a simple R-module Mk ofdimension k . Every Ik = Ann(Mk) is an ideal of codimension k2 thereforeis idempotent. By a result of Whitehead every Ik is a trace ideal of aprojective module which cannot contain a �nitely generated directsummand.

Simple case

Theorem

(Weyl) Let L be a semisimple Lie algebra of �nite dimension over a �eld of

characteristic zero. Then every U(L)-module of �nite dimension is

completely reducible. In particular, if I is an ideal of �nite codimension

over U(L), then U(L)=I is semisimple artinian.

Corollary

(Kraft, Small, Wallach) If L is as above then every ideal in U(L) of �nitecodimension is idempotent.

Put R = U(sl2(C)). For every k there is a simple R-module Mk ofdimension k . Every Ik = Ann(Mk) is an ideal of codimension k2 thereforeis idempotent. By a result of Whitehead every Ik is a trace ideal of aprojective module which cannot contain a �nitely generated directsummand.

Problems in U(sl2(C))

One can show that fair-sized theory cannot work here. There exists acountably generated projective module P such thatfI � U(sl2(C)) j P=PI is �nitely generatedg has not the leastelement.

There are at least uncountably many countably generated projectiveU(sl2(C))-modules up to isomorphism.

Is every nonzero idempotent ideal in U(sl2(C)) of �nite codimension?

Is every indecomposable projective module over U(sl2(C)) �nitelygenerated? (A similar question is posed in a book by Fuchs and Salceover commutative domains.)

Let R = U(sl2(C)) and R̂ its completion in linear topology induced byideals of �nite codimension. Let P;Q be countably (�nitely)generated projective R-modules such that P R R̂ ' Q R R̂. Does itmean P ' Q?

Is there a way how to classify projective modules over U(sl2(C))?

Thanks for your attention.

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