GoBack - Temple Universitylorenz/talks/Raleigh.pdf · enveloppante, Bull. Soc. Math. France 109 (1981), 403–426. Sur la classiﬁcation des ideaux primitifs des alg´ ebres enveloppantes`,

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Algebraic group actions onnoncommutative spectra

Special Session on Brauer Groups, Quadratic Forms, AlgebraicGroups, and Lie Algebras — NCSU 04/04/2009

Martin LorenzTemple University, Philadelphia

http://math.temple.edu/~lorenz/

Overview

Algebraic group actions on noncommutative spectra NCSU 04/04/2009

• Background: enveloping algebras and quantized coordinatealgebras

Overview



• Tool: the Amitsur-Martindale ring of quotients

Overview




• Rational and primitive ideals

Overview




• Rational and primitive ideals

• Stratification of the prime spectrum (if time)

References


• “Group actions and rational ideals”,Algebra and Number Theory 2 (2008), 467-499

• “Algebraic group actions on noncommutative spectra”,Transformation Groups (to appear)

Both articles & the pdf file of this talk available on my web page:

http://math.temple.edu/˜lorenz/

http://math.temple.edu/~lorenz/


I will work / base field � = �

Background

Enveloping algebras


Goal: For R = U(g), the enveloping algebra of a finite-dim’l Liealgebra g, describe

Prim R = {primitive ideals of R}

kernels of irreducible (generally infinite-dimensional)

representations R → End�(V )

Enveloping algebras


Jacques Dixmier (* 1924)

in Reims, Dec. 2008

• former secretary of Bourbaki

• Ph.D. advisor of A. Connes, M. Duflo, . . .

• author of several influential monographs:

Les algebres d’operateurs dans l’espace hilbertien:algebres de von Neumann, Gauthier-Villars, 1957

Les C∗-algebres et leurs representations,Gauthier-Villars, 1969

Algebres enveloppantes, Gauthier-Villars, 1974

Dixmier’s Problem 11


from Algebres enveloppantes, 1974 :



• Problem 11 for k solvable, char � = 0

Dixmier, Sur les ideaux generiques dans les algebres enveloppantes,Bull. Sci. Math. (2) 96 (1972), 17–26. � existence: (a)

Borho, Gabriel, Rentschler, Primideale in Einhullenden auflosbarer Lie-Algebren,Springer Lect. Notes in Math. 357 (1973). � uniqueness: (b)

• for noetherian or Goldie rings R / char � = 0:Mœglin & Rentschler

Orbites d’un groupe algebrique dans l’espace des ideaux rationnels d’une algebreenveloppante, Bull. Soc. Math. France 109 (1981), 403–426.

Sur la classification des ideaux primitifs des algebres enveloppantes, Bull. Soc. Math.France 112 (1984), 3–40.

Sous-corps commutatifs ad-stables des anneaux de fractions des quotients desalgebres enveloppantes; espaces homogenes et induction de Mackey, J. Funct. Anal.69 (1986), 307–396.

Ideaux G-rationnels, rang de Goldie, preprint, 1986.



• Problem 11 for k solvable, char � = 0

Dixmier, Sur les ideaux generiques dans les algebres enveloppantes,Bull. Sci. Math. (2) 96 (1972), 17–26. � existence: (a)

Borho, Gabriel, Rentschler, Primideale in Einhullenden auflosbarer Lie-Algebren,Springer Lect. Notes in Math. 357 (1973). � uniqueness: (b)

• for noetherian or Goldie rings R / char � = 0:Mœglin & Rentschler

Orbites d’un groupe algebrique dans l’espace des ideaux rationnels d’une algebreenveloppante, Bull. Soc. Math. France 109 (1981), 403–426.

Sur la classification des ideaux primitifs des algebres enveloppantes, Bull. Soc. Math.France 112 (1984), 3–40.

Sous-corps commutatifs ad-stables des anneaux de fractions des quotients desalgebres enveloppantes; espaces homogenes et induction de Mackey, J. Funct. Anal.69 (1986), 307–396.

Ideaux G-rationnels, rang de Goldie, preprint, 1986.



• under weaker Goldie hypotheses / char � arbitrary:N. Vonessen

Actions of algebraic groups on the spectrum of rational ideals,J. Algebra 182 (1996), 383–400.

Actions of algebraic groups on the spectrum of rational ideals. II,J. Algebra 208 (1998), 216–261.

Quantum groups


Goal: For R = Oq(�

n),Oq(Mn),Oq(G) . . . a quantized coordinatering, describe

Spec R = {prime ideals of R} ⊇ Prim R

Quantum groups


Goal: For R = Oq(�

n),Oq(Mn),Oq(G) . . . a quantized coordinatering, describe

Spec R = {prime ideals of R} ⊇ Prim R

Typically, some algebraic torus T acts rationally by �-algebraautomorphisms on R; so have

Spec R −→ SpecT R = {T -stable primes of R}

P �→ P : T =⋂g∈T

g.P

Quantum groups


� T -stratification of Spec R( Goodearl & Letzter, . . . ; see the monograph by Brown & Goodearl )

Spec R =⊔

I∈SpecT R

SpecI R

{P ∈ Spec R | P : T = I}

Quantum groups


� T -stratification of Spec R( Goodearl & Letzter, . . . ; see the monograph by Brown & Goodearl )

Spec R =⊔

I∈SpecT R

SpecI R

?

Notation


For the remainder of this talk,

R denotes an associative �-algebra (with 1)

G is an affine algebraic �-group acting rationally on R;so R is a �[G]-comodule algebra.

Equivalently, we have a rational representation

ρ = ρR : G→ Aut�-alg(R) ⊆ GL(R)

Tool: The Amitsur-Martindale ringof quotients

Original references


for prime rings R:

W. S. Martindale, III, Prime rings satisfying a generalizedpolynomial identity, J. Algebra 12 (1969), 576–584.

for general R:

S. A. Amitsur, On rings of quotients, Symposia Math., Vol. VIII,Academic Press, London, 1972, pp. 149–164.

The definition


In brief,

Qr(R) = lim−→I∈E

Hom(IR, RR)

where E = E (R) is the filter of all I � R such that l. annR I = 0.

The definition


In brief,


Hom(IR, RR)


Explicitly, elements of Qr(R) are equivalence classes of rightR-module maps

f : IR → RR (I ∈ E ) ,

the map f being equivalent to f ′ : I ′R → RR (I ′ ∈ E ) if f = f ′ on

some J ⊆ I ∩ I ′, J ∈ E .

The definition


In brief,


Hom(IR, RR)


Addition and multiplication of Qr(R) come from addition andcomposition of R-module maps.

The definition


In brief,


Hom(IR, RR)


Sending r ∈ R to the equivalence class of λr : R→ R, x �→ rx,yields an embedding of R as a subring of Qr(R).

Extended centroid


Defn: The extended centroid of R is defined by

C(R) = Z Qr(R)

center

Fact: If R is prime then C(R) is a �-field.

Examples


• If R is simple, or a finite product of simple rings, then

Qr(R) = R

Examples


• If R is simple, or a finite product of simple rings, then

Qr(R) = R

• For R semiprime right Goldie,

Qr(R) = {q ∈ Qcl(R) | qI ⊆ R for some I � R with annR I = 0}

classical quotient ring of R

In particular,

C(R) = ZQcl(R)

Examples


• R = U(g)/I a semiprime image of the enveloping algebra of afinite-dimensional Lie algebra g. Then

Qr(R) = { ad g-finite elements of Qcl(R) }

Rational Ideals

Definition


Want: an intrinsic characterization of “primitivity”, ideally

in detail . . .

Definition


“coeur”

“Herz”

“heart”

“core”

Definition: • Recall that C(R/P ) is a �-field for any P ∈Spec R. We call P rational if C(R/P ) = �.

Definition


“coeur”

“Herz”

“heart”

“core”

Definition: • Recall that C(R/P ) is a �-field for any P ∈Spec R. We call P rational if C(R/P ) = �.

• Put RatR = {P ∈ Spec R | P is rational }; so

RatR ⊆ Spec R

Connection with irreducible representations


Given an irreducible representation f : R→ End�(V ), let P = Ker fbe the corresponding primitive ideal of R.




• There always is an embedding of �-fields

C(R/P ) ↪→ Z (EndR(V ))




• There always is an embedding of �-fields

C(R/P ) ↪→ Z (EndR(V ))

• Typically, EndR(V ) = � (“weak Nullstellensatz”); in this case

PrimR ⊆ RatR

Examples


The weak Nullstellensatz holds for

• R any affine �-algebra, � uncountable Amitsur

• R an affine PI-algebra Kaplansky

• R = U(g) “Quillen’s Lemma”

• R = �Γ with Γ polycyclic-by-finite Hall, L.

• many quantum groups: Oq(�n), Oq(Mn(�)), Oq(G), . . .

Examples


The weak Nullstellensatz holds for

• R any affine �-algebra, � uncountable Amitsur

• R an affine PI-algebra Kaplansky

• R = U(g) “Quillen’s Lemma”

• R = �Γ with Γ polycyclic-by-finite Hall, L.

• many quantum groups: Oq(�n), Oq(Mn(�)), Oq(G), . . .

In fact, in all these examples except the first, it has been shownthat, under mild restrictions on � or q,

PrimR = RatR

Group action: G-prime and G-rational ideals


The G-action on R induces actions on {ideals of R}, Spec R, Rat R,. . . . As usual, G\? denotes the orbit sets in question.

Definition: A proper G-stable ideal I � R is called G-prime ifA,B �

G-stabR , AB ⊆ I ⇒ A ⊆ I or B ⊆ I. Put

G-Spec R = {G-prime ideals of R}



Propn The assignment γ : P �→ P :G =⋂g∈G

g.P yields

surjections

Spec R

can.��

γ �� G-Spec R

G\ Spec R

��



Definition: Let I ∈ G-Spec R. The group G acts on C(R/I)and the invariants C(R/I)G are a �-field. We call IG-rational if C(R/I)G = �. Put

G-RatR = {G-rational ideals of R}



Definition: Let I ∈ G-Spec R. The group G acts on C(R/I)and the invariants C(R/I)G are a �-field. We call IG-rational if C(R/I)G = �. Put

G-RatR = {G-rational ideals of R}

The following result solves Dixmier’s Problem # 11 (a),(b) forarbitrary algebras.

Theorem 1 G\RatRbij.−→ G-RatR

∈ ∈

G.P �→⋂g∈G

g.P

Noncommutative spectra


Spec R

can.

��

γ : P �→P :G=⋂

g∈G g.P

��

Rat R

��

��

��

��

��

��

� �

��

G\ Spec R �� G-Spec R

G\RatR ∼=��

� �

��

G-RatR� �

��



Spec R

can.

��

γ : P �→P :G=⋂

g∈G g.P

��

Rat R

��

��

��

��

��

��

� �

��


G\RatR ∼=��

� �

��

G-RatR� �

��

Spec R carries the Jacobson-Zariski topology: closed subsets arethose of the form V(I) = {P ∈ Spec R | P ⊇ I} where I ⊆ R.



Spec R

can.

��

γ : P �→P :G=⋂

g∈G g.P

��

Rat R

��

��

��

��

��

��

� �

��


G\RatR ∼=��

� �

��

G-RatR� �

��

� is a surjection whose target has the final topology,

↪→ is an inclusion whose source has the induced topology, and∼= is a homeomorphism, from Thm 1

Local closedness


Recall: locally closed = open ∩ closed

Local closedness


Theorem 2 If P ∈ RatR then:

{P} loc. cl. in Spec R ⇐⇒ {P :G} loc. cl. in G-Spec R

Cor If P ∈ Rat R is loc. closed in Spec R then the orbit G.Pis open in its closure in RatR.

Local closedness


Theorem 2 If P ∈ RatR then:

{P} loc. cl. in Spec R ⇐⇒ {P :G} loc. cl. in G-Spec R

Cor If P ∈ Rat R is loc. closed in Spec R then the orbit G.Pis open in its closure in RatR.

Proof of Cor: P :G ∈ G-Spec R is locally closed by Theorem 2, andhence so is its preimage under f : RatR ↪→ Spec R

γ→ G-Spec R.Finally, f−1(P:G) = G.P by Theorem 1.

Stratification of the primespectrum

Goal


Spec R

can.

��

γ : P �→P :G=⋂

g∈G g.P

��

Rat R

��

��

��

��

��

��

� �

��


G\RatR ∼=��

� �

��

G-RatR� �

��

Next, we turn to the map γ . . .

Goal


Recall: γ : Spec R� G-Spec R yields the G-stratification of Spec R

Spec R =⊔

I∈G-Spec R

SpecI R

Main goal: describe the G-strata

SpecI R = γ−1(I) = {P ∈ Spec R | P :G = I}

Goal


For simplicity, I assume G to be connected; so �[G] is a domain.In particular,

G-Spec R = SpecG R = {G-stable primes of R}

The rings TI


For a given I ∈ G-Spec R, put Fract �[G]

TI = C(R/I)⊗ �(G)

This is a commutative domain, a tensor product of two fields.

The rings TI



TI = C(R/I)⊗ �(G)


G-actions: • on C(R/I) via the given action ρ : G →Aut�-alg(R)

• on �(G) by the right and left regular actionsρr : (x.f)(y) = f(yx) and ρ� : (x.f)(y) = f(x−1y)

• on TI by ρ⊗ ρr and Id⊗ρ� ←− commute

The rings TI



TI = C(R/I)⊗ �(G)


PutSpecG TI = {(ρ⊗ ρr)(G)-stable primes of TI}

Stratification Theorem


Theorem 3 There is a bijection

c : SpecI R −→ SpecG TI

having the following properties:

(a) G-equivariance: c(g.P ) = (Id⊗ρ�)(g)(c(P ));

(b) inclusions: P ⊆ P ′ ⇐⇒ c(P ) ⊆ c(P ′);

(c) hearts: C (TI/c(P )) ∼= C (R/P ⊗ �(G)) as �(G)-fields;

(d) rationality: P is rational ⇐⇒ TI/c(P ) = �(G).

Stratification Theorem


Theorem 3 There is a bijection

c : SpecI R −→ SpecG TI

having the following properties:

(a) G-equivariance: c(g.P ) = (Id⊗ρ�)(g)(c(P ));

(b) inclusions: P ⊆ P ′ ⇐⇒ c(P ) ⊆ c(P ′);

(c) hearts: C (TI/c(P )) ∼= C (R/P ⊗ �(G)) as �(G)-fields;

(d) rationality: P is rational ⇐⇒ TI/c(P ) = �(G).

Cor: Rational ideals are maximal in their strata

Rudolf Rentschler (PhD 1967 Munich)


GoBack - Temple Universitylorenz/talks/Raleigh.pdf · enveloppante, Bull. Soc. Math. France 109 (1981), 403–426. Sur la classiﬁcation des ideaux primitifs des alg´ ebres enveloppantes`,

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