prime ideals and group actions in noncommutative algebralorenz/talks/usc2013.pdf · 2013-02-20 ·...

Prime ideals and group actions

in noncommutative algebra

Martin Lorenz

Temple University, Philadelphia

Colloquium USC 2/20/2013

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

Overview

Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013

• Prime ideals: historical background, first examples,

Jacobson-Zariski topology . . .

Overview




• Representations: primitive ideals, Nullstellensatz,

Dixmier-Mœglin equivalence

Overview






• Groups actions: stratification, orbits, finiteness question

Overview






• Groups actions: stratification, orbits, finiteness question

• Torus actions: some examples

Prime ideals

Prime ideals


R = (R,+, ·, 1) a ring

Definition The ring R is called prime if R 6= 0 and the product

of any two nonzero ideals (!) of R is nonzero.

Prime ideals


R = (R,+, ·, 1) a ring

Definition The ring R is called prime if R 6= 0 and the product

of any two nonzero ideals (!) of R is nonzero.

An ideal I of R is called prime if R/I is a prime ring

SpecR = {prime ideals of R}

First examples (commutative)


(1) SpecZ 1-1←→ {prime numbers} ∪ {0}

from Mumford’s “Red Book” (mid 1960s, reprinted as Springer Lect. Notes # 1358)



(2) Spec k[x, y] 1-1←→ k2∪{monic irreducible polynomials}∪{0}k some algebraically closed field



(3) SpecZ[x]

from Mumford’s original mimeographed Harvard notes



(3) SpecZ[x]

Pioneers (number theory)


Richard Dedekind (1831 – 1916)

Introduced “ideals” and “prime ideals”

into number theory

Pioneers (number theory)


Richard Dedekind (1831 – 1916)

Introduced “ideals” and “prime ideals”

into number theory

David Hilbert (1862 – 1943)

Introduced the term “ring”

“Zahlring, Ring oder Integritatsbereich”; Dedeking used “Ordnung”

Reference Die Theorie der algebraischen Zahlkorper,Jahresbericht DMV (1897), 175-546; see §31

Pioneers (noncommutative algebra)


Emmy Noether (1882 – 1935)

in the 1920s in Gottingen

Gave the current definition of “prime” in

terms of products of ideals.

Reference Idealtheorie in Ringbereichen,Math. Ann. 83 (1921), 24-66;see Definition III a, p. 38

Pioneers (noncommutative algebra)


Wolfgang Krull (1899 – 1971)

as a student in Gottingen (1920)

First to investigate prime ideals in a

noncommutative setting.

References Zur Theorie der zweiseitigen Ideale in nichtkommutativenBereichen, Math. Zeitschr. 28 (1928), 481-503

Primidealketten in allgemeinen Ringbereichen, Sitzungsber.d. Heidelberger Akad. d. Wissensch. (1928), 3-14

The Jacobson-Zariski topology


Original references

on SpecR for commutative R:

O. Zariski, The fundamental ideas of abstract algebraic geometry,

Proceedings of the ICM, Cambridge, Mass., 1950

on PrimR for general R: special prime ideals (later)

N. Jacobson, A topology for the set of primitive ideals in an arbi-

trary ring, Proc. Nat. Acad. Sci. USA 31 (1945), 333–338



Definition Closed subsets of SpecR are those of the form

V(I) = {P ∈ SpecR | P ⊇ I} where I ⊆ R .





Bad separation properties !

{closed points of SpecR} = {maximal ideals of R}






R prime ⇒ SpecR is irreducible: all nonempty

open subsets are dense






R prime ⇒ SpecR is irreducible: all nonempty

open subsets are dense

But topological notions become available . . .



e.g., Definition The (Krull) dimension of a top. space X is the

supremum of the lengths ℓ of all chains

Y0 $ Y1 $ · · · $ Yℓ

with closed irreducible subsets Yi ⊆ X.



e.g., Definition The (Krull) dimension of a top. space X is the

supremum of the lengths ℓ of all chains

Y0 $ Y1 $ · · · $ Yℓ

with closed irreducible subsets Yi ⊆ X.

Dimension Thm(classical)

If R is an affine commutative k-algebra then

dim SpecR = maxP

tr.degk Fract(R/P )

where P runs over the minimal primes of R .

Affine algebraic varieties


R := k[x2, y2, xy] ∼= k[r, s, t]/(rs− t2)

cone

k[x, y]

plane

algebra-geometry dictionary

Representations

Representations and primitive ideals


From now on: R a k-algebra

k some alg. closed field

e.g., • R = kG the group algebra of the group G

• R = U(g) the enveloping algebra of the Lie algebra g

• . . .



Definition A (linear) representation of R is an algebra homo-

morphism ρ : R → Endk(V ), r 7→ rV , where V is a

k-vector space.

The representation is called irreducible if 0 and V are

the only two subspaces of V that are stable under all

operators rV .

In this case, Ker ρ = {r ∈ R | rV = 0V } ∈ SpecR;

such primes are called primitive.



Goal: For a given algebra R, describe

IrrRepR = {irreducible representations of R}/ ∼=



Goal: For a given algebra R, describe

IrrRepR = {irreducible representations of R}/ ∼=

Unfortunately, this is generally too hard; so . . .



Modified goal: For a given algebra R, describe

PrimR = {primitive ideals of R} ⊆ SpecR

Recall: kernels of (generally infinite-dimensional)irreducible reps R → Endk(V )



Modified goal: For a given algebra R, describe

PrimR = {primitive ideals of R} ⊆ SpecR

This will at least give a coarse classification of IrrRepR

Some examples


(1) Finite-dimensional R:

SpecR =PrimR1-1←→ IrrRepR = a finite set

∈ ∈

Ker ρ ←→ ρ

Some examples


(1) Finite-dimensional R:

SpecR =PrimR1-1←→ IrrRepR = a finite set

∈ ∈

Ker ρ ←→ ρ

(2) The polynomial algebra R = k[x1, . . . , xn]:

MaxR = PrimR1-1←→ IrrRepR

︸︷︷︸

1-1←→ n-space kn

for any commutative R

Some examples


(3) The Weyl algebra R = k{x, y}/(yx = xy + 1) with chark = 0:

SpecR = PrimR = {0} but #IrrRepR =∞

Some examples




Moreover, all reps R→ Endk(V ) are infinite-dimensional:

Some examples




Moreover, all reps R→ Endk(V ) are infinite-dimensional:

dimk V = n <∞ ⇒ yV xV = xV yV + 1V

✭✭✭✭✭✭✭

trace(yV xV ) =✭✭✭✭✭✭✭

trace(xV yV ) + n · 1k0 = n · 1k a contradiction!

Enveloping algebras


Jacques Dixmier (* 1924)

in Reims, Dec. 2008

Enveloping algebras


Recall: for R = kG, the group algebra of a finite group G, one has

SpecR1-1←→ IrrRepR

Enveloping algebras


Recall: for R = kG, the group algebra of a finite group G, one has

SpecR1-1←→ IrrRepR

Clifford’s Thm Given P ∈ Spec kG and N E G , there is a

Q ∈ Spec kN , unique up to G-conjugacy, with

P ∩ kN = Q :G =def

⋂

g∈G

gQg−1

“G-core” of Q

Enveloping algebras


Dixmier’s Problem 11 aims for an analog of Clifford’s Thm for

R = U(g), the enveloping algebra of a finite-dim’l Lie algebra g

from J. Dixmier, Algebres enveloppantes (1974)

Enveloping algebras


solved!

• for chark = 0 by Mœglin & Rentschler, even for noetherian or

Goldie algebras ROrbites 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.

• for chark arbitrary and under weaker finiteness hypotheses by

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.

The Nullstellensatz


Want: an intrinsic characterization of “primitivity”

Classical example: R an affine commutative k-algebra, P ∈ SpecR

P is primitive ⇐⇒ P is maximal ⇐⇒ R/P = k

Hilbert’s “weak Nullstellensatz”(special case of Dimension Thm)

The Nullstellensatz


A “typical” noncommutative algebra R sats the following version of

the weak Nullstellensatz:

EndR(V ) = k for all V ∈ IrrRepR

The Nullstellensatz





Example: R any affine k-algebra, k uncountable Amitsur

The Nullstellensatz





. . . or even the Nullstellensatz: weak Nullstellensatz &

Jacobson property

semiprime ≡⋂

primitives

equivalently: the inclusion PrimR → SpecR is aquasi-homeomorphism

The Nullstellensatz





. . . or even the Nullstellensatz: weak Nullstellensatz &

Jacobson property

Examples: • R affine noetherian / uncountable k (Amitsur)

• R an affine PI-algebra (Kaplansky, Procesi)

• R = U(g) (Quillen, Duflo)

• R = kΓ with Γ polycyclic-by-finite (Hall, L., Goldie & Michler)

• Oq(kn), Oq(Mn(k)), . . .

Rational ideals


Want: a noncommutative generalization of Fract(R/P )

“coeur”

“Herz”

“heart”

This is provided by the extended center C(R/P ) = Z Qr(R/P ) . . .

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

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

Rational ideals


Qr(R) = lim−→

I∈E

Hom(IR, RR)

where E = {I E R | l. annR I = 0}, a filter of ideals of R.

• Elements are equivalence classes of right R-module maps

f : IR → RR (I ∈ E ) ,

with f ∼ f ′ : I ′R → RR if f = f ′ on some J ⊆ I ∩ I ′, J ∈ E .

• + and · come from addition and composition of maps.

• R → Qr(R) via r 7→ (x 7→ rx).

Rational ideals


Connection with irreducible representations:

Lemma(W.S. Martindale)

Given V ∈ IrrRepR, there is an embedding

C(R/ annR V ) → Z (EndR(V ))

Rational ideals






Consequently, if R sats the weak Nullstellensatz then

PrimR ⊆ RatR =def{P ∈ SpecR | C(R/P ) = k}

Rational ideals






. . . and if R sats the full Nullstellensatz then

{P ∈ SpecR | P is locally closed in SpecR} ⊆ PrimR ⊆ RatR

Rational ideals


In many of the aforementioned examples, it has been shown that

equality holds (under mild restrictions on k or the def. param. q)


locally closed = primitive = rational

topology

representation theory

geometry (Nullstellensatz)

Group actions

Notations and hypotheses


For the remainder of this talk,

k denotes an algebraically closed base field

R is an associative k-algebra

G is an affine algebraic k-group acting rationally on R;

equivalently, R is a k[G]-comodule algebra

Example: torus actions


G ∼= (k×)d an algebraic torus

• k[G] = kΛ, the group algebra of the “character lattice” Λ ∼= Zd

• kΛ-comodule algebras are the same as Λ-graded algebras

Example: torus actions


G ∼= (k×)d an algebraic torus

• k[G] = kΛ, the group algebra of the “character lattice” Λ ∼= Zd

• kΛ-comodule algebras are the same as Λ-graded algebras

Thus: a rational G-action on R is equivalent to a Zd-grading

R =⊕

λ∈Zd

Rλ , RλRλ′ ⊆ Rλ+λ′

G-prime and G-rational ideals


G-action on R G-actions on SpecR, PrimR, RatR

G\? will denote the orbit sets in question



G-action on R G-actions on SpecR, PrimR, RatR

G\? will denote the orbit sets in question

Definition The algebra R is called G-prime if R 6= 0 and the

product of any two nonzero G-stable (!) ideals of Ris nonzero.

A G-stable ideal I of R is called G-prime if R/I is

G-prime

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



Proposition(W. Chin)

(a) The assignment γ : P 7→ P :G =⋂

g∈G g.Pyields surjections

SpecR

can.��

γ // // G-SpecR

G\ SpecR

77 77♦♦♦♦♦♦♦♦♦♦♦

(b) If G is connected then all G-primes are in fact

prime; so

G-SpecR = {G-stable prime ideals of R}



Given I ∈ G-SpecR, the group G acts on C(R/I)and the invariants C(R/I)G are a k-field.

Definition: We call I G-rational if C(R/I)G = k and put

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

Noncommutative spectra


SpecR

can.

{{{{①①①①①①①①①①①①①①①①①①①①①

γ : P 7→P :G=⋂

g∈G g.P

"" ""❋❋❋

❋❋❋❋

❋❋❋❋

❋❋❋❋

❋❋❋❋

❋❋

RatR

①①①①①①①①①①①

||||①①①①①①①①①①

❊❊❊❊

❊❊❊❊

❊❊❊

"" ""❊❊❊

❊❊❊❊

❊❊❊

� ?

OO

G\ SpecR // // G-SpecR

G\RatR ∼=

Thm //� ?

OO

G-RatR� ?

OO

։ is a surjection whose target has the final topology,

→ is an inclusion whose source has the induced topology,∼= is a homeomorphism ( Dixmier’s Problem 11 for any R)

Noncommutative spectra


A sample geometric result:

Theorem: Let P ∈ RatR.

(a) {P} is loc. closed in SpecR iff {P : G} is loc. closed in

G-SpecR.

(b) In this case, the orbit G.P is open in its closure in RatR.

Pf of (b) from (a): Since {P :G} is loc. closed in G-SpecR, the fiber of

f : RatR → SpecRγ։ G-SpecR

over P :G is loc. closed in RatR. But f−1(P :G) = G.P by Thm.

The Goodearl-Letzter stratification


“strata” of SpecR = fibres of γ : SpecR։ G-SpecR

SpecR =⊔

I∈G-SpecR

SpecI R

=def

γ−1(I) = {P ∈ SpecR | P :G = I}

The Goodearl-Letzter stratification


“strata” of SpecR = fibres of γ : SpecR։ G-SpecR

SpecR =⊔

I∈G-SpecR

SpecI R

?

Finiteness of G-Spec R


Heuristic fact: For numerous algebras R, there is a natural

choice of G such that G-SpecR is finite — and

interesting!

?Find conditions on R and G that imply

finiteness of G-SpecR . . .

Finiteness of G-Spec R


Theorem: Assume that R sats the Nullstellensatz. Then the

following are equivalent:

(a) G-SpecR is finite;

(b) G\RatR is finite;

(c) R sats (1) ACC for G-stable semiprime ideals,

(2) the Dixmier-Mœglin equivalence, and

(3) G-RatR = G-SpecR.

If these conditions are satisfied then rational ideals of R are

exactly the primes that are maximal in their G-strata.

Recall: locally closed = primitive = rational

Examples

Torus actions


Recall: a rational action of the algebraic torus G = (k×)d on Ramounts to a Zd-grading

R =⊕

λ∈Zd


Torus actions


Recall: a rational action of the algebraic torus G = (k×)d on Ramounts to a Zd-grading

R =⊕

λ∈Zd


G-SpecR = {homogeneous primes of R}

Quantum n-space and quantum tori


Work of• . . .

• McConnell & Pettit (1988)

• De Concini, Kac & Procesi

• Brown & Goodearl

• Hodges

• Goodearl & Letzter (1998)



Quantum n-space is the algebra

R = Oq(kn) = k{x1, . . . , xn}/(xixj = qi,jxjxi | i < j)

for given parameters q = {qi,j} ⊆ k×; it has the degree grading:

Rλ = k xλ11 xλ2

2 · · · xλnn for λ = (λ1, . . . , λn) ∈ Zn






Rλ = k xλ11 xλ2

2 · · · xλnn for λ = (λ1, . . . , λn) ∈ Zn

This corresponds to a rational action of G = (k×)n:

(α1, . . . , αn).xi = αixi






Rλ = k xλ11 xλ2

2 · · · xλnn for λ = (λ1, . . . , λn) ∈ Zn

Easy: G-SpecR1-1←→ {subsets of [1..n]}

∈ ∈IS = 〈xi | i ∈ S〉 ←→ S






Rλ = k xλ11 xλ2

2 · · · xλnn for λ = (λ1, . . . , λn) ∈ Zn

Strata: SpecIS R1-1←→ SpecOqS

((k×)nS)

= k{x±1

i| i /∈ S}/(xixj = qi,jxjxi | i < j) a quantum torus



Quantum n-torus:

R = Oq((k×)n) = k{x±1

1 , . . . , x±1

n }/(xixj = qi,jxjxi | i < j)



Quantum n-torus:

R = Oq((k×)n) = k{x±1

1 , . . . , x±1


Always (!): SpecR1-1←→ SpecZ(R)

commutative!

But the nature of Z(R) depends very much on the choice of q!



Quantum n-torus:

R = Oq((k×)n) = k{x±1

1 , . . . , x±1


Always (!): SpecR1-1←→ SpecZ(R)

commutative!

But the nature of Z(R) depends very much on the choice of q!

Example: n = 2 • q a root of unity: Z(R) ∼= k[x±1, y±1]

• q not a root of unity: Z(R) = k

Quantum plane vs. ordinary affine plane


SpecOq(k2) (q 6= •√1): SpecO1(k2) = Spec k[x, y]:

Quantum n× n matrices


R = Oq(Mn(k)) = k

x1,1 . . . x1,n

......

xn,1 . . . xn,n

(q ∈ k×, q 6= •√1)



R = Oq(Mn(k)) = k

x1,1 . . . x1,n

a b...

...

c dxn,1 . . . xn,n

(q ∈ k×, q 6= •√1)

For each 2× 2-submatrix, there are relations:

ab = q ba ac = q ca bc = cb

bd = q db cd = q dc ad− da = (q − q−1)bc



R = Oq(Mn(k)) = k

x1,1 . . . x1,n

......

xn,1 . . . xn,n

(q ∈ k×, q 6= •√1)

The relations express the fact that quantum n× n-matrices act on

quantum n-space by matrix multiplication from both sides.

Explicitly: the following maps are k-algebra homomorphisms

Oq(kn) −→ Oq(k

n) ⊗ Oq(Mn(k)) xi 7→

∑

j

xj ⊗ xi,j

Oq(kn) −→ Oq(Mn(k)) ⊗ Oq(k

n) xi 7→

∑

j

xj,i ⊗ xj



R = Oq(Mn(k)) = k

x1,1 . . . x1,n

......

xn,1 . . . xn,n

(q ∈ k×, q 6= •√1)

Torus action: G = (k×)2n acts rationally by k-algebra auto’s on R,

with (α1, . . . , αn, β1, . . . , βn) ∈ G acting by

x1,1 . . . x1,n

......

xn,1 . . . xn,n

7−→ (α1, . . . , αn)

x1,1 . . . x1,n

......

xn,1 . . . xn,n

β1

...

βn



Theorem: R = Oq(Mn(C)) (q ∈ C× transcendental/Q)

(a) There is an (explicit) bijection between G-SpecR and a

certain collection of diagrams, called Cauchon diagrams

or -diagrams (“le”). Cauchon

(b) #G-SpecR =∑n

t=0(t!)2S(n+1, t+1)2, where the S(−,−)

are Stirling numbers of the 2nd kind.

Cauchon, Goodearl, Lenagan, McCammond

(c) There is an order isomorphism between (G-SpecR,⊆)and the following set of permutations

S ={σ ∈ S2n

∣∣ |σ(i)− i| ≤ n for all i = 1, . . . , 2n

}

w.r.t. the Bruhat order on S2n. Launois



Cauchon diagrams

These are n× n arrays of black and white boxes satisfying the

following requirement: if a box is colored black then all boxes on top

of it or all boxes to the left must be black as well.



“Pipe dreams”

n× n Cauchon

diagrams restricted permutations

σ ∈ S2n: |σ(i)− i| ≤ n



“Pipe dreams”

n× n Cauchon

diagrams restricted permutations

σ ∈ S2n: |σ(i)− i| ≤ n

1

2

3

4

5

6

7 8 9 10 11 12

1 2 3 4 5 6

7

8

9

10

11

12

σ = (3 7 6 5)(4 9)(11 12)

The case n = 2


The case n = 2


G-SpecOq(M2(k)):

⟨

0 0

0 0

⟩

⟨

0 0

c 0

⟩

〈Dq〉⟨

0 b0 0

⟩

⟨

a 0

c 0

⟩⟨

0 0

c d

⟩

⟨

0 bc 0

⟩ ⟨

a b0 0

⟩⟨

0 b0 d

⟩

⟨

a 0

c d

⟩

⟨

0 bc d

⟩ ⟨

a bc 0

⟩ ⟨

a b0 d

⟩

⟨

a bc d

⟩

Dq = ad − qbc the quantum determinant

The case n = 2


G-SpecOq(M2(k)):

⟨

0 0

0 0

⟩

⟨

0 0

c 0

⟩

〈Dq〉⟨

0 b0 0

⟩

⟨

a 0

c 0

⟩⟨

0 0

c d

⟩

⟨

0 bc 0

⟩ ⟨

a b0 0

⟩⟨

0 b0 d

⟩

⟨

a 0

c d

⟩

⟨

0 bc d

⟩ ⟨

a bc 0

⟩ ⟨

a b0 d

⟩

⟨

a bc d

⟩


2× 2 Cauchon diagrams:

The case n = 2


G-SpecOq(M2(k)):

⟨

0 0

0 0

⟩

⟨

0 0

c 0

⟩

〈Dq〉⟨

0 b0 0

⟩

⟨

a 0

c 0

⟩⟨

0 0

c d

⟩

⟨

0 bc 0

⟩ ⟨

a b0 0

⟩⟨

0 b0 d

⟩

⟨

a 0

c d

⟩

⟨

0 bc d

⟩ ⟨

a bc 0

⟩ ⟨

a b0 d

⟩

⟨

a bc d

⟩


restricted permutations ∈ S4:

1234

2134 1324 1243

2314 3124 2143 1423 1342

3214 2413 3142 1432

3412


Thank you!

prime ideals and group actions in noncommutative algebralorenz/talks/usc2013.pdf · 2013-02-20 ·...

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