Prime ideals and group actions
in noncommutative algebra
Martin Lorenz
Temple University, Philadelphia
Colloquium USC 2/20/2013
Overview
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
• Prime ideals: historical background, first examples,
Jacobson-Zariski topology . . .
Overview
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
• Prime ideals: historical background, first examples,
Jacobson-Zariski topology . . .
• Representations: primitive ideals, Nullstellensatz,
Dixmier-Mœglin equivalence
Overview
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
• Prime ideals: historical background, first examples,
Jacobson-Zariski topology . . .
• Representations: primitive ideals, Nullstellensatz,
Dixmier-Mœglin equivalence
• Groups actions: stratification, orbits, finiteness question
Overview
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
• Prime ideals: historical background, first examples,
Jacobson-Zariski topology . . .
• Representations: primitive ideals, Nullstellensatz,
Dixmier-Mœglin equivalence
• Groups actions: stratification, orbits, finiteness question
• Torus actions: some examples
Prime ideals
Prime ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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)
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(1) SpecZ 1-1←→ {prime numbers} ∪ {0}
from Mumford’s “Red Book” (mid 1960s, reprinted as Springer Lect. Notes # 1358)
First examples (commutative)
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(2) Spec k[x, y] 1-1←→ k2∪{monic irreducible polynomials}∪{0}k some algebraically closed field
First examples (commutative)
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(3) SpecZ[x]
from Mumford’s original mimeographed Harvard notes
First examples (commutative)
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(3) SpecZ[x]
Pioneers (number theory)
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Richard Dedekind (1831 – 1916)
Introduced “ideals” and “prime ideals”
into number theory
Pioneers (number theory)
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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)
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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)
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
The Jacobson-Zariski topology
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Definition Closed subsets of SpecR are those of the form
V(I) = {P ∈ SpecR | P ⊇ I} where I ⊆ R .
The Jacobson-Zariski topology
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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}
The Jacobson-Zariski topology
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Definition Closed subsets of SpecR are those of the form
V(I) = {P ∈ SpecR | P ⊇ I} where I ⊆ R .
Bad separation properties !
R prime ⇒ SpecR is irreducible: all nonempty
open subsets are dense
The Jacobson-Zariski topology
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Definition Closed subsets of SpecR are those of the form
V(I) = {P ∈ SpecR | P ⊇ I} where I ⊆ R .
Bad separation properties !
R prime ⇒ SpecR is irreducible: all nonempty
open subsets are dense
But topological notions become available . . .
The Jacobson-Zariski topology
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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.
The Jacobson-Zariski topology
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
R := k[x2, y2, xy] ∼= k[r, s, t]/(rs− t2)
cone
k[x, y]
plane
algebra-geometry dictionary
Representations
Representations and primitive ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
• . . .
Representations and primitive ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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.
Representations and primitive ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Goal: For a given algebra R, describe
IrrRepR = {irreducible representations of R}/ ∼=
Representations and primitive ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Goal: For a given algebra R, describe
IrrRepR = {irreducible representations of R}/ ∼=
Unfortunately, this is generally too hard; so . . .
Representations and primitive ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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 )
Representations and primitive ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(1) Finite-dimensional R:
SpecR =PrimR1-1←→ IrrRepR = a finite set
∈ ∈
Ker ρ ←→ ρ
Some examples
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(3) The Weyl algebra R = k{x, y}/(yx = xy + 1) with chark = 0:
SpecR = PrimR = {0} but #IrrRepR =∞
Some examples
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(3) The Weyl algebra R = k{x, y}/(yx = xy + 1) with chark = 0:
SpecR = PrimR = {0} but #IrrRepR =∞
Moreover, all reps R→ Endk(V ) are infinite-dimensional:
Some examples
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
(3) The Weyl algebra R = k{x, y}/(yx = xy + 1) with chark = 0:
SpecR = PrimR = {0} but #IrrRepR =∞
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Jacques Dixmier (* 1924)
in Reims, Dec. 2008
Enveloping algebras
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Recall: for R = kG, the group algebra of a finite group G, one has
SpecR1-1←→ IrrRepR
Enveloping algebras
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
A “typical” noncommutative algebra R sats the following version of
the weak Nullstellensatz:
EndR(V ) = k for all V ∈ IrrRepR
The Nullstellensatz
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
A “typical” noncommutative algebra R sats the following version of
the weak Nullstellensatz:
EndR(V ) = k for all V ∈ IrrRepR
Example: R any affine k-algebra, k uncountable Amitsur
The Nullstellensatz
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
A “typical” noncommutative algebra R sats the following version of
the weak Nullstellensatz:
EndR(V ) = k for all V ∈ IrrRepR
. . . or even the Nullstellensatz: weak Nullstellensatz &
Jacobson property
semiprime ≡⋂
primitives
equivalently: the inclusion PrimR → SpecR is aquasi-homeomorphism
The Nullstellensatz
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
A “typical” noncommutative algebra R sats the following version of
the weak Nullstellensatz:
EndR(V ) = k for all V ∈ IrrRepR
. . . 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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Connection with irreducible representations:
Lemma(W.S. Martindale)
Given V ∈ IrrRepR, there is an embedding
C(R/ annR V ) → Z (EndR(V ))
Rational ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Connection with irreducible representations:
Lemma(W.S. Martindale)
Given V ∈ IrrRepR, there is an embedding
C(R/ annR V ) → Z (EndR(V ))
Consequently, if R sats the weak Nullstellensatz then
PrimR ⊆ RatR =def{P ∈ SpecR | C(R/P ) = k}
Rational ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Connection with irreducible representations:
Lemma(W.S. Martindale)
Given V ∈ IrrRepR, there is an embedding
C(R/ annR V ) → Z (EndR(V ))
. . . and if R sats the full Nullstellensatz then
{P ∈ SpecR | P is locally closed in SpecR} ⊆ PrimR ⊆ RatR
Rational ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
In many of the aforementioned examples, it has been shown that
equality holds (under mild restrictions on k or the def. param. q)
Dixmier-Mœglin equivalence
locally closed = primitive = rational
topology
representation theory
geometry (Nullstellensatz)
Group actions
Notations and hypotheses
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
G-action on R G-actions on SpecR, PrimR, RatR
G\? will denote the orbit sets in question
G-prime and G-rational ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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}
G-prime and G-rational ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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}
G-prime and G-rational ideals
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
“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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
“strata” of SpecR = fibres of γ : SpecR։ G-SpecR
SpecR =⊔
I∈G-SpecR
SpecI R
?
Finiteness of G-Spec R
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Recall: a rational action of the algebraic torus G = (k×)d on Ramounts to a Zd-grading
R =⊕
λ∈Zd
Rλ , RλRλ′ ⊆ Rλ+λ′
Torus actions
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Recall: a rational action of the algebraic torus G = (k×)d on Ramounts to a Zd-grading
R =⊕
λ∈Zd
Rλ , RλRλ′ ⊆ Rλ+λ′
G-SpecR = {homogeneous primes of R}
Quantum n-space and quantum tori
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Work of• . . .
• McConnell & Pettit (1988)
• De Concini, Kac & Procesi
• Brown & Goodearl
• Hodges
• Goodearl & Letzter (1998)
Quantum n-space and quantum tori
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Quantum n-space and quantum tori
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
This corresponds to a rational action of G = (k×)n:
(α1, . . . , αn).xi = αixi
Quantum n-space and quantum tori
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Easy: G-SpecR1-1←→ {subsets of [1..n]}
∈ ∈IS = 〈xi | i ∈ S〉 ←→ S
Quantum n-space and quantum tori
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Strata: SpecIS R1-1←→ SpecOqS
((k×)nS)
= k{x±1
i| i /∈ S}/(xixj = qi,jxjxi | i < j) a quantum torus
Quantum n-space and quantum tori
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Quantum n-torus:
R = Oq((k×)n) = k{x±1
1 , . . . , x±1
n }/(xixj = qi,jxjxi | i < j)
Quantum n-space and quantum tori
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Quantum n-torus:
R = Oq((k×)n) = k{x±1
1 , . . . , x±1
n }/(xixj = qi,jxjxi | i < j)
Always (!): SpecR1-1←→ SpecZ(R)
commutative!
But the nature of Z(R) depends very much on the choice of q!
Quantum n-space and quantum tori
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Quantum n-torus:
R = Oq((k×)n) = k{x±1
1 , . . . , x±1
n }/(xixj = qi,jxjxi | i < j)
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
SpecOq(k2) (q 6= •√1): SpecO1(k2) = Spec k[x, y]:
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
R = Oq(Mn(k)) = k
x1,1 . . . x1,n
......
xn,1 . . . xn,n
(q ∈ k×, q 6= •√1)
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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.
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
“Pipe dreams”
n× n Cauchon
diagrams restricted permutations
σ ∈ S2n: |σ(i)− i| ≤ n
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
“Pipe dreams”
n× n Cauchon
diagrams restricted permutations
σ ∈ S2n: |σ(i)− i| ≤ n
Quantum n× n matrices
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
“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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
The case n = 2
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
The case n = 2
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
2× 2 Cauchon diagrams:
The case n = 2
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
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
restricted permutations ∈ S4:
1234
2134 1324 1243
2314 3124 2143 1423 1342
3214 2413 3142 1432
3412
Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013
Thank you!