Prime ideals and group actions in noncommutative algebra

Transcription

1 Prime ideals and group actions in noncommutative algebra Martin Lorenz Temple University, Philadelphia Colloquium USC 2/20/2013

2 Overview Prime ideals: historical background, first examples, Jacobson-Zariski topology...

3 Overview Prime ideals: Representations: historical background, first examples, Jacobson-Zariski topology... primitive ideals, Nullstellensatz, Dixmier-Mœglin equivalence

4 Overview Prime ideals: Representations: historical background, first examples, Jacobson-Zariski topology... primitive ideals, Nullstellensatz, Dixmier-Mœglin equivalence Groups actions: stratification, orbits, finiteness question

5 Overview Prime ideals: Representations: historical background, first examples, Jacobson-Zariski topology... primitive ideals, Nullstellensatz, Dixmier-Mœglin equivalence Groups actions: stratification, orbits, finiteness question Torus actions: some examples

6 Prime ideals

7 Prime ideals R = (R,+,,1) a ring Definition The ring R is called prime if R 0 and the product of any two nonzero ideals (!) of R is nonzero.

8 Prime ideals R = (R,+,,1) a ring Definition The ring R is called prime if R 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}

9 First examples (commutative) (1) 1-1 Spec Z {prime numbers} {0} from Mumford s Red Book (mid 1960s, reprinted as Springer Lect. Notes # 1358) Prime ideals and group actions in noncommutative algebra Colloquium USC 2/20/2013

10 First examples (commutative) (2) Spec k[x, y] 1-1 k 2 {monic irreducible polynomials} {0} k some algebraically closed field

11 First examples (commutative) (3) Spec Z[x] from Mumford s original mimeographed Harvard notes

12 First examples (commutative) (3) Spec Z[x]

13 Pioneers (number theory) Richard Dedekind ( ) Introduced ideals and prime ideals into number theory

14 Pioneers (number theory) Richard Dedekind ( ) Introduced ideals and prime ideals into number theory David Hilbert ( ) Introduced the term ring Zahlring, Ring oder Integritätsbereich ; Dedeking used Ordnung Reference Die Theorie der algebraischen Zahlkörper, Jahresbericht DMV (1897), ; see 31

15 Pioneers (noncommutative algebra) Emmy Noether ( ) 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 in the 1920s in Göttingen

16 Pioneers (noncommutative algebra) Wolfgang Krull ( ) First to investigate prime ideals in a noncommutative setting. References Zur Theorie der zweiseitigen Ideale in nichtkommutativen Bereichen, Math. Zeitschr. 28 (1928), Primidealketten in allgemeinen Ringbereichen, Sitzungsber. d. Heidelberger Akad. d. Wissensch. (1928), 3-14 as a student in Göttingen (1920)

17 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 Prim R for general R: special prime ideals (later) N. Jacobson, A topology for the set of primitive ideals in an arbitrary ring, Proc. Nat. Acad. Sci. USA 31 (1945),

18 The Jacobson-Zariski topology Definition Closed subsets of Spec R are those of the form V(I) = {P SpecR P I} where I R.

19 The Jacobson-Zariski topology Definition Closed subsets of Spec R 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}

20 The Jacobson-Zariski topology Definition Closed subsets of Spec R 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

21 The Jacobson-Zariski topology Definition Closed subsets of Spec R 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...

22 The Jacobson-Zariski topology e.g., Definition The (Krull) dimension of a top. space X is the supremum of the lengths l of all chains Y 0 Y 1 Y l with closed irreducible subsets Y i X.

23 The Jacobson-Zariski topology e.g., Definition The (Krull) dimension of a top. space X is the supremum of the lengths l of all chains Y 0 Y 1 Y l with closed irreducible subsets Y i X. Dimension Thm (classical) If R is an affine commutative k-algebra then dim SpecR = max P tr.deg k Fract(R/P) where P runs over the minimal primes of R.

24 Affine algebraic varieties plane k[x, y] algebra-geometry dictionary R := k[x 2,y 2,xy] = k[r,s,t]/(rs t 2 ) cone

25 Representations

26 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...

27 Representations and primitive ideals Definition A (linear) representation of R is an algebra homomorphism ρ: R End k (V), r r V, where V is a k-vector space. The representation is called irreducible if 0 and V are the only two subspaces ofv that are stable under all operators r V. In this case, Kerρ = {r R r V = 0 V } SpecR; such primes are called primitive.

28 Representations and primitive ideals Goal: For a given algebra R, describe IrrRepR = {irreducible representations of R}/ =

29 Representations and primitive ideals Goal: For a given algebra R, describe IrrRepR = {irreducible representations of R}/ = Unfortunately, this is generally too hard; so...

30 Representations and primitive ideals Modified goal: For a given algebra R, describe Prim R = {primitive ideals of R} Spec R Recall: kernels of (generally infinite-dimensional) irreducible rep s R End k (V )

31 Representations and primitive ideals Modified goal: For a given algebra R, describe Prim R = {primitive ideals of R} Spec R This will at least give a coarse classification of IrrRepR

32 Some examples (1) Finite-dimensional R: SpecR =PrimR 1-1 IrrRep R = a finite set Ker ρ ρ

33 Some examples (1) Finite-dimensional R: SpecR =PrimR 1-1 IrrRep R = a finite set Ker ρ ρ (2) The polynomial algebra R = k[x 1,...,x n ]: MaxR = PrimR 1-1 IrrRepR }{{} 1-1 n-space k n for any commutative R

34 Some examples (3) The Weyl algebra R = k{x,y}/(yx = xy +1) with chark = 0: SpecR = PrimR = {0} but #IrrRepR =

35 Some examples (3) The Weyl algebra R = k{x,y}/(yx = xy +1) with chark = 0: SpecR = PrimR = {0} but #IrrRepR = Moreover, all rep s R End k (V) are infinite-dimensional:

36 Some examples (3) The Weyl algebra R = k{x,y}/(yx = xy +1) with chark = 0: SpecR = PrimR = {0} but #IrrRepR = Moreover, all rep s R End k (V) are infinite-dimensional: dim k V = n < y V x V = x V y V +1 V trace(y V x V ) = trace(x V y V )+n 1 k 0 = n 1 k a contradiction!

37 Enveloping algebras Jacques Dixmier (* 1924) in Reims, Dec. 2008

38 Enveloping algebras Recall: for R = kg, the group algebra of a finite group G, one has SpecR 1-1 IrrRepR

39 Enveloping algebras Recall: for R = kg, the group algebra of a finite group G, one has SpecR 1-1 IrrRepR Clifford s Thm Given P SpeckG and N G, there is a Q SpeckN, unique up to G-conjugacy, with P kn = Q:G = gqg 1 def g G G-core of Q

40 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, Algèbres enveloppantes (1974)

41 Enveloping algebras solved! for chark = 0 by Mœglin & Rentschler, even for noetherian or Goldie algebras R Orbites d un groupe algébrique dans l espace des idéaux rationnels d une algèbre enveloppante, Bull. Soc. Math. France 109 (1981), Sur la classification des idéaux primitifs des algèbres enveloppantes, Bull. Soc. Math. France 112 (1984), Sous-corps commutatifs ad-stables des anneaux de fractions des quotients des algèbres enveloppantes; espaces homogènes et induction de Mackey, J. Funct. Anal. 69 (1986), Idéaux G-rationnels, rang de Goldie, preprint, 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), Actions of algebraic groups on the spectrum of rational ideals. II, J. Algebra 208 (1998),

42 The Nullstellensatz Want: an intrinsic characterization of primitivity Classical example: R an affine commutative k-algebra, P Spec R P is primitive P is maximal R/P = k Hilbert s weak Nullstellensatz (special case of Dimension Thm)

43 The Nullstellensatz A typical noncommutative algebra R sat s the following version of the weak Nullstellensatz: End R (V) = k for all V IrrRepR

44 The Nullstellensatz A typical noncommutative algebra R sat s the following version of the weak Nullstellensatz: End R (V) = k for all V IrrRepR Example: R any affine k-algebra, k uncountable Amitsur

45 The Nullstellensatz A typical noncommutative algebra R sat s the following version of the weak Nullstellensatz: End R (V) = k for all V IrrRepR... or even the Nullstellensatz: weak Nullstellensatz & Jacobson property semiprime primitives equivalently: the inclusion PrimR SpecR is a quasi-homeomorphism

46 The Nullstellensatz A typical noncommutative algebra R sat s the following version of the weak Nullstellensatz: End R (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) O q (k n ), O q (M n (k)),...

47 Rational ideals Want: a noncommutative generalization of Fract(R/P) heart This is provided by the extended center C(R/P) = ZQ r (R/P)... coeur Herz References: W. S. Martindale, III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), S. A. Amitsur, On rings of quotients, Symposia Math., Vol. VIII, Academic Press, London, 1972, pp

48 Rational ideals Q r (R) = lim I E Hom(I R,R R ) where E = {I R l.ann R I = 0}, a filter of ideals of R. Elements are equivalence classes of right R-module maps f: I R R R (I E), with f f : I R R R if f = f on some J I I, J E. + and come from addition and composition of maps. R Q r (R) via r (x rx).

49 Rational ideals Connection with irreducible representations: Lemma (W.S. Martindale) Given V IrrRepR, there is an embedding C(R/ann R V) Z(End R (V))

50 Rational ideals Connection with irreducible representations: Lemma (W.S. Martindale) Given V IrrRepR, there is an embedding C(R/ann R V) Z(End R (V)) Consequently, if R sat s the weak Nullstellensatz then PrimR RatR = def {P SpecR C(R/P) = k}

51 Rational ideals Connection with irreducible representations: Lemma (W.S. Martindale) Given V IrrRepR, there is an embedding C(R/ann R V) Z(End R (V))... and if R sat s the full Nullstellensatz then {P SpecR P is locally closed in SpecR} PrimR RatR

52 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) Dixmier-Mœglin equivalence locally closed = primitive = rational topology representation theory geometry (Nullstellensatz)

53 Group actions

54 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

55 Example: torus actions G = (k ) d an algebraic torus k[g] = kλ, the group algebra of the character lattice Λ = Z d kλ-comodule algebras are the same as Λ-graded algebras

56 Example: torus actions G = (k ) d an algebraic torus k[g] = kλ, the group algebra of the character lattice Λ = Z d kλ-comodule algebras are the same as Λ-graded algebras Thus: a rational G-action on R is equivalent to a Z d -grading R = λ Z d R λ, R λ R λ R λ+λ

57 G-prime and G-rational ideals G-action on R G-actions on SpecR, PrimR, RatR G\? will denote the orbit sets in question

58 G-prime and G-rational ideals 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 0 and the product of any two nonzero G-stable (!) ideals of R is 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}

59 G-prime and G-rational ideals Proposition (W. Chin) (a) The assignment γ: P P:G = g G g.p yields surjections SpecR can. G\SpecR γ G-SpecR (b) IfGis connected then allg-primes are in fact prime; so G-SpecR = {G-stable prime ideals of R}

60 G-prime and G-rational ideals GivenI G-SpecR, the groupgacts onc(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}

61 Noncommutative spectra G\SpecR can SpecR RatR Thm G\RatR = γ: P P:G= g G g.p G-SpecR G-Rat R 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)

62 Noncommutative spectra A sample geometric result: Theorem: Let P Rat R. (a) {P} is loc. closed in SpecR iff {P : G} is loc. closed in G-Spec R. (b) In this case, the orbit G.P is open in its closure in Rat R. Pf of (b) from (a): Since {P :G} is loc. closed in G-SpecR, the fiber of f : RatR SpecR γ G-SpecR over P :G is loc. closed in RatR. But f 1 (P :G) = G.P by Thm.

63 The Goodearl-Letzter stratification strata of Spec R = fibres of γ: Spec R G-Spec R SpecR = I G-Spec R Spec I R = def γ 1 (I) = {P SpecR P:G = I}

64 The Goodearl-Letzter stratification strata of Spec R = fibres of γ: Spec R G-Spec R SpecR = I G-Spec R Spec I R?

65 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-Spec R...

66 Finiteness of G-Spec R Theorem: Assume that R sat s the Nullstellensatz. Then the following are equivalent: (a) (b) G-Spec R is finite; G\ Rat R is finite; (c) R sat s (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

67 Examples

68 Torus actions Recall: a rational action of the algebraic torus G = (k ) d on R amounts to a Z d -grading R = λ Z d R λ, R λ R λ R λ+λ

69 Torus actions Recall: a rational action of the algebraic torus G = (k ) d on R amounts to a Z d -grading R = λ Z d R λ, R λ R λ R λ+λ G-SpecR = {homogeneous primes of R}

70 Quantum n-space and quantum tori Work of... McConnell & Pettit (1988) De Concini, Kac & Procesi Brown & Goodearl Hodges Goodearl & Letzter (1998)

71 Quantum n-space and quantum tori Quantum n-space is the algebra R = O q (k n ) = k{x 1,...,x n }/(x i x j = q i,j x j x i i < j) for given parameters q = {q i,j } k ; it has the degree grading: R λ = kx λ 1 1 x λ 2 2 x λ n n for λ = (λ 1,...,λ n ) Z n

72 Quantum n-space and quantum tori Quantum n-space is the algebra R = O q (k n ) = k{x 1,...,x n }/(x i x j = q i,j x j x i i < j) for given parameters q = {q i,j } k ; it has the degree grading: R λ = kx λ 1 1 x λ 2 2 x λ n n for λ = (λ 1,...,λ n ) Z n This corresponds to a rational action of G = (k ) n : (α 1,...,α n ).x i = α i x i

73 Quantum n-space and quantum tori Quantum n-space is the algebra R = O q (k n ) = k{x 1,...,x n }/(x i x j = q i,j x j x i i < j) for given parameters q = {q i,j } k ; it has the degree grading: R λ = kx λ 1 1 x λ 2 2 x λ n n for λ = (λ 1,...,λ n ) Z n Easy: G-Spec R 1-1 {subsets of [1..n]} I S = x i i S S

74 Quantum n-space and quantum tori Quantum n-space is the algebra R = O q (k n ) = k{x 1,...,x n }/(x i x j = q i,j x j x i i < j) for given parameters q = {q i,j } k ; it has the degree grading: R λ = kx λ 1 1 x λ 2 2 x λ n n for λ = (λ 1,...,λ n ) Z n Strata: Spec IS R 1-1 SpecO qs ((k ) n S ) = k{x ±1 i i / S}/(x i x j = q i,j x j x i i < j) a quantum torus

75 Quantum n-space and quantum tori Quantum n-torus: R = O q ((k ) n ) = k{x ±1 1,...,x ±1 n }/(x i x j = q i,j x j x i i < j)

76 Quantum n-space and quantum tori Quantum n-torus: R = O q ((k ) n ) = k{x ±1 1,...,x ±1 n }/(x i x j = q i,j x j x i i < j) Always (!): SpecR 1-1 SpecZ(R) commutative! But the nature of Z(R) depends very much on the choice of q!

77 Quantum n-space and quantum tori Quantum n-torus: R = O q ((k ) n ) = k{x ±1 1,...,x ±1 n }/(x i x j = q i,j x j x i i < j) Always (!): SpecR 1-1 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

78 Quantum plane vs. ordinary affine plane SpecO q (k 2 ) (q 1): SpecO 1 (k 2 ) = Speck[x,y]:

79 Quantum n n matrices R = O q (M n (k)) = k x 1,1... x 1,n.. x n,1... x n,n (q k,q 1)

80 Quantum n n matrices R = O q (M n (k)) = k x 1,1... x 1,n a b.. c d x n,1... x n,n (q k,q 1) For each 2 2-submatrix, there are relations: ab = qba ac = qca bc = cb bd = qdb cd = qdc ad da = (q q 1 )bc

81 Quantum n n matrices R = O q (M n (k)) = k x 1,1... x 1,n.. x n,1... x n,n (q k,q 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 O q (k n ) O q (k n ) O q (M n (k)) x i j x j x i,j O q (k n ) O q (M n (k)) O q (k n ) x i j x j,i x j

82 Quantum n n matrices R = O q (M n (k)) = k x 1,1... x 1,n.. x n,1... x n,n (q k,q 1) Torus action: G = (k ) 2n acts rationally by k-algebra auto s on R, with (α 1,...,α n,β 1,...,β n ) G acting by x 1,1... x 1,n x 1,1... x 1,n β 1.. (α 1,...,α n )... x n,1... x n,n x n,1... x n,n β n

83 Quantum n n matrices Theorem: (a) (b) R = O q (M n (C)) (q C transcendental/q) There is an (explicit) bijection between G-Spec R and a certain collection of diagrams, called Cauchon diagrams or -diagrams ( le ). Cauchon #G-SpecR = n t=0 (t!)2 S(n+1,t+1) 2, where thes(, ) are Stirling numbers of the 2 nd kind. Cauchon, Goodearl, Lenagan, McCammond (c) There is an order isomorphism between (G-Spec R, ) and the following set of permutations S = { σ S 2n σ(i) i n for all i = 1,...,2n } w.r.t. the Bruhat order on S 2n. Launois

84 Quantum n n matrices 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.

85 Quantum n n matrices Pipe dreams n n Cauchon diagrams restricted permutations σ S 2n : σ(i) i n

86 Quantum n n matrices Pipe dreams n n Cauchon diagrams restricted permutations σ S 2n : σ(i) i n

87 Quantum n n matrices Pipe dreams n n Cauchon diagrams restricted permutations σ S 2n : σ(i) i n σ = (3765)(49)(1112)

88 The case n = 2

89 The case n = 2

90 The case n = 2 G-SpecO q (M 2 (k)): a b c d a 0 0 b a b a b c d c d c 0 0 d 0 0 a 0 0 b 0 b a b c d c 0 c 0 0 d c 0 D q 0 b D q = ad qbc the quantum determinant

91 The case n = 2 G-SpecO q (M 2 (k)): 2 2 Cauchon diagrams: a b c d a 0 0 b a b a b c d c d c 0 0 d 0 0 a 0 0 b 0 b a b c d c 0 c 0 0 d c 0 D q 0 b D q = ad qbc the quantum determinant

92 The case n = 2 G-SpecO q (M 2 (k)): restricted permutations S 4 : a b c d a 0 0 b a b a b c d c d c 0 0 d 0 0 a 0 0 b 0 b a b c d c 0 c 0 0 d c 0 D q 0 b D q = ad qbc the quantum determinant

93 Thank you!