Algebraic group actions on noncommutative spectra

Transcription

1 GoBack

2 Algebraic group actions on noncommutative spectra Special Session on Brauer Groups, Quadratic Forms, Algebraic Groups, and Lie Algebras NCSU 04/04/2009 Martin Lorenz Temple University, Philadelphia

3 Overview Background: enveloping algebras and quantized coordinate algebras

4 Overview Background: enveloping algebras and quantized coordinate algebras Tool: the Amitsur-Martindale ring of quotients

5 Overview Background: enveloping algebras and quantized coordinate algebras Tool: the Amitsur-Martindale ring of quotients Rational and primitive ideals

6 Overview Background: enveloping algebras and quantized coordinate algebras Tool: the Amitsur-Martindale ring of quotients Rational and primitive ideals Stratification of the prime spectrum (if time)

7 References Group actions and rational ideals, Algebra and Number Theory 2 (2008), Algebraic group actions on noncommutative spectra, Transformation Groups (to appear) Both articles & the pdf file of this talk available on my web page: lorenz/

8 I will work / base field k = k

9 Background

10 Enveloping algebras Goal: For R = U(g), the enveloping algebra of a finite-dim l Lie algebra g, describe Prim R = {primitive ideals of R} kernels of irreducible (generally infinite-dimensional) representations R End k (V )

11 Enveloping algebras Jacques Dixmier (* 1924) former secretary of Bourbaki Ph.D. advisor of A. Connes, M. Duflo,... author of several influential monographs: in Reims, Dec Les algèbres d opérateurs dans l espace hilbertien: algèbres de von Neumann, Gauthier-Villars, 1957 Les C -algèbres et leurs représentations, Gauthier-Villars, 1969 Algèbres enveloppantes, Gauthier-Villars, 1974

12 Dixmier s Problem 11 from Algèbres enveloppantes, 1974 :

13 Dixmier s Problem 11 Problem 11 for k solvable, char k =0 Dixmier, Sur les idéaux génériques dans les algèbres enveloppantes, Bull. Sci. Math. (2) 96 (1972), existence: (a) Borho, Gabriel, Rentschler, Primideale in Einhüllenden auflösbarer Lie-Algebren, Springer Lect. Notes in Math. 357 (1973). uniqueness: (b) for noetherian or Goldie rings R / char k =0: Mœglin & Rentschler 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, 1986.

14 Dixmier s Problem 11 Problem 11 for k solvable, char k =0 Dixmier, Sur les idéaux génériques dans les algèbres enveloppantes, Bull. Sci. Math. (2) 96 (1972), existence: (a) Borho, Gabriel, Rentschler, Primideale in Einhüllenden auflösbarer Lie-Algebren, Springer Lect. Notes in Math. 357 (1973). uniqueness: (b) for noetherian or Goldie rings R / char k =0: Mœglin & Rentschler 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, 1986.

15 Dixmier s Problem 11 under weaker Goldie hypotheses / char k arbitrary: 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),

16 Quantum groups Goal: ForR = O q (k n ), O q (M n ), O q (G)...aquantized coordinate ring, describe Spec R = {prime ideals of R} Prim R

17 Quantum groups Goal: ForR = O q (k n ), O q (M n ), O q (G)...aquantized coordinate ring, describe Spec R = {prime ideals of R} Prim R Typically, some algebraic torus T acts rationally by k-algebra automorphisms on R; so have Spec R Spec T R = {T -stable primes of R} P P : T = g T g.p

18 Quantum groups T -stratification of Spec R ( Goodearl & Letzter,...;seethemonograph by Brown & Goodearl ) Spec R = I Spec T R Spec I R {P Spec R P : T = I}

19 Quantum groups T -stratification of Spec R ( Goodearl & Letzter,...;seethemonograph by Brown & Goodearl ) Spec R = I Spec T R Spec I R?

20 Notation For the remainder of this talk, R denotes an associative k-algebra (with 1) G is an affine algebraic k-group acting rationally on R; so R is a k[g]-comodule algebra. Equivalently, we have a rational representation ρ = ρ R : G Aut k-alg (R) GL(R)

21 Tool: The Amitsur-Martindale ring of quotients

22 Original references for prime rings R: W. S. Martindale, III, Prime rings satisfying a generalized polynomial identity, J. Algebra 12 (1969), for general R: S. A. Amitsur, On rings of quotients, Symposia Math., Vol. VIII, Academic Press, London, 1972, pp

23 The definition In brief, Q r (R) = lim I E Hom(I R,R R ) where E = E (R) is the filter of all I R such that l. ann R I =0.

24 The definition In brief, Q r (R) = lim I E Hom(I R,R R ) where E = E (R) is the filter of all I R such that l. ann R I =0. Explicitly, elements of Q r (R) are equivalence classes of right R-module maps f : I R R R (I E ), the map f being equivalent to f : I R R R (I E ) if f = f on some J I I, J E.

25 The definition In brief, Q r (R) = lim I E Hom(I R,R R ) where E = E (R) is the filter of all I R such that l. ann R I =0. Addition and multiplication of Q r (R) come from addition and composition of R-module maps.

26 The definition In brief, Q r (R) = lim I E Hom(I R,R R ) where E = E (R) is the filter of all I R such that l. ann R I =0. Sending r R to the equivalence class of λ r : R R, x rx, yields an embedding of R as a subring of Q r (R).

27 Extended centroid Def n : The extended centroid of R is defined by C(R) =Z Q r (R) center Fact: If R is prime then C(R) is a k-field.

28 Examples If R is simple, or a finite product of simple rings, then Q r (R) =R

29 Examples If R is simple, or a finite product of simple rings, then Q r (R) =R For R semiprime right Goldie, Q r (R) ={q Q cl (R) qi R for some I R with ann R I =0} In particular, classical quotient ring of R C(R) =ZQ cl (R)

30 Examples R = U(g)/I a semiprime image of the enveloping algebra of a finite-dimensional Lie algebra g. Then Q r (R) ={ ad g-finite elements of Q cl (R) }

31 Rational Ideals

32 Definition Want: an intrinsic characterization of primitivity, ideally in detail...

33 Definition coeur Herz heart core Definition: Recall that C(R/P) is a k-field for any P Spec R. We call P rational if C(R/P) =k.

34 Definition coeur Herz heart core Definition: Recall that C(R/P) is a k-field for any P Spec R. We call P rational if C(R/P) =k. Put Rat R = {P Spec R P is rational }; so Rat R Spec R

35 Connection with irreducible representations Given an irreducible representation f : R End k (V ), let P =Kerf be the corresponding primitive ideal of R.

36 Connection with irreducible representations Given an irreducible representation f : R End k (V ), let P =Kerf be the corresponding primitive ideal of R. There always is an embedding of k-fields C(R/P) Z(End R (V ))

37 Connection with irreducible representations Given an irreducible representation f : R End k (V ), let P =Kerf be the corresponding primitive ideal of R. There always is an embedding of k-fields C(R/P) Z(End R (V )) Typically, End R (V )=k ( weak Nullstellensatz ); in this case Prim R Rat R

38 Examples The weak Nullstellensatz holds for R any affine k-algebra, k uncountable R an affine PI-algebra R = U(g) Amitsur Kaplansky Quillen s Lemma R = kγ with Γ polycyclic-by-finite Hall, L. many quantum groups: O q (k n ), O q (M n (k)), O q (G),...

39 Examples The weak Nullstellensatz holds for R any affine k-algebra, k uncountable R an affine PI-algebra R = U(g) Amitsur Kaplansky Quillen s Lemma R = kγ with Γ polycyclic-by-finite Hall, L. many quantum groups: O q (k n ), O q (M n (k)), O q (G),... In fact, in all these examples except the first, it has been shown that, under mild restrictions on k or q, Prim R =RatR

40 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 if A, B G-stab R, AB I A I or B I. Put G-Spec R = {G-prime ideals of R}

41 Group action: G-prime and G-rational ideals Prop n The assignment γ : P P : G = g.p yields g G surjections Spec R can. G\ Spec R γ G-Spec R

42 Group action: G-prime and G-rational ideals Definition: Let I G-Spec R. The group G acts on C(R/I) and the invariants C(R/I) G are a k-field. We call I G-rational if C(R/I) G = k. Put G-Rat R = {G-rational ideals of R}

43 Group action: G-prime and G-rational ideals Definition: Let I G-Spec R. The group G acts on C(R/I) and the invariants C(R/I) G are a k-field. We call I G-rational if C(R/I) G = k. Put G-Rat R = {G-rational ideals of R} The following result solves Dixmier s Problem # 11 (a),(b) for arbitrary algebras. Theorem 1 G\ Rat R bij. G-Rat R G.P g G g.p

44 Noncommutative spectra can. Spec R Rat R γ : P P :G= g G g.p G\ Spec R G-Spec R G\ Rat R = G-Rat R

45 Noncommutative spectra can. Spec R Rat R γ : P P :G= g G g.p G\ Spec R G-Spec R G\ Rat R = G-Rat R Spec R carries the Jacobson-Zariski topology: closed subsets are those of the form V(I) ={P Spec R P I} where I R.

46 Noncommutative spectra can. Spec R Rat R γ : P P :G= g G g.p G\ Spec R G-Spec R G\ Rat R = G-Rat R 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

47 Local closedness Recall: locally closed = open closed

48 Local closedness Theorem 2 If P Rat R 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.P is open in its closure in Rat R.

49 Local closedness Theorem 2 If P Rat R 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.P is open in its closure in Rat R. Proof of Cor: P:G G-Spec R is locally closed by Theorem 2, and hence so is its preimage under f : RatR Spec R γ G-Spec R. Finally, f 1 (P:G) =G.P by Theorem 1.

50 Stratification of the prime spectrum

51 Goal can. Spec R Rat R γ : P P :G= g G g.p G\ Spec R G-Spec R G\ Rat R = G-Rat R Next, we turn to the map γ...

52 Goal Recall: γ : SpecR G-Spec R yields the G-stratification of Spec R Spec R = Spec I R I G-Spec R Main goal: describe the G-strata Spec I R = γ 1 (I) ={P Spec R P:G = I}

53 Goal For simplicity, I assume G to be connected; sok[g] is a domain. In particular, G-Spec R =Spec G R = {G-stable primes of R}

54 The rings T I For a given I G-Spec R, put Fract k[g] T I = C(R/I) k(g) This is a commutative domain, a tensor product of two fields.

55 The rings T I For a given I G-Spec R, put Fract k[g] T I = C(R/I) k(g) This is a commutative domain, a tensor product of two fields. G-actions: on C(R/I) via the given action ρ: G Aut k-alg (R) on k(g) by the right and left regular actions ρ r :(x.f)(y) =f(yx) and ρ l :(x.f)(y) =f(x 1 y) on T I by ρ ρ r and Id ρ l commute

56 The rings T I For a given I G-Spec R, put Fract k[g] T I = C(R/I) k(g) This is a commutative domain, a tensor product of two fields. Put Spec G T I = {(ρ ρ r )(G)-stable primes of T I }

57 Stratification Theorem Theorem 3 There is a bijection c: Spec I R Spec G T I having the following properties: (a) G-equivariance: c(g.p) =(Id ρ l )(g)(c(p )); (b) inclusions: P P c(p ) c(p ); (c) hearts: C (T I /c(p )) = C (R/P k(g)) as k(g)-fields; (d) rationality: P is rational T I /c(p )=k(g).

58 Stratification Theorem Theorem 3 There is a bijection c: Spec I R Spec G T I having the following properties: (a) G-equivariance: c(g.p) =(Id ρ l )(g)(c(p )); (b) inclusions: P P c(p ) c(p ); (c) hearts: C (T I /c(p )) = C (R/P k(g)) as k(g)-fields; (d) rationality: P is rational T I /c(p )=k(g). Cor: Rational ideals are maximal in their strata

59 Rudolf Rentschler (PhD 1967 Munich)