Introduction to Depth Two and a Galois Theory for Bialgebroids. Lars Kadison

Transcription

1 Introduction to Depth Two and a Galois Theory for Bialgebroids Lars Kadison Leeds, May 2006

2 Origins of depth two and finite depth in C - and von Neumann algebras Bratteli diagram for inclusion of fin. dim. C - algebras: e.g. S 2 < S 3 and their corresp. group C -algebras is unitarily ( equiv. ) to inclusion mapping (λ, µ) (λ,, µ) λ 0 0 µ Inclusion matrix is induction-restriction table for irreducible characters: S 2 S 3 χ 1 χ 2 χ 3 ψ ψ Jones tower of von Neumann factors: N M M 1 = End M N M 2 (iteration by basic End construction). Basic construction of f.d. C -algebras has a mirror image Bratteli diagram.

3 Derived tower of centralizers or relative commutants: C N (N) C M (N) C M1 (N) C M2 (N) Draw Bratteli diagrams of each inclusion : N M is depth n if at n th level it begins to repeat itself via reflections. The notion depth two (D2) extends to any ring extension (Nikshych-L.K., Szlachanyi-L.K.) Let B A be a subring with 1 B = 1 A or B A a unital ring homomorphism. A B is D2 if A B A Add(A) as natural A-B-bimodules (right D2) and B-A-bimodules (left D2). Dress equiv. of cat s: Add(M C ) = F GP End MC for any module M over ring C,

4 Add(A) = F GP R where R = End B A A is centralizer C A (B). Since Hom (A, A B A) = (A B A) B := T and Hom (A B A, A) = End B A B := S we have equiv. def. of D2 extension: (left D2 quasibase) β i S, t i T : a a = n i=1 t i β i (a)a (right D2 quasibase) γ j S, u j T : a a = m j=1 aγ j (a )u j Note: T R and R S are f.g. projective by Dress equivalence or the left D2 eq. above. Also Add(A) = Add(A B A) and indeed obtain via Hirata that R and (End A B A A ) B are Morita equivalent.

5 Example. A f.g. projective algebra A. If x i A and p i A satisfies id A = n i x i p i, then a a = n i=1 ap i (a )1 x i. Example. A normal subgroup of a finite group: N G. If G = n i=1 g i N, A = C G and B = C N, then t i = g i gi 1 and β i ( g G a g g) = g g i N a g g is a left D2 quasibase since for all g G g N 1 = n i=1 g i gi 1 β i (g) Example. Hopf-Galois Ext s. H = bialgebra of dim H = n, A = an H-comodule algebra w/ coaction A A H s.t. B = A co H and isomorphism β : A B A A H, β(a a ) = aa (0) a (1) Then A A B A B = n A A B. H has an antipode [Sch], so β (a a ) = a (0) a a (1) is isomorphism, and A B is left D2 as well. E.g. normal Hopf subalg s are D2.

6 C A (B) IS NORMAL SUBALG [R, LK5] Given A B D2, ideal I A (I R)A = A(I R) (1) Reason: r R I, A B A I, x B y xry, apply to 1 B a = j γ j (a)u 1 j B u 2 j to get ra = j γ j (a)u 1 j ru2 j A(I R) Similarly ar = i t 1 i rt2 i β i(a) (I R)A, whence eq. (1). OBS. Using the similar equations a (1) τ(a (2) )a (3) = a 1 a (1) τ(a (2) ) a (3) = 1 a for a a Hopf algebra H w/ antipode τ, a normal Hopf subalgebra K H (i.e. every x H satisfying τ(x (1) )Kx (2) K and x (1) Kτ(x (2) ) K) is a normal subring. Converse follows from HK + = K + H characterization (where K + = ker ε K).

7 THEOREM (Sz.-L.K.) Given left or right D2 ext A B, the ring S = End B A B is a (f.g. projective left) bialgebroid over R = C A (B) and T = (A B A) B is its R-dual (f.g. proj. right) bialgebroid. Right R-bialgebroid structure on T : 1. Note T := (A B A) B = End (A A B A A ) via F F (1 1), which induces the multiplication on T : tu = u 1 t 1 B t 2 u 2, 1 T = 1 A 1 A 2. dual groupoid set-up: R s R T s R (r ) = 1 B r and t R (r) = r B 1 satisfying s R (r )t R (r) = t R (r)s R (r ). t R R op via 3. bimodule R T R = T sr,t R : r t r = rt 1 t 2 r.

8 4. (T,, ε) is R-coring where comultiplication T : T T R T = (A B A B A) B, (t) = t (1) t (2) = t 1 B 1 B t 2 and counit ε T (t) = t 1 t 2 R. 5. bialgebroid properties: (1 T ) = 1 T R 1 T, (tu) = (t) (u), ε(1 T ) = 1 R, and right properties s R (r)u (1) u (2) = u (1) t R (r)u (2), ε(tu) = ε(t R (ε(t))u) = ε(s R (ε(t))u). Left R-bialgebroid structure on S = End B A B : briefly, left and right mult. R λ S ρ R op, coproduct is dual of mult. S (α)(a B a ) = α(aa ) Formula works because of cup product result for relative Hochschild cochains of D2 ext., C 2 (A, B; A) = C 1 (A, B; A) C 1 (A, B; A). [LK4] Counit ε S (α) = α(1) R. S is R-dual to T : pairing α, t = α(t 1 )t 2 R nondegenerate if left D2.

9 Theorem [NK, SK, LK] An algebra extension A B is a right T -Galois ext. for some left fin. projective right R-bialgebroid T A B is right D2 and balanced. Explanation. is rather like the example above. ( ) A is a right T -comodule algebra: coaction δ : A A R T, a (0) R a (1) = j γ j (a) R u j satisfies δ(1 A ) = 1 A R 1 T and δ(xy) = δ(x)δ(y). Coinvariants A co T = {x A x (0) R x (1) = x 1} = B where follows from A B balanced. Finally A R T is Galois A-coring [BW] with bimodule a(a t)a = aa a (0) R ta (1), comult. A T and counit A ε T. Grouplike elt. 1 A 1 T and isomorphism A R T = A B A via a R t at 1 B t 2 w/ inverse a B a j aγ j (a ) R u j = aa (0) a (1) the Galois isomorphism.

10 Different Hopf algebroids that turn up via depth two theory: D2 Ext. A B Fin. proj. alg. A [Lu] Irred. Frob. ext. [KN1] Frob. ext. w/ sep. cent. [KS] H-separable ext. [LK0] Hopf-Galois ext. [LK1] Pseudo-Galois ext. [LK2] Bialgebroid End B A B or Dual (A B A) B End A and A e dual Hopf algs. dual weak Hopf algs. [KN2] Hopf algebroid R e Lu Geometric Hopf algebroid Connes-Mosc. Hopf algebroid Example. An H-extension A B w/ split injective Galois (A-B-)mapping β : A B A A H is D2, therefore T -Galois. If β is =, then T op = R#H op is a Lu geometric Hopf algebroid where H acts by r h = a 1 ra 2 and 1 h = β(a 1 B a 2 ). Anchor maps Ping Xu quantizes certain Lie algebroids like tangent bundle T X over Poisson manifold X

11 to obtain noncocommutative Hopf algebroids such as twisted differential operators D q (X). A Lie algebroid is a vector bundle over X equipped with a Lie bracket and a Lie homomorphism and bundle map into T X called anchor map. For a Hopf algebroid H over R, the anchor map µ is the unit representation H End R in the tensor category of H-modules. Interesting anchor maps in depth two theory: µ : S End R is evaluation of End B A B on the centralizer R. µ : T End R given by r t = t 1 rt 2, a generalized Miyashta-Ulbrich action of Hopf algebra on centralizer of Hopf-Galois extension, or quotient group conjugation action on centralizer of normal subgroup [KK]. By comparing anchors of Hopf algebroid in [LK2] and in [CM] you may write down an isomorphism between these two objects.

12 Some questions and problems related to depth two: 1. Chirality Problem. Is there a left D2 algebra extension that is not right D2? 2. Normality Problem. Is a D2 Hopf subalgebra normal? 3. Galois Correspondence Problem. Find new conditions under which D2 subextensions correspond to Hopf subalgebroids. 4. Inverse Galois Problem. Given a f.g. projective left bialgebroid, is it the endomorphism ring of a D2 extension?

13 ARTICLES BB G. Böhm and T. Brzeziński, Strong connections and the relative Chern-Galois character for corings, Int. Math. Res. Not. 42 (2005), BoSz G. Böhm and K. Szlachányi, Hopf algebroids with bijective antipodes: axioms, integrals and duals, J. Algebra 274 (2004), BM T. Brzeziński and G. Militaru, Bialgebroids, A - bialgebras and duality, J. Algebra 251 (2002), BW T. Brzeziński and R. Wisbauer, Corings and Comodules, LMS 309, Cambridge University Press, CM A. Connes and H. Moscovici, Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Moscow Math. J. 4 (2004),

14 EN P. Etingof and D. Nikshych, Dynamical quantum groups at roots of 1, Duke Math. J., 108 (2001), GS M. Gerstenhaber and S.D. Schack, Algebraic cohomology and deformation theory, in: Deformation Theory of Algebras and Structures, and Applications, eds. M. Hazewinkel and M. Gerstenhaber, Kluwer, 1988, LK0 L. Kadison, Hopf algebroids and H-separable extension, Proc. A.M.S. 131 (2003), LK L. Kadison, Galois theory for bialgebroids, depth two and normal Hopf subalgebras, Ann. Univ. Ferrara - Sez. VII - Sc. Mat. 51 (2005), LK1 L. Kadison, Hopf algebroids and Galois extensions, Bulletin Belgian Math. Soc. - Simon Stevin 12 (2005), LK2 L. Kadison, Pseudo-Galois extensions and Hopf algebroids, QA/

15 LK3 L. Kadison, An endomorphism ring theorem for depth two and Galois extensions, J. Algebra, to appear. LK4 L. Kadison, Codepth two and related topics, Appl. Cat. Struct., to appear, QA/ LK5 L. Kadison, Centralizers and induction, preprint, QA/ KK L. Kadison and B. Külshammer, Depth two, normality and a trace ideal condition for Frobenius extensions, Comm. Algebra, to appear. GR/ KN1 L. Kadison and D. Nikshych, Hopf algebra actions on strongly separable extensions of depth two, Adv. in Math. 163 (2001), KN2 L. Kadison and D. Nikshych, Frobenius extensions and weak Hopf algebras, J. Algebra 244 (2001), KS L. Kadison and K. Szlachányi, Bialgebroid actions on depth two extensions and duality, Adv. in Math. 179 (2003),

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