Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Generalized Symmetry In Noncommutative Complex Geometry joint work with Indranil Biswas and Debashish Goswami Suvrajit Bhattacharjee Indian Statistical Institute, Kolkata Quantum Flag Manifolds, Prague, 19 September, 2019 Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Outline 1 Hopf Algebroids 2 Hopf Algebroid Equivariant Kähler Structures 3 The Hodge Decomposition Theorem Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Motivation Transversely Kähler foliations A Kähler manifold with a pseudogroup action such that the Kähler structure is invariant. Question How does it fit into the framework of [Ó Buachalla, 2017]? Use the foliation groupoid and the groupoid algebra. Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Aim We introduce Hopf algebroid covariant Kähler structures. We show that the framework of [Ó Buachalla, 2017] is a special case. We present a version of Hodge decomposition theorem. Using the groupoid algebra, we reprove the theorem for transversely Kähler foliations (and orbifolds), which is relatively recent. Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Takeuchi Product Definition An (s; t)-ring over a C-algebra A is a C-algebra H with homomorphisms s : A ! H and t : Aop ! H whose images commute in H. The Takeuchi product of H is the subspace X 0 X 0 H ×A H := f hi ⊗A hi 2 H ⊗A H j hi t(a) ⊗ hi i i X 0 = hi ⊗ hi s(a) 8a 2 Ag i of H ⊗A H, where the tensor product ⊗A is defined with respect to the following (A; A)-bimodule structure on H: a1 · h · a2 := s(a1)t(a2)h; a1; a2 2 A; h 2 H: (1) Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Bialgebroid Definition A left bialgebroid over a C-algebra Al is a tuple (Hl ; Al ; sl ; tl ; ∆l ;"l ) such that Hl is an (sl ; tl )-ring equipped with the structure of an Al -coalgebra (∆l ;"l ) with respect to the Al -bimodule structure (1), subject to the following conditions: i) the (left) coproduct ∆l : Hl ! Hl ⊗Al Hl maps into the subset Hl ×Al Hl and defines a morphism ∆l : Hl ! Hl ×Al Hl of unital C-algebras; ii) the (left) counit has the property: 0 0 0 0 "l (hh ) = "l (hsl ("l h )) = "l (htl ("l h )) h; h 2 Hl : A right bialgebroid over a C-algebra Ar is a tuple (Hr ; Ar ; sr ; tr ; ∆r ;"r ) defined similarly. Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Hopf Algebroid Definition A Hopf algebroid is given by a triple (Hl ; Hr ; S), where Hl and Hr are left Al and right Ar -bialgebroids on the same C-algebra H, and S : H ! H is invertible C-linear, subject to the following conditions: sl "l tr = tr ; tl "l sr = sr ; sr "r tl = tl ; tr "r sl = sl (∆l ⊗ idH )∆r = (idH ⊗ ∆r )∆l (∆r ⊗ idH )∆l = (idH ⊗ ∆l )∆r S(tl (al )htr (ar )) = sr (ar )S(h)sl (al ) µH (S ⊗ idH )∆l = sr "r µH (idH ⊗ S)∆r = sl "l Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Modules over a Hopf Algebroid Definition A left module over a Hopf algebroid H is a left module M over the underlying C-algebra. An Al -bimodule structure on M can be given by a1 · m · a2 = sl (a1) · tl (a2) · m; 8a1; a2 2 Al ; 8m 2 M: An H-module algebra B is a C-algebra as well as a left H-module such that the multiplication in B is Al -balanced and for b; b0 2 B, h 2 H i) h · 1B = sl "l (h) · 1B; ii) 0 0 h · (bb ) = (h1 · b)(h2 · b ). Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Examples Étale Groupoid A Lie groupoid G = (G; G0; s; t) is called étale if all the structure maps are local diffeomorphisms. For an étale groupoid G, with 1 1 compact G0, Cc (G) forms a Hopf algebroid over C (G0). Enveloping Algebra op Given an arbitrary C-algebra A, let H = A ⊗C A . The left bialgebroid structure over A is given by sl (a) = a ⊗C 1; tl (b) = 1 ⊗C b; ∆l (a ⊗ b) = (a ⊗C 1) ⊗A (1 ⊗C b); op "l (a ⊗C b) = ab; and the right bialgebroid structure over A is given similarly. Finally, the antipode S(a ⊗C b) = b ⊗C a makes H into a Hopf algebroid. Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Examples Connes-Moscovici left bialgebroid Let Q be a Hopf algebra over C with antipode T satisfying 2 T = id and A a Q-module algebra. Let H = A ⊗C Q ⊗C A with multiplication given by 0 0 0 0 0 0 (a ⊗C q ⊗C b)(a ⊗C q ⊗C b ) = a(q1a ) ⊗C q2q ⊗C (q3b )b; for a; b; a0; b0 2 A and q; q0 2 Q. A left bialgebroid structure over A is given as follows. For a; b 2 A and q 2 Q, define sl (a) = a ⊗C 1 ⊗C 1; tl (b) = 1 ⊗C 1 ⊗C b "l (a ⊗C q ⊗C b) = a"(q)b; ∆l (a ⊗C q ⊗C b) = (a ⊗C q1 ⊗C 1) ⊗A (1 ⊗C q2 ⊗C b): Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Examples Theorem ( _, Biswas, Goswami, 2019) A right bialgebroid structure on the (left) Connes-Moscovici bialgebroid H can be given by sr (b) = 1 ⊗C 1 ⊗C b; tr (a) = a ⊗C 1 ⊗C 1; ∆r (a ⊗C q ⊗C b) = (a ⊗C q1 ⊗C 1) ⊗Aop (1 ⊗C q2 ⊗C b); "r (a ⊗C q ⊗C b) = T (q)(ba); for a; b 2 A and q 2 Q,which makes the map S defined by S(a ⊗C q ⊗C b) = T (q3)b ⊗C T (q2) ⊗C T (q1)a; an antipode. Thus H becomes a Hopf algebroid which we will call the Connes-Moscovici Hopf algebroid. Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Covariant DGA Definition ( _, Biswas, Goswami, 2019) A pair (B; d) is called an H-covariant differential graded algebra if B is an N0-graded H-module algebra (the action preserves the grading) and d is homogeneous of degree one with d 2 = 0 such that i) the graded Leibniz rule holds: d(bb0) = d(b)b0 + (−1)k bd(b0) b 2 Bk ; b0 2 B: ii) Al and H0 generate H as algebra, where H0 := fh 2 H j [h − sl "l (h); d] = [h − tl "l (h); d] = 0g : Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Covariant Differential Calculus Definition An H-covariant differential calculus over an H-module algebra B (with unit map ιB) is an H-covariant differential graded 0 algebra (Ω; d) (with unit map ιΩ) such that Ω = B, the two H-action on B coming from B itself and Ω0 coincide, and Ωk = f ^ · · · ^ j ; ··· ; 2 g: spanC b0db1 dbk b0 bk B Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Examples Proposition ( _, Biswas, Goswami, 2019) For an étale groupoid G, the differential d on G0 satisfies d(a · !) = d("l (a)) ^ ! + a · d(!) 1 for a 2 Cc (G) and ! 2 Ω(G0). Hence [a − "l (a); d] = 0 for all 1 1 a 2 Cc (G), thus implying H0 = Cc (G). Proposition ( _, Biswas, Goswami, 2019) Let (Ω; d) be a Q-covariant differential calculus on A. Then (Ω; d) can be made into an H-covariant differential calculus on A, H being the Connes-Moscovici Hopf Algebroid. Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Covariant Complex Structures Definition An H-covariant almost complex structure for an H-covariant ∗-differential calculus (Ω; d) over an H-module ∗-algebra B is 2 (k;l) an -algebra grading ⊕ 2 Ω for Ω such that N0 (k;l)2N0 i) 2 the H-action preserves the N0-grading; ii) n (k;l) Ω = ⊕k+l=nΩ , for all n 2 N0; iii) Ω(k;l) = (Ω(l;k))∗. It is integrable if d = @ + @: We call an integrable almost complex structure a complex structure. Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Covariant Kähler Structures Definition A symplectic form for Ω is a d-closed central real H-invariant 2-form σ (h · σ = sl "l (h) · σ for all h 2 H) such that, the Lefschetz operator L :Ω ! Ω , ! 7! σ ^ ! satisfies the following condition: the maps Ln−k :Ωk ! Ω2n−k are isomorphisms for all 0 ≤ k < n. Definition An (Hermitian) Kähler structure for Ω is a pair (Ω(·;·); σ), where Ω(·;·) is an H-covariant complex structure, and σ is an (almost) symplectic form, called the (Hermitian) Kähler form, such that σ 2 Ω(1;1). Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Hodge Star Operator And Metric Definition The Hodge map associated to an Hermitian structure is the morphism uniquely defined by j k(k+1) a−b j! n−j−k ?(L (!)) = (−1) 2 i L (!) (n − j − k)! for ! 2 P(a;b) ⊂ Pk = f! 2 Ωk j σ(n−k+1) ^ ! = 0g: Definition k l Define g :Ω ⊗B Ω ! B by g(! ⊗ η) = 0 for ! 2 Ω , η 2 Ω , k 6= l, and for !; η 2 Ωk , g(! ⊗ η) = vol(! ^ ?(η∗)): Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem The Laplacian And An Inner Product Inner Product For a positive definite Hermitian structure and a positive linear functional τ on B, an inner product is given by Z h!; ηi = τg(! ⊗ η) = ! ^ ?(η∗): τ Corollary d is adjointable and d ∗ := − ? d?; Definition ∗ 2 The Laplacian is defined as ∆d := (d + d ) .