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 := { hi ⊗A hi ∈ H ⊗A H | hi t(a) ⊗ hi i i X 0 = hi ⊗ hi s(a) ∀a ∈ A} 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 ∈ A, h ∈ 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 , εl ) such that Hl is an (sl , tl )-ring equipped with the structure of an Al -coalgebra (∆l , εl ) 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 εl (hh ) = εl (hsl (εl h )) = εl (htl (εl h )) h, h ∈ Hl .

A right bialgebroid over a C-algebra Ar is a tuple (Hr , Ar , sr , tr , ∆r , εr ) 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 εl tr = tr , tl εl sr = sr , sr εr tl = tl , tr εr 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 εr

µH (idH ⊗ S)∆r = sl εl

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, ∀a1, a2 ∈ Al , ∀m ∈ 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 ∈ B, h ∈ H

i) h · 1B = sl εl (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 ∞ ∞ 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 εl (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 ∈ A and q, q0 ∈ Q. A left bialgebroid structure over A is given as follows. For a, b ∈ A and q ∈ Q, define

sl (a) = a ⊗C 1 ⊗C 1, tl (b) = 1 ⊗C 1 ⊗C b

εl (a ⊗C q ⊗C b) = aε(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);

εr (a ⊗C q ⊗C b) = T (q)(ba); for a, b ∈ A and q ∈ 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 ∈ Bk , b0 ∈ B.

ii) Al and H0 generate H as algebra, where

H0 := {h ∈ H | [h − sl εl (h), d] = [h − tl εl (h), d] = 0} .

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 = { ∧ · · · ∧ | , ··· , ∈ }. 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 · ω) = d(εl (a)) ∧ ω + a · d(ω)

∞ for a ∈ Cc (G) and ω ∈ Ω(G0). Hence [a − εl (a), d] = 0 for all ∞ ∞ a ∈ 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)∈N0 i) 2 the H-action preserves the N0-grading; ii) n (k,l) Ω = ⊕k+l=nΩ , for all n ∈ 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 εl (h) · σ for all h ∈ H) such that, the Lefschetz operator L :Ω → Ω , ω 7→ σ ∧ ω 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 ∈ 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 (ω)) = (−1) 2 i L (ω) (n − j − k)!

for ω ∈ P(a,b) ⊂ Pk = {ω ∈ Ωk | σ(n−k+1) ∧ ω = 0}.

Definition

k l Define g :Ω ⊗B Ω → B by g(ω ⊗ η) = 0 for ω ∈ Ω , η ∈ Ω , k 6= l, and for ω, η ∈ Ωk ,

g(ω ⊗ η) = vol(ω ∧ ?(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 hω, ηi = τg(ω ⊗ η) = ω ∧ ?(η∗). τ

Corollary d is adjointable and d ∗ := − ? d?;

Definition ∗ 2 The Laplacian is defined as ∆d := (d + d ) .

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem A Special Case

[Beggs-Smith, 2013] defines NC complex structures in the nonequivariant setting.

[Ó Buachalla, 2017] defines NC complex and Kähler structures in the Hopf algebra covariant setting.

If Al equals C, then a Hopf algebroid over Al is just a Hopf algebra. Thus, we recover the set up in [Ó Buachalla, 2017].

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Regularity of Hermitian Structures

Definition ( _, Biswas, Goswami, 2019) The Hermitian structure is said to be d-regular if the following are satisfied:

i) k Let η ∈ Ω , and let l be a weak solution of ∆d (ω) = η (a bounded linear functional l :Ωk → C such that k l(∆d (φ)) = hη, φi, for all φ ∈ Ω ). Then there exists an element ω ∈ Ωk such that l(ν) = hω, νi for every ν ∈ Ωk . k ii) For a sequence {ηn} in Ω such that kηnk ≤ c and k∆d (ηn)k ≤ c for all n and for some constant c > 0, there k exists a Cauchy subsequence of {ηn} in Ω .

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Hodge Decomposition

Theorem ( _, Biswas, Goswami, 2019) Assume that the Hermitian structure is regular. Then for each k k with 0 ≤ k ≤ 2n, the space Hd of d-harmonic forms is finite dimensional and we have the following orthogonal direct sum decomposition of Ωk called the Hodge decomposition:

k k k Ω = ∆d (Ω ) ⊕ Hd ∗ ∗ k k = (dd ⊕ d d)(Ω ) ⊕ Hd k−1 ∗ k+1 k = d(Ω ) ⊕ d (Ω ) ⊕ Hd .

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem A Sufficient Condition For Regularity

An Assumption Define the Hilbert space of forms L2(Ω) to be the completion of Ω with respect to the inner product. Then ∆d becomes a non-negative densely defined symmetric operator on L2(Ω). Thus ∆d has a canonical self-adjoint extension. k Assume: ∩k dom(∆d ) = Ω.

Theorem ( _, Biswas, Goswami, 2019)

Assume that ∆d has purely discrete spectrum, i.e., there is an 2 orthonormal basis {ωj } for the Hilbert space L (Ω) consisting of forms ωj ∈ Ω which are eigenforms for ∆d , i.e., ∆d (ωj ) = λj ωj , for some scalar λj such that 0 = λ0 < λ1 < · · · → ∞ as j → ∞. Then the Hermitian structure is d-regular.

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Invariant Forms

Definition

We define the space of invariant forms Ω0 of Ω as

Ω0 = {ω ∈ Ω | h · ω = sl εl (h) · ω = tl εl (h) · ω for all h ∈ H0}.

Proposition ( _, Biswas, Goswami, 2019) For the space of invariant forms we have,

i) (Ω0, d|Ω0 ) is a differential graded algebra; ii) Ω0 is a ∗-algebra; iii) ∗ ∗ d|Ω0 satisfies d|Ω0 (ω ) = (d|Ω0 ω) for all ω ∈ Ω0;

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Hodge Decomposition For the Invariant Subalgebra

Definition ( _, Biswas, Goswami, 2019) We say that H acts on (Ω, d) properly (or (Ω, d) is a proper H-module) if there is a graded C-linear morphism π :Ω → Ω which is a self-adjoint idempotent with range Ω0.

Theorem ( _, Biswas, Goswami, 2019) For a d-regular Hermitian structure on (Ω, d) which is also a k proper H-module, any ω ∈ Ω0 can be written as

ω = ∆d (η) + ν ,

k k k where η ∈ Ω0 and ν ∈ Hd ∩ Ω0. Hence Hodge decomposition hold for (Ω0, d) under d-regularity.

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem A Sufficient Condition For Proper Action

Notation 2 2 Let L (Ω0) be the closure of Ω0 in L (Ω), and let P be the 2 orthogonal projection onto L (Ω0)

Proposition ( _, Biswas, Goswami, 2019) 2 Suppose ∆d |Ω0 is essentially self-adjoint on L (Ω0). Then P takes dom(∆d ) into dom(∆d ), and ∆d P(ω) = P∆d (ω) for all ω ∈ dom(∆d ). Moreover, P takes Ω into Ω. Hence P|Ω gives a projection in the sense of the definition above.

Associated Results One can further weaken the above proposition.

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Examples

Foliations It can be shown that the Laplacian restricted to the invariant forms is essentially-selfadjoint. So that the holonomy groupoid algebra acts properly.

Orbifolds The associated proper groupoid admits a "Haar system", so one has averaging at disposal.

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem Some Classical Results

[Cordero-Wolak, 1991] proves a Hodge decomposition theorem for a transversely Kähler foliation.

[Bazzoni, Biswas, Fernandez,´ Munoz, Tralle, 2017] proves a Hodge decomposition theorem for an orbifold.

Our method gives new proofs of these results, if we take the Hopf algebroid to be the groupoid algebra coming from the holonomy groupoid and the proper groupoid associated to the orbifold, respectively.

Bhattacharjee, Biswas, Goswami Hopf Algebroids Hopf Algebroids Hopf Algebroid Equivariant Kähler Structures The Hodge Decomposition Theorem

Thank You!

Bhattacharjee, Biswas, Goswami Hopf Algebroids