1. Algebraic Morita Equivalence

Home , Module (mathematics), Morita equivalence

MORITA EQUIVALENCE IN ALGEBRA AND GEOMETRY

RALF MEYER

Abstract We study the notion of Morita equivalence in various categories

We start with Morita equivalence and Morita duality in pure algebra Then

we consider strong Morita equivalence for C algebras and Morita equivalence

for W algebras Finally we lo ok at the corresp onding notions for group oids

with structure and Poisson manifolds

Algebraic Morita Equivalence

The main idea of Morita equivalence in pure algebra can b e illustrated by the

following example Let R b e any ring with unit let M R b e the ring of n n

matrices over R for some n N If V is a left Rmo dule then V is a M R

mo dule in a canonical way matrixvector multiplication and the corresp ondence

V V is functorial Conversely every M Rmo dule can b e so obtained from

some Rmo dule Thus the rings R and M R have equivalent categories of left

mo dules

Denition We write M for the category of left Rmo dules Two unital rings

are called Morita equivalent if they have equivalent categories of left mo dules

There is also a useful theory of Morita equivalence for rings with a set of lo cal

units ie suciently many idemp otents cf but things b ecome far more

complicated Unless we have useful top ologies around as in the case of C alge

bras we assume all our rings to b e unital

Let R and S b e rings with unit There is a standard way to get a functor from

M to M If Q is any S Rbimo dule and V is an Rmo dule then Q

R S S R S R R

V carries a natural S mo dule structure Thus every S Rbimo dule induces a

functor from M to M Taking the tensor pro duct of bimo dules corresp onds

R S

to the comp osition of these functors Conversely under some hyp otheses every

covariant functor must b e of this form

Theorem Watts Let T be a rightexact covariant functor from M to M

R S

which commutes with direct sums Then there is an S Rbimodule Q such that

the functors T and Q xy are natural ly equivalent Moreover Q is unique up to

isomorphism of bimodules

This result was discovered simultaneously by Eilenb erg Gabriel and Watts

around As usual in homological algebra the pro of is trivial Notice that

every equivalence of categories has to preserve direct sums and exact sequences and

thus satises the hyp otheses of Theorem Hence we obtain

This article has b een prepared for the Spring Math course at the University of

California at Berkeley taught by Alan Weinstein

RALF MEYER

Corollary Two rings R and S are Morita equivalent if and only if there are

S R and Q P bimodules P and Q such that P Q

S S R R S R RR S R S S R R S SS R

as bimodules

This result implies that Morita equivalent rings also have equivalent categories of

right mo dules and bimo dules It is also easy to see that they have equivalent lattices

of ideals so that the prop erties of b eing No etherian Artinian or simple are Morita

invariant cf They have isomorphic categories of pro jective mo dules and thus

equivalent Ktheories More generally a decent cohomology theory should b e

Morita invariant and this is indeed true for cyclic homology Ho chschild homology

for k algebras cf

Moreover Morita equivalent rings have isomorphic centers This implies that

Morita equivalent Ab elian rings are already isomorphic Thus Morita equivalence

is essentially a noncommutative phenomenon This gives another reason why so

many homology functors are Morita invariant Usually they arise as extensions

of functors dened on a category of commutative algebras to a category of non

commutative algebras But Morita invariance imp oses no restrictions whatso ever

on functors dened on a category of commutative algebras so that we can hop e

for a Morita invariant extension Examples show that if a functor can b e extended

naturally then the extension tends to b e indeed Morita invariant

An imp ortant problem is to nd conditions when two rings are equivalent Notice

that we do not have to nd two bimo dules P and Q b ecause one of them determines

the other In general if the bimo dules P and Q implement a Morita equivalence

b etween R and S we have

Q Hom P S Hom P R P Hom Q S Hom Q R

S R S R

This means that Q and P are in some sense dual to each other In the purely

algebraic setting there is no natural way to turn an S Rbimo dule into an R S

bimo dule the nearest we can get is the ab ove relation b etween P and Q For C

algebras or group oids we can turn left actions into right actions using the adjoint

op eration or inversion which will slightly simplify matters there

We let EndQ b e the ring of endomorphisms of the additive group Q ie Q

without the S Rbimo dule structure Then the rightleft op erations of R and

S on Q induce injective homomorphisms R EndQ and S EndQ The

bimo dule prop erty asserts that the images of R and S in EndQ commute and in

order to have a Morita equivalence they must b e the full commutants of each other

0 0 0

ie R S S R This is clear b ecause Q V is an R mo dule for all V M

R R

and if our inducing pro cess gives all S mo dules we need R S Thus R and the

right mo dule structure of Q determine S as the commutant of R in EndQ Of

course not every mo dule Q induces a Morita equivalence eg the zero mo dule

do es not work

Theorem Morita Let R be a ring with unit and Q a right Rmodule

Then Q induces a Morita equivalence between R and R EndQ if and only if Q

is a nitely generated projective generator

Of course the idea of representation equivalence is older than Moritas work

His main contribution was to make formal denitions and to put the various uses of

An ob ject X in an Ab elian category is a generator i every ob ject is a quotient of a direct

sum of copies of X

MORITA EQUIVALENCE

this idea into a general theory Besides the notion of equivalence Morita also stud

ied a corresp onding duality Formally this consists of replacing covariant functors

by contravariant functors Since we have V V for an innitedimensional vector

space duality can only hold if one restricts attention to nitely generated mo dules

and assumes that the underlying ring is No etherian Under these assumptions the

theory go es through smo othly and yields

Denition Morita Let F b e the category of nitely generated left R

mo dules If R and S are unital No etherian rings a duality is a pair T F F

R S

U F F of contravariant equivalence functors

S R

Theorem Morita Let R and S be Noetherian rings If there is a du

ality T U between F and F then there exists a bimodule Q such that T

R S S R

Hom xy Q U Hom xy Q Moreover the maps R S End Q are injective

R S

0 0

and R S S R

Morita also has a necessary and sucient condition for Q to induce a Morita

duality

Morita equivalence for C algebras and W algebras

In these categories we have considerably more structure and therefore restrict

our categories of mo dules

Denition Rieel A Hermitian module over a C algebra A is the Hilb ert

space H of a nondegenerate representation A H together with the action

a a for a A H If A is even a W algebra we assume in addition

that is a normal map and call H a normal Amodule In b oth cases morphisms

are the intertwining op erators ie Amo dule homomorphisms in the usual algebraic

sense

We call two C algebras Morita equivalent if they have equivalent categories

of Hermitian mo dules and if the equivalence functors T are functors ie if

f V V is a morphism then T f T f Similarly we call two W al

gebras Morita equivalent if they have equivalent categories of normal mo dules and

if the equivalence is implemented by functors

The category of Hermitian mo dules over a C algebra A is equivalent to the

category of normal mo dules over the enveloping von Neumann algebra nA Hence

Morita equivalence of C algebras is really a von Neumann algebra concept and

to o weak for most applications We will so on dene the more restrictive concept

of strong Morita equivalence for C algebras As in the purely algebraic case we

need more concrete criteria in terms of bimo dules for two algebras to b e equivalent

Since we have to transp ort the Hilb ert space inner pro ducts we need to put more

structure on our bimo dules

Denition Paschke Rieel Let B b e a C algebra A preHilbert

B module is a right B mo dule X with a compatible C vector space structure

equipp ed with a conjugatebilinear map linear in the second variable hxy xyi X

X B satisfying

In a W algebra every b ounded increasing net of p ositive elements has a least upp er b ound

A p ositive map f A ! B b etween W algebras is called normal if for any b ounded increasing

net p of p ositive elements of A with least upp er b ound p f p is the least upp er b ound of

1 1

the net f p j

RALF MEYER

hx xi for all x X

hx xi only if x

hx y i hy xi for all x y X

hx y bi hx y i b for all x y X b B

B B

The map hxy xyi is called a B valued inner product on X

It can b e shown that kxk khx xi k denes a norm on X If X is complete

with resp ect to this norm it is called a Hilbert B module If not all the structure

can b e extended to its completion to turn it into a Hilb ert B mo dule Actually

in Paschkes pap er the inner pro duct is linear in the rst variable and in Rieels

pap er this ob ject is called a right preB rigged space

This contains enough structure to transp ort Hilb ert space inner pro ducts If V

is a Hermitian B mo dule and X is a Hilb ert B mo dule we can equip the algebraic

tensor pro duct X V with an inner pro duct

0 0 0 0

hx v x v i hhx xi v v i

B V

where hxy xyi is the inner pro duct on V It can b e shown that this is nonnegative

denite and thus denes a preinner pro duct on X V Thus factoring out the

V This vectors of lenth zero and completing gives a new Hilb ert space X

construction is functorial If f V V is a morphism of Hermitian B mo dules

then id f extends to a b ounded map X V X V

B B

If e linX X is a b ounded op erator commuting with the action of B by right

multiplication then e id extends to a b ounded op erator on X V However

V B

the commutant of B in general is not a C algebra b ecause b ounded op erators

may fail to have an adjoint If T linX X an op erator T linX X is called

an adjoint for T if hT x y i hx T y i for all x y X Let E b e the algebra

of all adjointable op erators on X ie op erators that have an adjoint It is easy

to see that an adjointable op erator is necessarily b ounded and commutes with

the action of B Moreover E with the natural norm b ecomes a C algebra It

is easy to see that X V is a Hermitian E mo dule as exp ected Moreover the

mappings X V X V induced by B mo dule homomorphisms are E mo dule

B B

homomorphisms as desired so that X induces a functor from B to E mo dules

Let B B b e the closed linear span of hX X i fhx y i x y X g If B acts

trivially on V then X V is the zero mo dule so that the functor induced by X

fails to b e faithful Similarly the algebra E may b e to o big

This can easily b e seen from the example B C X H innitedimensional

In this case E H is the algebra of all b ounded op erators on H But H

B B

has the nontrivial ideal H of compact op erators It is wellknown that H is

K K

Morita equivalent to C This means that every irreducible representation of H

is a p ossibly innite direct sum of copies of the standard representation But

H as a C algebra has more complicated representations coming from the Calkin

algebra H H Here we have to b e careful As a W algebra H is Morita

B K B

equivalent to C but not if we view it as a C algebra

This example suggests to lo ok for an analogue of the ideal of compact op erators

for Hilb ert mo dules The right approach is to let E b e the closed linear span of

the rank one op erators hx y i E given by hx y i z xhy z i for x y z X

E E B

It is easily seen that E is an ideal in E Moreover now the roles of E and B

are symmetric We have just dened an E valued inner pro duct on X and X is

an E mo dule by denition only that we have exchanged left and right

MORITA EQUIVALENCE

Denition Rieel Let E and B b e C algebras By an E B equiv

alence bimodule we mean an E B bimo dule which is equipp ed with E and B val

ued inner pro ducts with resp ect to which X is a right Hilb ert B mo dule and a left

Hilb ert E mo dule such that

hx y i z xhy z i for all x y z X

E B

hX X i spans a dense subset of B and hX X i spans a dense subset of E

B E

We call E and B strongly Morita equivalent if there is an E B equivalence bimo dule

If X is an E B equivalence bimo dule it is easy to endow the conjugate space X

which is X as a set with the same addition and scalar multiplication x x with

the structure of a B E equivalence bimo dule For examplexe e x Moreover

it is not dicult to see that strong Morita equivalence is an equivalence relation

Theorem Rieel Let X be an E B equivalence bimodule Then X

xy induces an equivalence between the category of Hermitian B modules and the

category of Hermitian E modules the inverse being given by X xy This functor

preserves weak containment and direct integrals

The reason for Rieel to intro duce strong Morita equivalence was to improve

the understanding of induced representations of lo cally compact groups Let G

b e a lc group and let H b e a closed subgroup Then unitary representations of H

induce representations of G Moreover the representations of G obtained by

this pro cess are precisely those that admit a system of imprimitivity In more

mo dern language the representations that can b e obtained by inducing from H are

the covariant representations of C GH G where C GH are the functions

1 1

on GH vanishing at innity and the action of G on C GH is obtained from

the left translation action of G on GH These results are due to Mackey for the

separable case but his pro ofs were based on rather unintuitive measure theoretic

arguments In Rieel gave a new pro of by showing that the group algebra

C H is strongly Morita equivalent to the crossed pro duct C GH o G Actu

ally he worked with the dense subalgebras C xy of functions of compact supp ort

and showed that C G can b e given the structure of a preHilb ert C H mo dule

c c

Then he identied the algebra of nite rank op erators on C G with a dense

subalgebra of the crossed pro duct C GH o G

But there are also other more noncommutative applications For example if G

is a compact group acting on a C algebra A by automorphisms G AutA

we can dene a conditional exp ectation p A A where A is the xed p oint

algebra by averaging pa a dx with resp ect to Haar measure dx Then

pa b turns A into a preHilb ert A mo dule It can b e shown that ha bi

this gives us a strong Morita equivalence of A with a certain ideal of the crossed

pro duct algebra A o G In the commutative case this ideal is the whole crossed

pro duct algebra if and only if the action of G is free Actually the case where G

is not compact is very imp ortant but also much more subtle cf

Another more elementary example is the following Let p A then the corre

ApApApequivalence bimo dule with sp onding left ideal Ap can b e made into an

inner pro ducts hx y i xy hx y i x y Subalgebras of the form pAp

ApA pAp

are the prototyp e of hereditary subalgebras and the corresp onding hereditary sub

ApA A This example is of considerable theoretical algebra is called full if

imp ortance b ecause every strong Morita equivalence is of this form If A and B

are strongly Morita equivalent C algebras there is a C algebra C that contains

RALF MEYER

b oth A and B as full hereditary subalgebras Together with a result of Brown on

hereditary subalgebras in this gives the following remarkable theorem

Theorem BrownGreenRieel Let A and B be C algebras with a count

able approximate identity eg separable or unital Then they are strongly Morita

equivalent if and only if they are stably equivalent ie A B where is

K K K

the algebra of compact operators on a separable Hilbert space

Thus stable equivalence which is of considerable imp ortance in Ktheory can

b e viewed as a separable version of Morita equivalence Moreover since the class of

separable or unital algebras is already rather large one can exp ect that prop erties

that are invariant under stable equivalence are also Morita invariant For example

Morita equivalent C algebras have isomorphic lattices of ideals and the same K

E and KKtheory In it is shown how to induce traces b etween Morita equiv

alent C algebras In Morita equivalence for group actions on C algebras is

dened and it is shown that equivalent group actions give rise to Morita equivalent

group C algebras and reduced group C algebras

Now let us briey discuss the situation for von Neumann algebras If M is a von

Neumann algebra a further requirement for normal Hilb ert M N bimo dules

is that the maps m hx my i b e weakly continous for all x y X On the

other hand we can weaken the requirements for an equivalence bimo dule replacing

density by weak density That this is p ossible is illustrated by the example C H

With these changes the analogue of the Eilenb ergGabrielWatts theorem is again

true

Theorem Rieel Let M and N be W algebras Then every normal equiv

alence bimodule implements an equivalence between the categories of normal M and

N modules by a functor Conversely every such equivalence is implemented by

some normal equivalence bimodule

It is easy to see that Morita equivalent von Neumann algebras have isomorphic

centers and isomorphic lattices of weakly closed ideals Moreover if M and N are

Morita equivalent and if M is of typ e X fI II IIIg then the same holds for N

ie Morita equivalence resp ects the typ e of a von Neumann algebra For typ es I

and I I I the classication up to Morita equivalence is very easy

Theorem Rieel Two W algebras of type I are Morita equivalent if and

only if they have isomorphic centers Two von Neumann algebras of type III on

separable Hilbert spaces are Morita equivalent if and only if they are isomorphic

As p ointed out to me by Dimitri Shlyakhtenko two factors M N of typ e I I on

separable Hilb ert spaces are equivalent if and only if they are stably equivalent as

von Neumann algebras ie M H N H this tensor pro duct is in the

B B

category of von Neumann algebras and is dened to b e the weak closure of the

spatial tensor pro duct Thus every typ e II factor is equivalent to a I I factor

and conversely The idea of the pro of is to turn a M N Hilb ert bimo dule for two

II factors into a genuine preHilb ert space using the trace on one of them The

actions extend to the completion and it turns out that M and N are commutants

of one another Hence we obtain a corresp ondence in the sense of and can

apply the theory for those

It should b e remarked that Morita equivalence of von Neumann algebras is not

an imp ortant technical to ol but at most a convenient way of formulating some of

MORITA EQUIVALENCE

the known results For example a von Neumann algebra is of typ e I i it is Morita

equivalent to a commutative von Neumann algebra

Morita equivalence for topological and symplectic groupoids

Now we lo ok at geometric analogues of Morita equivalence rst for lo cally com

pact top ological group oids The bimo dule version still makes sense

Denition MuhlyRenaultWilliams Let G b e a lo cally compact top olog

ical group oid with unit space G and source and range maps s and r A lo cally

compact space X with a continuous op en map X G which we call the

momentum map and an action G X X where G X fg x G X j

sg xg is called a left Gspace if

g x r g for all g x G X

x x x for all x X and

g h x g h x whenever g h G G and h x G X

We write g x g x g x A right Gspace is dened similarly

The action is called free if g x G X and g x x implies g G ie only

units have xed p oints

The action is called proper if the map id G X X X sending g x to

g x x is prop er

If H is another group oid and if X is at the same time a left Gspace and a

right H space with momentum maps X G and X H we call it a

GH bimodule if the actions commute ie

x h x for all x h X H and similarly g x x for all g x

in G X and

g x h g x h for all g x G X x h X H

We say that a GH bimo dule X is an equivalence bimodule if

it is free and prop er b oth as a G and an H space

the momentum map X G induces a bijection of XH to G and

the momentum map X H induces a bijection of GnX to H

We call G and H Morita equivalent if a GH equivalence bimo dule exists

The orbit space for a prop er group oid action is always lo cally compact Hausdor

and the pro jection onto the orbit space is op en Thus for an equivalence bimo dule

the bijections XH G GnX H are automatically homeomorphisms

The action of a group oid on itself by left and right multiplication turns it into

a GGequivalence bimo dule so that Morita equivalence is a reexive relation It

is easy to see that it is also symmetric and transitive For the latter one uses

the analogue X Y of the bimo dule tensor pro duct If X and Y are GH and

H K bimo dules resp ectively then X Y fx y X Y j x y g

X Y

where X H and Y H are the momentum maps In order to

X Y

get X Y identify x h y x h y when this is dened It is not dicult to

endow this with the structure of a lo cally compact GK space and to see that this

pro cess pro duces equivalence bimo dules if X and Y were equivalence bimo dules

Moreover this tensor pro duct is functorial for equivariant continuous maps as

morphisms

Corollary Let G and H be Morita equivalent local ly compact groupoids Then

the categories of left right G and H spaces are equivalent

RALF MEYER

As in the algebraic case under suitable hyp otheses a left Gspace determines a

group oid H such that it b ecomes a GH equivalence bimo dule To b e more

sp ecic let X b e a free prop er Gspace with a surjective momentum map Let

X X fx y X X j x y g Then G acts freely and prop erly on X X

by the diagonal action g x y g x g y The orbit space H GnX X can b e

endowed naturally with a group oid structure over GnX by putting x y y z

x z and this multiplication is continuous There is an obvious right action of H

on X dened by x x y y It can b e checked that this turns X into a GH

equivalence bimo dule Moreover if X was a GH equivalence bimo dule to start

with then we get H H

There are many examples of Morita equivalent group oids If G is a transitive

group oid u G then r u is an equivalence bimo dule for G and the isotropy

group r u s u at u if r and s are op en maps A similar statement holds

if U G is a subset meeting every Gorbit This applies esp ecially to foliations

transverse submanifold meeting every leaf Moreover we get that the group oid

asso ciated to a Cartan principal bundle cf is equivalent to the structure

group of the bundle

Another typical example is the following situation Let H and K b e lo cally

compact groups acting freely and prop erly on a lo cally compact Hausdor space P

such that the actions commute Let H act on the left and K act on the right

The commutativity assumption means that we get an action of K on H nP and an

action of H on P K Then the space P is an equivalence for the transformation

group oids H P K and K H nP

How is Morita equivalence of group oids related to the algebraic notion Fix

Haar systems and for G and H Then we can form the full group oid C alge

bras C G and C H with resp ect to these Haar systems For the group oids

coming from the last example ab ove it was already discovered by Green cf

that the asso ciated group oid C algebras are Morita equivalent In it is shown

that this remains true in general with a pro of similar to Rieels argument in

Theorem MuhlyRenaultWilliams Let G and H be local ly compact sec

ond countable Hausdor groupoids with Haar systems and If there is a G H

equivalence bimodule X then the ful l groupoid C algebras C G and C H

are strongly Morita equivalent

The denition of the group oid C algebra dep ends on the choice of a Haar system

However the denition of a representation of a group oid do es not In the group

case Haar measure is essentially unique but for group oids this is no longer the

case Due to the corresp ondence of group oid representations and representations of

the group oid C algebra dierent choices of Haar system certainly pro duce Morita

equivalent C algebras This still leaves op en whether we actually get isomorphic

C algebras At least in the case of transitive group oids this is indeed the case

Theorem MuhlyRenaultWilliams Let G be a second countable local ly

compact transitive groupoid let u G and let H be the isotropy group at u

Let be a Haar system for G Then there is a positive measure on G of ful l

support such that C G is isomorphic to C H L G

It is easy to see that C G must b e strongly Morita equivalent to C H

But the ab ove renement shows that we do not have to tensor C G with the

MORITA EQUIVALENCE

compact op erators This shows that the group oid algebra is stable and do es not

dep end on the choice of Haar system

By the way it probably is not very interesting to lo ok for criteria on group oids

that are necessary and sucient for the group oid C algebras to b e Morita equiv

alent This can already b e seen by lo oking at groups It is easy to see that two

groups are equivalent in the sense of Denition i they are isomorphic as top o

logical groups However if K is a compact group then by the PeterWeyl theorem

its group oid C algebra is a direct sum of copies of full matrix algebras and there

are innitely many such copies if and only if K is has innitely many elements

Thus any two innite compact groups have strongly Morita equivalent even stably

equivalent group C algebras However there seems to b e no natural equivalence

bimo dule in this situation that can b e written down without knowing the full rep

resentation theory of the involved groups

If we drop all continuity assumptions we get a notion of Morita equivalence

for algebraic group oids without any further structure More imp ortantly if our

group oids carry additional dierentiability structure we should strengthen our re

quirements on equivalences by asserting that the actions are smo oth in order to

get an equivalence of the categories of smo oth actions Moreover the bijections of

the orbit spaces XH with G and GnX with H should b e dieomorphic This

follows if the momentum maps are ful l ie surjective submersions

Morita equivalence for symplectic groupoids and Poisson

manifolds

Denition Xu Two symplectic group oids G and H with unit spaces

G and H are called Morita equivalent if there are a symplectic manifold X

and surjective submersions X G and X H such that

G has a free prop er symplectic left action on X with momentum map

H has a free prop er symplectic right action on X with momentum map

the two actions commute with each other

induces a dieomorphism XH G

induces a dieomorphism GnX H

X is called an equivalence bimodule between G and H

As exp ected Morita equivalence is an equivalence relation among symplectic

group oids Since the notion is stronger than equivalence for top ological group oids

Morita equivalent symplectic group oids still have Morita equivalent group oid C

algebras Equivalence bimo dules for symplectic group oids were also studied from a

slightly dierent viewp oint and under the name of an anoid structure by Weinstein

The p oint of intro ducing the stronger relation ab ove is that we now get results

ab out the category of symplectic actions of our group oids that are completely anal

ogous to the results for lo cally compact group oids In the pro ofs it only has to

b e checked that the symplectic structure can b e transp orted On the other hand

top ological problems almost disapp ear in this category It is easy to see that Morita

equivalence of symplectic group oids is an equivalence relation

Theorem Xu Let G be a symplectic groupoid over G and let

X G be a ful l symplectic free and proper left Gmodule Then GnX

is a Poisson manifold and H GnX X is a symplectic groupoid over GnX G

RALF MEYER

in a natural way Moreover X GnX natural ly becomes a symplectic right

H module such that X is an equivalence bimodule between G and H

Conversely if X is any equivalence bimodule between symplectic groupoids

G and H then H GnX X as symplectic groupoids

As usual if P is a Poisson manifold with bracket then P denotes the same

manifold with bracket

Let G b e a symplectic group oid We can consider the category of symplectic

left mo dules over G in which morphisms b etween symplectic mo dules F and F

are Lagrangian submanifolds of F F invariant under the diagonal action of G

and the comp osition of morphisms is the set theoretic comp osition of relations

This is not a true category since the comp osition of two morphisms need not b e a

submanifold Then we obtain

Theorem Xu Morita equivalent symplectic groupoids have equiva

lent categories of symplectic left modules

One motivation for intro ducing Morita equivalence of symplectic group oids is

the corresp ondence b etween integrable Poisson manifolds and r simply connected

symplectic group oids It is p ossible to pull back the group oid equivalence to the

base Poisson manifolds

Denition Xu Two Poisson manifolds P and P are Morita equiv

alent if there exists a symplectic manifold X together with complete Poisson mor

phisms X P and X P that form a full dual pair with connected and

simply connected b ers Then X is called an equivalence bimo dule

The reason for requiring connected simply connected b ers is to exclude certain

cases that we do not want to b e Morita equivalences For example with this

denition two connected symplectic manifolds are Morita equivalent i they have

the same fundamental group This somewhat complicated denition is b orne to

make true the following theorem

Theorem Xu Let P and P be integrable Poisson manifolds Then

P and P are Morita equivalent if and only if their r simply connected symplectic

groupoids are Morita equivalent

Let G b e an r simply connected group oid over P The main step in the pro of of

Theorem is to show that if X is a symplectic left Gmo dule then its momentum

map X P is a complete symplectic realization and that conversely every

complete symplectic realization of P carries a natural left Gaction This also

proves the following theorem

Theorem Xu Equivalent integrable Poisson manifolds have equiv

alent categories of complete symplectic realizations

Furthermore Theorem immediately implies that Morita equivalence is an

equivalence relation among integrable Poisson manifolds This is not true for arbi

trary Poisson manifolds Already reexivity fails Currently it is not even known

whether every Poisson manifold has a complete symplectic realization

It is not dicult to see that an equivalence bimo dule b etween two Poisson man

ifolds induces a bijection b etween their leaf spaces Moreover Morita equivalence

A group oid is called r if all r b ers have the prop erty Many authors call the range

and source map somewhat unintuitively and and thus write instead

MORITA EQUIVALENCE

takes into account the variation of the symplectic structures on the leaves which

is measured by the fundamental class cf theorem This idea allows

very precise statements ab out Morita equivalence of regular Poisson manifolds For

example

Theorem Xu Let P be a regular Poisson manifold with symplectic

bration P Q Then P is Morita equivalent to Q with the zero Poisson

structure if and only if al l the symplectic leaves of P are connected and simply

connected and the fundamental class vanishes

Theorem Xu Let P M be a local ly trivial bund le of con

nected simply connected symplectic manifolds Then P is Morita equivalent to M

with zero Poisson structure

Another interesting result is that Morita equivalent Poisson manifolds have the

same zeroth and rst cohomology groups The example of symplectic manifolds

shows that no results ab out higher cohomologies can b e exp ected

Theorem is an imp ortant to ol for computing symplectic realizations of Poisson

manifolds by reducing the problem to an apparently simpler Morita equivalent

manifold see For example Theorem reduces the study of a lo

cally trivial bundle of symplectic manifolds to that of symplectic realizations of

the base manifold with zero Poisson structure Thus in order to classify the com

plete symplectic realizations of a simply connected connected symplectic manifold

we have to classify symplectic realizations of a single p oint which is rather easy

Careful b o okkeeping shows that every symplectic realization of S is of the form

pr S X S for a symplectic manifold X where pr is the canonical pro jection

S S

Another case that can b e treated in this way is the orbit space of a Hamiltonian

action of a Lie group on a symplectic manifold and the crossed pro duct of an

integrable Poisson manifold with a Lie group acting on it

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Email address rameyermathunimuensterde