Communications
in
Commun.
Math. Phys.
Ill,
613-665
(1987)
Mathematical
Physics
©
Springer-Verlag
1987
Compact
Matrix
Pseudogroups
S.
L.
Woronowicz
Department
of
Mathematical Methods
of
Physics, Faculty
of
Physics, University
of
Warsaw,
Hoza
74,
PL-00-682
Warszawa,
Poland
Abstract.
The
compact matrix
pseudogroup
is a
non-commutative compact
space endowed with
a
group structure.
The
precise
definition
is
given
and a
number
of
examples
is
presented. Among them
we
have compact group
of
matrices, duals
of
discrete groups
and
twisted (deformed) SU(N) groups.
The
representation theory
is
developed.
It
turns
out
that
the
tensor product
of
representations depends essentially
on
their order.
The
existence
and the
uniqueness
of the
Haar
measure
is
proved
and the
orthonormality relations
for
matrix elements
of
irreducible representations
are
derived.
The
form
of
these
relations
differs
from
that
in the
group case. This
is due to the
fact
that
the
Haar
measure
on
pseudogroups
is not
central
in
general.
The
corresponding
modular properties
are
discussed.
The
Haar
measures
on the
twisted SU(2)
group
and on the
finite
matrix pseudogroup
are
found.
0.
Introduction
Let
G be a Lie
group.
A
family
(G
τ
)
τe[0
ε[
of Lie
groups
is
said
to be a
deformation
of
G
if
GO
= G and
G
τ
depends continuously
on τ. The
latter should
be
understood
in a
natural
sense.
For
example
one may
require that
all
G
τ
are of the
same dimensions
and
that
it is
possible
to
choose bases
in
g
τ
(g
τ
is the Lie
algebra
of
G
τ
)
such that
the
corresponding structure constants depend continuously
on τ.
Assume that
the
group
G is
involved
in a
theory (e.g.
it is a
symmetry group)
describing
a
physical reality.
As we
well know
any
physical theory describes well
only
a
limited class
of
phenomena,
for the
phenomena beyond this class
the
theoretical
predictions disagree with
the
experimental results.
In
order
to
obtain
the
adequate description
of a
larger class
of
phenomena
one
must
modify
the
theory.
In
certain cases such
a
modification although revolutionary
from
the
conceptual point
of
view consists
in
replacing
G by one of the
group
G
τ
.
Then
the
value
of τ is one of the
fundamental
constants (small parameter)
of the
new, more
general theory. Within this
new
theory
the
group
G
retains
its
validity only
in the
approximate sense (e.g.
it
describes
a
broken symmetry).
The old
theory
can be
recovered
in the
limit
τ->0.
614 S. L.
Woronowicz
The
history
of
physics provides many examples
of
developments that
fit
into
the
scheme described above. Births
of the
special theory
of
relativity
and the
theory
of
quanta
are the
most famous. Another example
we get by
considering
the
symmetry
group
of the
flat
Minkowski space-time
of
special theory
of
relativity,
i.e.
the
Poincare
group.
It
admits
a
non-trivial deformation that leads
to the
theory
of
the
de
Sitter space-time.
The
above
consideration indicate that studying
all
possible deformations
of a
group involved
in a
physical theory,
one may
discover ways leading
to
more
general theories that might better describe
the
reality.
This procedure seems
to be
especially
useful
if
already
in the
existing theory
it is
known that
the
symmetry described
by the
considered group
is
broken. Such
situations
are
constantly
met in
elementary particle physics where
we
mainly deal
with
compact
semisimple
Lie
groups. Unfortunately these groups
are
rigid:
they
admit only trivial deformations
(a
deformation
(G
τ
)
τe[0
ε[
is
said
to be
trivial
if all
G
τ
are
isomorphic
to G). If
however
we
extend
the
notion
of
compact group including
non-commutative compact spaces (compact pseudospaces
in the
sense
of
[17])
endowed with
a
group structure, then
the
class
of
deformations becomes richer
and
one can find
non-trivial deformations
for
symmetry groups
in
elementary particle
physics.
For
SU(2)
such
a
deformation
is
described
in
[18].
Let A be a
C*-algebra with unity.
If A is
commutative, then according
to the
Gelfand-Naimark
theory
A is
isomorphic
to the
algebra
of all
continuous
complex-valued
functions
defined
on a
compact topological space.
No
corre-
sponding result exists
in the
non-commutative case (see however
[7]).
Nevertheless
in
the
general case
it is of
great inspirational value
to
treat elements
of A as
"continuous complex-valued functions"
defined
on a
topological space-like object.
The
latter
is
called
a
non-commutative space
or
pseudospace. From
the
formal
point
of
view
one may
introduce non-commutative spaces (pseudospaces)
as
objects
of the
category dual
to the
category
of
C*-algebras.
See
[17]
for the
details.
In
the
present
paper
we do not use
explicitly
the
pseudospace language (using
instead C*-algebra language),
one
should stress however that this concept stays
behind many
definitions
and
considerations presented
in the
following
sections.
The
theory
of
group structures
on
non-commutative spaces
is now
more than
20
years old.
It was
originated
in [6] by Kac in an
attempt
to
unify
in one
category
locally compact
groups
and
group duals
and to
consider generalized Pontryagin
duality
as a
contravariant
functor
acting within this category.
The
theory
was
then developed
by
Takesaki
[11]
and
Schwartz
and
Enock
[5].
In
[17]
it was
pointed
out
that
the
right approach
to the
theory
is the one
based
on
the
C*-algebra language
(in
earlier works
von
Neumann algebras were used
instead).
In
[13] Vallin developed
the
C*-algebra version
of the
theory. Entirely
different
approaches
are
contained
in
recent papers
of
Ocneanu
[10],
Drinfeld
[4],
and
Vaksman
and
Soibelman
[14].
Despite
the
long history
the
theory seems
to be
still
in the
introductory stage.
In
particular
the
basic notions
are not
fixed
yet.
In our
opinion this state
was
caused
by the
lack
of
interesting examples.
For a
long time
the
only examples
of
pseudogroups
were:
locally compact groups, group duals, their cartesian products
and
crossed
products
of a
group dual
by an
automorphism group.
The
first
example
of a
(not
finite)
compact pseudogroup
of
different
nature
was
found
in
[18].
Compact
Matrix
Pseudogroups
615
The aim of
this paper
is to
develop
the
theory
of
compact
pseudogroups
in a
way
completely analogous
to the
classical theory
of
compact groups
of
matrices.
In
particular
our
main
definition
says that compact matrix pseudogroup
is a
compact
pseudospace
of
TV
x N
matrices closed under matrix multiplication
and
under
taking inverses.
The
examples presented
in
Sect.
1
show that this
definition
gives
the
direct
and
natural generalization
of the
concept
of the
compact group
of
matrices.
It
also contains
the
Pontryagin duals
of
discrete
finitely
generated
groups.
In the
world
of
pseudogroups SU(2)
and
SU(3)
admit non-trivial
deformations.
At the end of
Sect.
1 we
show
the
existence
of the
neutral element
(it
is
represented
by
*-character
e
introduced
in
Proposition 1.8)
and
prove
elementary properties linking
e
with other basic notions.
In
Sect.
2 we
present
the
representation theory.
The
standard notions
of
intertwining
operators, equivalent representations, irreducible representations,
complex conjugate representations,
the
direct
sum and the
tensor product
of
representations
are
introduced
and
investigated.
The
non-commutativity
of the
tensor product turns
out to be the
property distinguishing pseudogroups
from
groups.
Section
3 is
devoted
to the
concept
of
contragradient representation which
is
the
main tool
of the
generalized
Peter-Weyl
theory presented
in
Sects.
4 and 5.
The
Haar measure
is the
main subject
of
Sect.
4. We
prove
its
existence
and
uniqueness
and
derive elementary properties. Using
the
Haar
measure
and the
machinery
built
in
Sect.
3 we
prove
that
any
smooth representation
can be
decomposed into
a
direct
sum of
irreducible representations.
The
limitation
to the
smooth representations
is
forced
by the
fact
that
our
axiomatic admits cases where
the
Haar
measure
is not
faithful.
It
becomes
faithful
when restricted
to the
subalgebra
of
smooth
functions.
In
Sect.
5 we
present
the
Peter-Weyl
theory
for
compact matrix pseudogroups.
We
prove that
the
matrix elements
of
inequivalent irreducible representations
are
orthogonal with respect
to the two
scalar product induced
by the
Haar
measure.
It
turns
out
that
the
analogous formulae
for
matrix elements
of the
same irreducible
representation
are
more complicated than
in the
group case. This
is due to the
fact
that
the
Haar measure need
not be
central.
It
turns
out
that
the
modular properties
of
the
Haar
measure
are
described
by a
family
(/J
ze(C
of
linear multiplicative
functionals
defined
on the
sublagebra
of
smooth
functions
and
that
the
formulae
expressing
the
orthonormality
of
matrix elements
of
each irreducible represen-
tation
involve
f±
and
/_
x
.
The end of
Sect.
5 is
devoted
to the
theory
of
characters.
We
prove
the
basic
properties
and
show that
the
character determines
the
representation
up to
equivalence.
In
order
to
limit
the
volume
of the
paper
we
shift
two
sections devoted
to the
Tannaka-Krein duality
and to
differential
calculus
on
compact matrix pseudo-
groups
to
separate publications
[19,20].
The
paper contains
two
Appendices.
In the
first
one we
present
the
short proof
of
the
Haar
measure formula
for
S
μ
U(2)
given
in
[18],
the
second
is
devoted
to
finite
matrix pseudogroups.
616 S. L.
Woronowicz
1.
Definitions
and
Examples
In
this section
we
introduce
the
concept
of
compact matrix pseudogroup
and
present several examples.
In
particular
we
show that compact subgroups
of
GL(./V,
(C)
and
duals
of
discrete
finitely
generated groups
are
compact matrix
pseudogroups. Another example
can be
obtained
by a
deformation
of
compact
subgroups
of
GL(N,
(C).
At the end of
this section
we
introduce
the
convolution
product
and
derive simple formulae used constantly
in the
next sections.
We
start
with
the
following
basic:
Definition
1.1.
Let A be a
C*-algebra with unity,
u be a N x N
matrix with entries
belonging
to A:
u
=
(u
kl
)
k
ι
=
ίt2
,...,N>
u
ki
E
^
an
d
^
be
the
*-subalgebra
of A
generated
by the
entries
of u. We say
that
(A
9
u) is a
compact matrix pseudogroup
if
1)
stf
is
dense
in A.
2)
There exists
a
C*-homomorphism
Φ\A-*A®A,
(1.1)
such that
Φ(«*/)=
Σ
"*r®M
Γί
(1-2)
r= 1
for
any fe,
ί=l,2,...,JV.
3)
There exists
a
linear antimultiplicative mapping
K
:<*/-+<&,
(1.3)
such
that
φ(α*)*)
=
α
(1.4)
for
any a
e
stf
and
N
Σ
κ(u
kr
)u
rl
=
δ
kl
l,
(1.5)
e(M
r/
)
=
3
H
/,
(1.6)
for
any
k,l=ί,2,...,N.δ
kl
denotes
the
Kronecker symbol equal
to 1 for fc
=
/
and 0
otherwise,
/ is the
unity
of the
algebra
A.
Due to
Condition
1 the
C*-homomorphism (1.1)
is
uniquely determined.
It
will
be
called
the
comultiplication associated with
(A,
u). Let us
notice that
the
diagram
A
>
A®A
Φ
ΦΘid
(1.7)
id®Φ
A®
A
>A®A®A
is
commutative
(cf.
[6,
11,
5,
13]).
Indeed using (1.2),
one
easily verifies that
(id®
Φ)Φ(u
kl
)
=
Σ
u
kr
®u
rs
®u
sl
=
(Φ®id)ΦK).
r,s
Compact
Matrix Pseudogroups
617
Therefore
(id(x)Φ)Φ(α)
=
(Φ®id)Φ(β)
for any
αej/,
and
taking into account
Condition
1 we
obtain
the
same formula
for any
aeA.
It
follows immediately
from
(1.2)
that
.
(1.8)
Equations (1.5)
and
(1.6) show that
u is an
invertible element
of
M
N
(A)
and
that
Φw)
=
«"'«,
(1-9)
where
u~
l
u
denote
the
matrix elements
of the
inverse
of u. For any
αe^/
we set
α*=φ*).
(1.10)
By
virtue
of
Condition
3,
#
is an
antilinear involution acting
on
s$
and
κ(a)*
=
α*
for
any
aestf.
Applying
#
to the
both sides
of
(1.5)
and
(1.6)
we
obtain
V,
(i.ii)
for
any fc,
/
=
1,
2,
. .
.,
Λf.
Let ΰ
denote
the N x N
matrix with matrix elements being
the
hermitian conjugate
of
elements
of u:
w
=
(w
k/
)
fcί
=
1>2
,...,jv
an
d
ΰ
kl
=
u
kl
*.
Equations (1.11)
and
(1.12) show that
ΰ is an
invertible element
oϊM
N
(A)
and
that
φ
w
*)
=
tΓ
1
w
,
(1.13)
where
ΰ~
l
kl
(fc,/=l,2,
...,IV)
denote
the
matrix elements
of the
inverse
of ΰ.
Remembering that
the
algebra
stf
is
generated
by
u
kl
and
u
kl
*
(fc,
/=
1,
2,
. .
.,
N)
and
taking into account (1.9)
and
(1.13)
we see
that
the
mapping (1.3)
is
uniquely
determined.
It
will
be
called
the
coin verse associated with
(A, u).
Let
σ
A
denote
the
flip
automorphism
of A® A:
σ
A
(a®b)
= b®a
(1.14)
for
any
a,
b e A and A ®
sym
A
=
{x
e
A ®
A
:
σ
A
(x)
=
x}.
We say
that
a
pseudogr
oup
(A,u)
is
abelian
if
Φ(A)cA®
sym
A.
Let
(>M
M
«)fcί
=
ι,2,...,Λr)
an
d
(^
/
5(
M
fc/)w=ι,2,...,N')
be
compact matrix pseudo-
groups.
We say
that they
are
identical
if N'
=
N,
and if
there exists
a
C*-isomorphism
s of A
onto
A'
such that
for
fe,
/=
1,
2,
.
.
.,
N.
In
that case
s is
uniquely determined,
it
maps
the
*-subalgebra
stf
of A
generated
by
matrix elements
of u
onto
the
corresponding
*-subalgebra
stf'
related
to
(A,
u'}.
Moreover denoting
by
Φ,
K and
Φ',
K' the
comultiplications
and
the
coinverses associated with
(A, u) and
(A,
u'}
respectively
we
have
the
following
commutative diagrams
-^->
A®
A