SPb. Math. Society Preprint 2001-06 19 Sep 2001
GRADED
LIE
ALGEBRAS
AND
DYNAMICAL
SYSTEMS
A.
M.
VERSHIK
Intr
oduction
W
econsider
aclass
ofinnite-dimen
sional
Lie
algebras
whic
his
asso
ciated
to
dynamical
systems
with
invarian
tm
easures.
There
are
two
constructions
of
the
algebras
{o
ne
based
on
the
asso
ciativ
ecross
pro
duct
algebra
whic
h
considered
as
Lie
algebra
and
then
extended
with
non
trivial
scalar
two-
co
cycle
the
second
description
is
the
sp
ecication
of
the
construction
of
the
graded
Lie
algebras
with
con
tin
uum
ro
ot
system
in
spirit
of
the
pap
ers
of
Sa
veliev-V
ershik
SV1,
SV2,
V]
whic
his
ageneralization
of
the
denition
of
classical
Cartan
nite-dimensional
algebras
as
well
as
Kac{Mo
ody
algebras.
In
the
last
paragraph
we
presen
tthe
third
construction
for
the
sp
ecial
case
of
dynamical
systems
with
discrete
spe
ctrum.
The
rst
example
of
suc
h
algebras
was
so
called
sine-algebras
whic
hw
as
disco
vered
indep
enden
tly
in
SV1
]and
FFZ
]and
had
been
studied
later
in
GKL]
from
poin
to
fview
Kac{Mo
ody
Lie
algebras.
In
the
last
paragraph
of
this
pap
erw
ealso
suggest
anew
examples
of
suc
ht
yp
ealgebras
app
eared
from
arithmetics:
adding
of
1in
the
additiv
egroup
Z
p
as
atransformation
ofthe
group
of
p
-adic
integers.
The
set
of
positiv
esimple
ro
ots
in
this
case
is
Z
p
Cartan
subalgebra
isthe
algebra
of
con
tin
uous
functions
on
the
group
Z
p
and
W
eyl
group
of
this
Lie
algebra
con
tains
the
innite
symmetric
group.
Remark
ably
this
algebra
is
the
inductiv
elimit
of
Kac{Mo
ody
ane
algebras
of
typ
e
A
1
p
n
.
1.
Lie
algebra
genera
ted
by
automorphism
1.1.
Asso
ciativ
ealgebra
A
(
X
T
)
.
Let
(
X
)b
ea
separable
compactum
with
Borel
probabilit
ymeasure
whic
his
positiv
efor
an
yo
pen
set
A
X
and
T
is
am
easure
preserving
homeomorphism
of
X
.
It
is
old
and
well-kno
wn
construction
of
W
-algebra
(von
Neumann)
and
C
-algebra
(Gel'fand)
generated
by(
X
T
),
see
for
example
D,
ZM].
Al-
gebraically
this
isan
asso
ciativ
ea
lgeb
ra
A
(
X
T
)w
hic
his
semidirect
pro
duct
of
C
(
X
)and
C
(
Z
)w
ith
the
action
of
Z
on
X
and
consequen
tly
on
C
(
X
).
St.
Petersburg
Division,
Steklo
v
M
athematical
Institute,
RAS,
27,
Fon
tank
a,
St.
Petersburg,
191011,
Russia.
Partially
supp
orted
by
gran
tRFBR
99-01-00098.
This
pap
er
con
tains
the
sub
ject
of
my
talk
on
the
Conference
in
Tw
en
te
in
Decem
be
r
2000.
1
2
A.
M.
VERSHIK
As
alinear
space
this
is
direct
sum
A
(
X
T
)=
M
n
2
Z
C
(
X
)
U
n
where
C
(
X
)i
sB
ana
ch
space
of
all
con
tin
uous
functions
on
X
,and
U
=
U
T
is
linear
op
erator
(
U
T
f
)(
x
)=
f
(
Tx
),
f
2
C
(
X
).
The
multiplication
of
the
monomials
is
dened
byf
orm
ula
(
'
U
n
)
(
U
m
)=(
'
U
n
)
U
n
+
m
n
m
2
Z
'
2
C
(
X
)
:
Inv
olution
on
A
(
X
T
)is
the
follo
wing:
(
'
U
n
)
=(
U
;
n
'
)
U
;
n
:
Completion
of
A
(
X
T
)w
ith
resp
ect
to
the
appropriate
C
-norm
giv
es
a
corresp
onding
C
-algebra.
It
is
possible
to
include
to
this
construction
a2-co
cycle
of
the
action
of
Z
with
values
in
C
(
X
)to
obtain
another
C
-algebra
whic
hare
unsplittable
extensions
(see
ZM
,V
Sh
]),
but
we
restrict
ourself
to
the
case
of
the
trivial
co
cycle.
Ifw
euse
ameasure
as
astate
on
A
(
X
T
)and
construct
-represen
tation
corresp
onding
to
this
state
then
W
-closure
of
image
of
A
(
X
T
)g
ives
us
W
-algebra
generated
by
triple
(
X
T
).
There
are
two
classical
represen
tations
of
the
algebra
A
(
X
T
)|
Ko
op-
mans
represen
tation
(in
L
2
(
X
))
and
von
Neumann
one
in
L
2
m
(
X
Z
)
(
m
is
Haar
measure
on
Z
).
This
area
called
\algebraic
theory
of
dynami-
cal
systems"
and
there
are
man
yp
ap
ers
on
this.
Extremally
popular
is
so
called
rotation
algebras
(also
called
as
"quan
tum
torus")
whic
his
asso
ciativ
e
C
-algebra
generated
by
irrational
rotation
of
the
unit
circle.
We
want
to
point
out
that
ther
e
is
another
remarkable
algebr
aic
obje
ct
which
is
asso
ciate
d
with
dynamic
al
systems|some
Lie
algebr
as
which
are
similar
to
the
classic
al
Cartan
Lie
algebr
as
and
to
ane
Kac-Mo
ody
alge-
bras.
Some
non
trivial
cen
tral
extension
included
in
the
denition
pla
ys
very
imp
oratan
trole
in
the
whole
theory
.Upto
no
w
only
the
Lie
algebras
cor-
resp
oning
to
rotation
algebras
were
considered
it
was
disco
vered
indep
en-
den
tly
in
SV1
]and
FFZ
](see
also
GKL]
were
the
shift
on
d-dimensional
torus
was
considered)
and
called
byp
hysisits
"sine
algebras"
-a
ll
those
dynamical
systems
ha
vea
discr
ete
spectrum
,
In
whole
generalit
ythe
Lie
algebras
generated
by
an
arbitrary
dynamical
system
with
invarian
tm
easure
was
briey
dened
in
SV2]
and
more
system-
atically
in
V].
It
is
interesting
that
we
started
not
form
the
generaltheory
of
dynamical
systems
as
in
the
rst
denition
belo
w
but
from
the
notions
presen
ted
in
our
series
of
pap
ers
with
M.
Sa
veliev
SV1,
SV2
]w
ere
we
had
de-
ned
so
called
\
Z
-graded
Lie
algebras
with
con
tin
uous
ro
ot
systems".
Those
algebras
whic
hw
ewill
discuss
here
were
one
oft
he
typ
eof
the
examples
and
asp
ecial
case
of
Z
-gr
aded
Lie
algebras
with
general
roo
tsystems.
Belo
ww
e
will
describ
eexplicitly
the
mo
dern
and
detailed
version
of
the
construction
of
Lie
algerbas
generated
by
an
arbitrary
discrete
dynamical
systems
with
GRADED
LIE
ALGEBRAS
AND
D
YNAMICAL
S
YSTEMS
3
invarian
tmeasure
and
then
will
giv
ethe
link
betw
een
various
denitions.
One
can
hop
ethat
this
typ
eo
falgebras
can
giv
ean
ew
typ
eo
fin
varian
ts
of
the
dynamical
systems
as
well
as
new
examples
ofcalssical
and
quan
tum
integrable
systems.
1.2.
Lie
algebras
e
A
(
X
T
)
.
Most
interesting
case
is
the
case
when
T
is
minimal
(=
eac
h
orbit
of
T
is
dense
in
X
)a
nd
ergo
dic
with
resp
ect
to
measure
(=
there
are
no
nonconstan
t
T
-in
varian
tmeasurable
functions).
W
eassume
this
in
further
considerations.
It
is
kno
wn
that
if
T
is
minimal
(=eac
h
T
-orbit
is
dense
in
X
)then
C
-algebra
is
simple
(=
has
no
prop
er
two-sided
ideals),
see
ZM
].
Algebra
A
(
X
T
)w
ith
brac
kets
a
b
]=
ab
;
ba
will
be
denoted
as
Lie
A
(
X
T
)
it
is
still
Z
-graded
Lie
algebra
and
the
brac
k-
ets
of
monomials
are
'
U
n
U
m
]=(
'
U
n
;
U
m
'
)
U
n
+
m
:
This
algebra
has
acen
ter.
Lemma
1.
The
center
of
Lie
A
(
X
T
)is
the
set
of
constants
functions
in
zer
oc
omp
onent
subsp
ac
e:
Z
=
c
U
0
,
c
2
C
.
The
complement
line
ar
subsp
ac
e
A
0
(
X
T
)=
M
n<
0
C
(
X
)
U
n
C
0
(
X
)
U
0
M
n>
0
C
(
X
)
U
n
wher
e
C
0
(
X
)=
'
2
C
(
X
):
R
X
'
(
x
)
d
=0
,is
Lie
sub
algebr
a
which
is
isomorphic
to
quotient
A
(
X
T
)
Z
over
center
Z
.
Remark.
The
cen
ter
is
not
ideal
of
asso
ciativ
ealgebra
consequen
tly
there
is
no
\asso
ciativ
e"
analogue
of
this
lemma
and
A
0
(
X
T
)is
not
asubalgebra
of
A
(
X
T
).
No
ww
ed
ene
a2
-co
cycle
on
A
0
(
X
T
)with
the
scalar
values
and
one-
dimensional
cen
tral
expansion
ofit.
Lemma
2.
The
fol
lowing
formula
denes
2-c
ocycle
on
A
0
(
X
T
):
(
'
U
n
U
m
)=
n
Z
X
'
U
n
d
n
+
m
so
(
'
U
n
U
m
)=
(
n
R
X
'
U
n
d
if
m
=
;
n
0
if
m
6
=
;
n:
Pr
oof.
W
eneed
to
chec
kt
hat
(
x
y
]
z
)+
(
y
z
]
x
)+
(
z
x
]
y
)=
0.
Let
k
+
l
+
n
=
0.
Then
;
'
U
k
U
l
]
U
n
+
=
;
(
'
U
k
;
U
l
'
)
U
k
+
l
U
;
k
;
l
+
:::
=(
k
+
l
)
Z
X
;
'
U
k
U
;
n
U
;
n
;
U
l
'
U
;
n
d
+
=0
4
A.
M.
VERSHIK
(Dots
mean
cyclic
perm
utation
of
indices,
we
used
here
the
invariance
of
measure
under
T
.)
Remark.
Co
cycle
is
not
cohomologous
to
zero
because
it
is
easy
to
chec
k
that
(
x
y
)can
not
be
represen
ted
as
f
(
x
y
]),
with
an
ylinear
functional
f
.o
f
C
(
X
).
Let
us
iden
tify
scalars
c
whic
h
are
extensions
of
A
0
(
X
T
)with
scalars
c
2
C
(
X
)
U
0
A
(
X
T
).
So
we
can
consider
again
linear
space
A
(
X
T
)
as
one
dimensional
non
trivial
extension
of
Lie
algebra
A
0
(
X
T
).
Denote
a
new
Lie
algebra
by
e
A
(
X
T
).
So,
Lie
algebra
e
A
(
X
T
)as
linear
space
is
the
same
as
A
(
X
T
)but
the
brac
kets
in
e
A
(
X
T
)dier
from
the
brac
kets
in
A
(
X
T
):
'
U
n
U
m
]=
(
'
U
n
;
U
m
'
)
U
n
+
m
+
Z
X
'
U
n
d
n
+
m
(1)
It
means
that
the
cen
ter
of
e
A
(
X
T
)is
again
scalars
C
1
C
(
X
)
U
0
Lie
A
(
X
T
),
but
no
w
subspace
A
0
(
X
T
)is
not
Lie
subalgebra
and
the
cen-
tral
extension
isn
ot
trivial.
Lie
algebra
e
A
(
X
T
)is
Z
-gra
ded
Lie
algebra.
W
ew
ill
giv
ea
new
denition
of
it
in
aframew
ork
of
Lie
algebras
with
con
tin
uous
ro
ot
systems.
W
ew
ill
call
the
subspace
of
e
A
(
X
T
)whic
h
consists
of
Span
f
U
;
1
g
Span
f
'
U
0
g
Span
f
U
1
g
,
'
2
C
(
X
),
a\lo
cal
subalgebra".
Here
are
the
brac
kets
for
local
part
of
e
A
(
X
T
)a
re
'
1
U
0
'
2
U
0
]=
0
'
U
0
U
1
]=
;
(
'
;
U'
)
U
1
=
;
(
I
;
U
)
'
U
1
'
U
+1
U
;
1
]=
(
'
U
;
U'
)
U
0
+
Z
X
(
'
U
)
d
c:
(2)
The
middle
term
of
local
algebra
(
f
'
U
0
'
2
C
(
X
)
g
)i
sb
y
denition
Cartan
subalgebra.
This
giv
es
the
rst|\dynamical"|description
of
the
Lie
algebra
e
A
(
X
T
).
1.3.
Lie
algebras
with
ro
ot
system
(
X
T
)
.
Denition
of
Lie
algebra
will
be
follo
wed
to
Kac{Mo
ody
pattern
but
with
imp
ortan
tc
hanges.
First
of
all
we
dene
a
local
algebr
a.L
et
'
2
C
(
X
)
we
consider
three
typ
es
of
uncoun
tably
man
yg
enerators:
X
;
1
(
'
),
X
0
(
'
),
X
+1
(
'
)where
'
runs
over
C
(
X
).
The
list
ofrelations
is
as
follo
ws:
X
0
(
'
)
X
0
(
)]
=
0
X
0
(
'
)
X
1
(
)]
=
X
1
(
K'
)
X
+1
(
'
)
X
;
1
(
)]
=
X
0
(
'
)
(3)
GRADED
LIE
ALGEBRAS
AND
D
YNAMICAL
S
YSTEMS
5
where
pro
duct
(
)is
the
pro
duct
in
asso
ciativ
ealgebra
C
(
X
)a
nd
K
is
a
linear
op
erator
in
C
(
X
)w
hichi
sc
alle
dC
arta
no
pera
tor:
(
K'
)(
x
)=
2
'
(
x
)
;
'
(
Tx
)
;
'
(
T
;
1
x
)
:
(4)
It
is
eviden
tthat
Jacobi
iden
tit
y
is
true
(if
it
mak
es
sense)
in
the
local
algebra
A
;
1
A
0
A
+1
where
A
i
=
Span
f
X
i
(
'
)
'
2
C
(
X
)
g
,
i
=
;
1
0
+1.
(
A
0
'
C
(
X
)is
Cartan
subalgebra.)
The
further
steps
are
the
same
as
in
Kac{Mo
ody
theory
K].
W
edene
free
Lie
algebra
whic
h
is
generated
by
the
local
algebra
and
factorize
it
over
the
maximal
ideal
whic
hh
as
zero
intersection
with
A
0
.The
resulting
Lie
algebra
isdenoted
by
A
(
X
T
).
The
fact
isthat
this
algebra
is
the
same
as
e
A
(
X
T
)of
subsection
2.
W
eomit
the
verication
that
A
(
X
T
)is
the
graded
Lie
algebra
with
the
graded
structure
(as
alinear
space)
as
follo
w
M
n
2
Z
A
n
and
eac
h
A
n
'
C
(
X
)(see
V
]).
So
we
can
denote
the
elemen
ts
of
A
n
as
X
n
(
'
),
'
2
C
(
X
).
Theorem
1.
The
fol
lowing
formulas
give
the
canonic
al
isomorphism
be-
twe
en
e
A
(
X
T
)and
dense
part
of
A
(
X
T
):
(
'
U
n
)=
8
>
<
>
:
X
;
n
(
U
;
n
'
)
(
n>
0)
X
0
(
'
;
U
;
1
'
)
n
=0
R
X
'd
=0
X
n
(
'
)
(
n>
0)
(
1
U
0
)=
X
0
(
1
)
:
Pr
oof.
The
kernel
of
is
0
.
Let
us
chec
k
that
(
a
b
])
=
a
b
].
It
is
enough
to
test
monomials
only
.
(
'
U
0
)
(
U
)
=
'
;
U
;
1
'
)
U
0
U
=
;
'
;
U
;
1
'
;
U
(
'
;
U
;
1
'
)
U
=(
K'
)
U
=
X
+1
(
K'
)=
;
'
U
0
U
]
X
0
(
'
)
X
;
1
(
)
=
(
'
;
U
;
1
'
)
U
0
U
;
1
U
;
1
=
;
U
;
1
(
'
;
U
;
1
'
;
U
;
1
'
+
U
;
2
'
)
U
;
1
=
;
U
;
1
(
(
U'
;
2
'
+
U
;
1
'
))
U
;
1
=
;
U
;
1
(
K'
)
U
;
1
=
;
X
;
1
(
K'
)
X
+1
(
)
X
;
1
(
)
=
U
U
;
1
U
;
1
=(
;
U
;
1
U
;
1
)
U
0
=
;
;
U
;
1
(
)
U
0
=
X
0
(
)
6
A.
M.
VERSHIK
X
+1
(
1
)
X
;
1
(
1
)
=
cX
0
(
1
)=
c
1
:
Remark.
1.
W
ecalculated
the
brac
ket
only
for
elemen
ts
of
local
algebra,
but
this
is
enough
because
it
generates
all
algebra.
2.
The
-image
of
e
A
(
X
T
)is
not
all
A
(
X
T
)but
dense
part
of
A
(
X
T
),
for
example,
the
set
of
functions
'
;
U'
is
dense
in
C
0
(
X
)only
,but
A
(
X
T
)i
s
the
extension
ofthe
image
of
e
A
(
X
T
)and
we
can
consider
A
(
X
T
)as
some
kind
of
completion
of
e
A
(
X
T
).
3.
Using
isomorphisms
we
can
rewrite
the
denition
ofc
ocycle
of
A
(
X
T
)(
(
'
U
;
n
)=
X
;
n
(
U
;
n
'
),
n>
0)
so
;
X
n
(
'
)
X
m
(
)
=
(
0
if
n
+
m
6
=0
,
n
R
X
'
d
n
+
m
=0
:
Lie
algebra
A
(
X
T
)dened
abo
ve
iso
ur
main
ob
ject.
Pr
oof.
If
n>
0then
(
X
n
(
'
)
X
;
n
(
))
=
n
R
X
'
U
n
(
U
;
n
)
d
=
n
R
X
'
d
.
If
n<
0t
he
(
X
;
n
(
'
)
X
n
(
))
=
;
n
R
X
U
;
n
'
U
;
n
d
=
;
n
R
X
'
d
(
U
is
an
unitary
op
erator).
This
consists
with
the
initial
form
ula
X
+1
;
'
X
;
1
;
(
'
)
;
1
=
X
0
(
1
)=
1
c
for
n
=
1.
No
ww
ecan
rewrite
the
brac
kets
for
all
monomials
(not
only
for
local
part).
Assume
n
m
>
0.
X
n
(
'
)
X
m
(
)
=
'
U
n
U
m
=
X
n
+
m
(
'
U
n
;
U
m
'
)
(+,+)
X
n
(
'
)
X
;
m
=
'
U
n
U
;
m
U
;
m
=
;
'
U
n
;
m
;
U
;
m
(
'
)
U
n
;
m
=
8
>
<
>
:
X
n
;
m
;
'
U
n
;
m
;
U
;
m
(
'
)
n>m>
0
X
0
;
(1
;
U
;
m
)(1
;
U
;
1
)
;
1
'
n
=
m
X
n
;
m
;
U
;
n
'
(
U
m
'
;
U
;
n
)
0
<n
<m
(+,
;
)
X
;
n
(
'
)
X
;
m
=
U
;
n
'
U
;
n
U
;
m
U
;
m
=(
U
;
n
'
U
;
n
;
m
;
U
;
m
U
;
n
;
m
'
)
U
;
n
;
m
=
X
;
n
;
m
(
U
m
'
;
'
U
n
)
=
;
X
;
n
;
m
(
'
U
n
;
U
m
'
)
(
;
,
;
)
X
0
(
'
)
X
n
(
)
=
(
'
;
U
;
1
'
)
U
0
U
n
=
;
(
'
;
U
;
1
'
+
U
n
;
1
'
;
U
n
'
)
U
n
=
X
n
(
K
n
'
)
(0,+)
GRADED
LIE
ALGEBRAS
AND
D
YNAMICAL
S
YSTEMS
7
where
K
n
=
I
;
U
;
1
+
U
n
;
1
;
U
n
=(
I
;
U
;
1
)(
I
;
U
n
)
X
0
(
'
)
X
;
n
=
(
'
;
U
;
1
'
)
U
0
U
;
n
U
;
n
=
;
U
;
n
(
'
;
U
;
1
'
;
U
;
n
'
+
U
;
n
;
1
'
)
U
;
n
=
X
;
n
((
U
;
n
)
(
U
n
'
;
U
n
;
1
'
;
'
+
U
;
1
'
)=
;
X
n
(
K
n
'
)
(0,
;
)
W
ecan
no
w
observ
ethat
the
form
ulas
(+,+)
and
(
;
,
;
)are
the
same,
as
well
as
(0,+)
and
(0,
;
).
Theorem
2.
The
formulas
for
the
brackets
of
monomials
in
the
sub
algebr
a
A
(
X
T
)ar
ethe
fol
lowing:
1)
X
n
(
'
)
X
m
(
)
=
X
n
+
m
(
'
U
n
;
U
m
'
),
wher
ethe
sign
is
\+
"i
f
n
m
>
0and
\
;
"i
f
n
m
<
0.
2)
X
0
(
'
)
X
n
(
)
=
X
n
(
K
n
'
).
3)
X
n
(
'
)
X
m
(
)
=
8
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
:
X
n
+
m
;
U
m
(
'
U
n
;
U
m
'
)
8
<
:
m<
0
n
+
m>
0
X
0
;
1
;
U
;
m
1
;
U
;
1
(
'
)
+
n
R
X
'
d
8
<
:
n
+
m
=0
n
6=0
X
n
+
m
;
U
;
n
'
(
U
;
m
'
;
U
;
n
)
8
<
:
m<
0
n
+
m<
0
4)
X
0
(
'
)
X
0
(
)
=0
.
Lie
algebra
A
(
X
T
)d
oes
not
asso
ciate
with
asso
ciativ
ea
lgebra
cocycle
has
nothing
to
do
with
asso
ciated
crosspro
duct
of
subsection
1.1.
The
role
of
cen
tral
extension
isv
ery
imp
ortan
t.
W
edened
Lie
algebra
A
(
X
T
)(
'
e
A
(
X
T
))
in
anew
terms,
compare
with
(3).
This
manner
giv
es
us
the
form
ulas
for
local
part
(1{2)
whic
ha
re
similar
to
classical
ones
(Cartan
simple
algebras
and
Kac{Mo
ody
algebras).
But
the
form
ulas
for
general
monomials
are
more
complicated
than
dynamical
(see
subsection
1.1)
description.
2.
General
Lie
algebras
w
ith
continuous
root
systems
and
new
examples
of
A
(
X
T
)
2.1.
General
denition.
W
erecall
SV1,
SV2,
V]
the
denition
ofg
raded
Lie
algebras
with
con
tin
uous
ro
ot
system.
Supp
ose
H
is
acomm
utativ
easso
ciativ
eLie
C
-algebra
with
unit
y(Cartan
subalgebra)
and
K
:
H
-
is
alinear
op
erator
(Cartan
op
erator).
The
local
algebra
K
]is,
as
alinear
space,
adirect
sum
H
;
1
H
0
H
+1
H
i
'H
i
=0
1
8
A.
M.
VERSHIK
with
brac
kets:
X
i
(
'
)
2H
i
i
=0
1
'
2H
X
0
(
'
)
X
0
(
)
=0
X
0
(
'
)
X
1
(
)
=
X
1
(
K'
)
X
+1
(
'
)
X
;
1
(
)
=
X
0
(
'
)
The
local
algebra
H
;
1
H
0
H
+1
generates
graded
Lie
algebra
A
(
H
K
).
in
the
same
spirit
as
in
Subsection
1.2
(and
as
in
the
theory
of
LKM-algebras).
Then
we
obtain
A
(
X
T
)from
Section
1.
The
spectrum
of
comm
utativ
ealgebra
H
(if
it
exists)
is
roo
tsystem
of
A
(
H
K
)b
yd
enition
(see
V]),
more
exactly
the
set
ofs
imple
positiv
ero
ots.
But
it
could
be
no
sp
ectra
(sa
y,
H
is
the
algebra
of
rational
functions)
so
weh
ave
Lie
algebras
without
simple
ro
ots
but
with
Cartan
op
erator.
The
condition
of
constan
tor
polynomial
gro
wth
of
the
dimension
(in
an
appropriate
sense)
puts
essen
tial
restriction
on
the
op
erator
K
.
Remark.
Let
E
H
isan
invarian
tunder
K
subalgebra
of
H
.T
hen
A
(
E
K
)
is
Lie
subalgebra
of
A
(
H
K
).
In
particular,
if
E
1
E
2
:::
,
i
E
i
=
H
,is
asequence
of
K
-in
varian
tsubalgebras
of
H
then
A
(
H
K
)=
1
i
=1
A
(
E
i
K
).
2.2.
New
examples
of
algebras
of
typ
e
A
(
X
T
)
.
The
rst
non
trivial
example
of
algebras
of
typ
e
A
(
X
T
)w
as
so
called
sine-algebra.
W
ew
ill
not
consider
it
because
itw
as
done
before
from
dieren
tp
oin
tof
view
(see
SV2
,F
FZ
,G
KL
,V]).
It
was
dened
indep
enden
tly
in
SV2]
and
FFZ
].
W
e
giv
en
ow
general
example
ofs
imilar
typ
e.
Let
(
X
T
)be
an
ergo
dic
system
with
discrete
spe
ctrum.
It
means
that
op
erator
U
=
U
T
has
spe
ctral
decomp
osition
Uf
=
X
f
where
f
=
X
f
sum
is
over
eigen
values
of
U
(
2
T
1
),
and
is
the
eigenfunction
corre-
sp
onding
to
.I
tis
well-kno
wn
(von
Neumann
theorem)
that
suc
h
system
can
be
realized
on
the
compact
abelian
group
G
=
X
with
Haar
measure
m
=
and
T
is
translation
on
some
elemen
t
g
0
2
G
,then
isac
haracter
of
G
(
2
G
^
)and
=
(
g
0
)
2
T
1
:
Sine-algebra
corresp
onds
to
the
case
X
=
T
1
and
T
is
translation
on
ir-
rational
num
be
r
2
S
1
=
R
=B
Z
.In
GKL]
was
considered
also
the
case
X
=
T
d
3
.
The
case
X
=
Z
p
Tx
=
x
+1
where
Z
p
is
additiv
eg
roup
of
p
-adic
integers,
p
is
aprime,
and
T
is
adding
of
unit
y,i
sm
ore
interesting
from
our
poin
tof
view.
The
measure
=
m
is
Haar
(additiv
e)
measure
on
Z
p
.
GRADED
LIE
ALGEBRAS
AND
D
YNAMICAL
S
YSTEMS
9
The
group
of
characters
Z
^
p
=
Q
p
1
is
a
group
of
all
roo
ts
of
unit
yo
f
the
degree
p
n
,
n
2
N
,and
the
characters
2
Q
p
1
(as
function
on
Z
p
with
values
in
T
1
)are
eigenfunctions
of
op
erator
U
=
U
T
.
W
egiv
ethe
description
of
the
general
case
when
ope
rator
U
=
U
T
has
discrete
sp
ectrum.
The
sp
ecic
prop
ert
yo
falgebra
A
(
X
T
)in
this
case
is
existence
of
the
natural
line
ar
basis
in
A
(
X
T
).
Supp
ose
G
is
ab
elian
(additiv
e)
compact
group
and
G
^
is
a
coun
table
group
of
the
(m
ultiplicativ
e)
characters
on
G
.W
e
x
the
elemen
t
2
G
with
dense
set
of
pow
ers:
Cl
f
n
n
2
Z
g
=
G
.
Then
T
g
=
Tg
=
g
+
,
U
T
=
U
,(
Uf
)(
g
)=
f
(
g
+
),
f
2
C
(
G
).
Eac
hc
haracter
2
G
^
is
an
eigenfunction
of
U
with
eigen
value
(
)
2
T
1
.
Theorem
3.
Line
ar
basis
in
the
Lie
algebr
a
A
(
G
T
)is
the
set
f
Y
n
:
2
G
^
n
2
Z
g
with
the
fol
lowing
brackets
Y
n
Y
1
n
1
=
;
1
(
)
n
;
(
)
n
1
Y
1
n
+
n
1
+
n
+
n
1
~
1
n
c
(5)
wher
e
n
=
(
1
n
=0
0
n
6
=0
, ~
=
(
1
=
1
0
6
=
1
.
Algebr
a
A
(
G
T
)is
Z
G
^
-gr
ade
dalgebr
a
the
sub
algebr
a
f
c
1:
c
2
C
g
in
Cartan
sub
algebr
a
A
0
=
C
(
G
)is
the
center
of
A
(
G
T
).
Pr
oof.
Assume
Y
n
=
U
n
as
an
elemen
to
f
A
n
,w
here
is
the
character
of
G
as
a
function
G
!
T
1
.
It
is
easy
to
chec
k
that
the
brac
kets
(see
form
ula
(1)
in
Section
1)
giv
eu
sform
ula
(5)
.N
ote
that
the
cen
ter
is
not
direct
summand,
so
weh
ave
Y
n
Y
;
1
;
n
=
n
c:
This
is
the
third
description
ofo
ur
algebra
A
(
X
T
)w
ith
linear
basis
this
description
is
valid
for
discrete
spe
ctrum
only
.
The
group
G
is
the
set
of
simple
ro
ots
for
A
(
G
T
)and
\Dynkin"
diagram
is
the
set
of
arro
ws
G
3
g
!
g
+
2
G
.In
opp
osite
to
Kac{Mo
ody
case
our
algebras
A
(
X
T
)h
ave
no
imaginary
ro
ots.
2.3.
The
case
of
p
-adic
integers.
Return
bac
kto
the
case
G
=
Z
p
,
Tx
=
x
+1
.In
this
case
A
(
Z
p
T
)is
Z
Q
p
1
-graded
algebra.
Cartan
subalgebra
is
space
C
(
Z
p
).
Consider
nite
dimensional
subspaces
L
n
C
(
Z
p
)of
functions
dep
ending
on
the
poin
ts
of
the
quotien
t
Z
p
!
Z
=p
n
it
isc
lear
that
subspace
L
n
is
U
T
-in
varian
t,
so
L
n
=
L
m
2
Z
L
n
U
m
is
asubalgebra
of
A
.
10
A.
M.
VERSHIK
Theorem
4.
The
Lie
algebr
a
L
n
is
canonic
ally
isomorphic
to
the
algebr
a
A
(1)
p
n
.Conse
quently,
A
(
Z
p
T
)is
(completion)
of
the
inductive
limit
of
Kac{
Mo
ody
Lie
algebr
as
1
A
(
Z
p
T
)
lim
ind
n
L
n
:
Remark.
It
is
possible
to
dene
W
eyl
group
W
for
this
algebra.
Group
W
con
tains
the
group
of
perm
utations
ofthe
co
ordinates
in
Z
p
.
It
is
very
instructiv
e
to
study
the
link
betw
een
theory
of
Kac{Mo
ody
ane
algebras
and
our
theory
looking
on
this
example.
References
SV1]
M.
Sa
veliev,
A.
Vershik.
Continuum
analo
gues
of
contr
agr
adien
tL
ie
algebr
as.
CMP
126
(1989),
367{378.
SV2]
M.
Sa
veliev,
A.
Vershik.
New
examples
of
continuum
grade
d
Lie
algebr
as.
Ph
ys.
Lett.
A
143
(1990),
121{128.
V]
A.
Vershik.
Lie
algebr
as
gener
ate
dby
dynamic
al
systems.
St.
Petersburg
Math.
J.
4
no.
6(1993).
K]
V.
Kac.
Innite
dimensional
Lie
algebr
as.
Cam
b.
Univ.
Press,
1991.
GKL]
M.
Golenishc
hev
a{Ku
znezo
va,
D.
Leb
edev.
Z
-gr
ad
ed
trigonometric
sub
algebr
as
and
its
repr
esentatio
ns
with
vertex
op
erator.
Func.
Anal.,
27
no.
1(1993),
12{24.
VSh]
A.
Vershik,
B.
Shoikhet.
Gr
ade
dL
ie
algebr
as
whose
Cartan
sub
algebr
as
is
the
alge-
brao
fp
olynomials
in
one
variable.
Th.
and
Math.
Ph
ys.
123
no.
2(2000),
345{352.
D]
J.
Dixmier.
C
-algebr
as
and
its
repr
esentation
s.
Paris,
1969.
ZM]
G.
Zeller{Mei
er.
Pr
oduits
crois
es
d'une
C
algebr
e
par
un
group
e
d'automorphis
mes.
J.
Math.
pures
et
appl.
47
(1968),
101{239.
FFZ]
D.
Fairlie,
P.Fletc
her,
C.
Zac
hos.
Trigonometric
structur
ec
onstants
for
new
in-
nite
dimensional
algebr
as.
Ph
ys.
Lett.
B
218
no.
2(1989),
203{206.
1
More
exactly
,inductiv
elimit
con
tains
only
linear
com
binations
of
monomials
oft
yp
e
'
U
n
,where
'
are
cylindric
functions
so
it
is
enough
to
extend
the
set
of
'
on
to
arbitrary
con
tin
uous
functions
whic
h
means
to
mak
ea
completion.
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