Download The quantum Heisenberg group H(1)q

The quantum Heisenberg group H(l),
Enrico Celeghini
Dipartimento di Fisica dell ‘Universith and I. N.F. N. Sezione di Firenze, Largo E. Fermi,
2-I 50125 Firenze, Italy
Riccardo
Giachetti
Dipartimento di Matematica,
50125 Firenze, Italy
Emanuele
Universitb di Bologna and I.N.F.N. Sezione di Firenze, Largo E. Fermi, 2-I
Sorace and Marco Tarlini
I. N.F. N. Sezione di Firenze, Largo E. Fermi, 2-I 50125 Firenze, Italy
(Received 18 July 1990; accepted for publication 21 November 1990)
The structure of the quantum Heisenberg group is studied in the two different frameworks of
the Lie algebra deformations and of the quantum matrix pseudogroups. The R-matrix
connecting the two approaches, together with its classical limit r, are explicitly calculated by
using the contraction technique and the problems connected with the limiting procedure
discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown.
I. INTRODUCTION
In a previous paper,’ we have shown how one can determine new quantum groups by using the technique of contraction. Indeed the contraction procedure defined in Ref. 1
as an extension of the usual one allows us to obtain quantum
groups (Hopf algebras) from semisimple quantum groups
making the following diagram commutative:
SQG
q-l
‘lo
CQG
1
I
SL
-t
q- 1,
CL
t--O
where SQG and CQG, SL and CL, respectively, mean simple
and contracted quantum group, simple and contracted Lie
algebras.
We note that in general, because of the nonlinearity of
the algebraic structure, the limit procedures produce q-deformations of Lie algebras that may be not Hopf algebras.2V3
Contracting the well-known SU(2),, we havedefinedin
Ref. 1 the quantum Heisenberg group H ( 1) 4, whose three
generators A, A + , and H satisfy the relations
[H,A ] =o,
[H,A+] =o,
[A,A ‘I = sh(Iy)
,
(1.1)
with coproduct:
wH/~~A
+ABeWH/4
hA=e
hAt=e-“‘H/4eAt+Atee”H/4,
hH=1@H+H@1,
(1.2)
the antipod and counit are
y(A)
=
-A,
y(A+)
E(A) =E(A+)
=E(H)
=
-A+,
=O,
y(H)
=
-H,
e(1) = 1.
The quantum content of this structure is clearly given by
the noncocommutative coproduct ( 1.2) while the algebraic
relations ( 1.1) can be obviously redefined, because His central, to yield the classical Heisenberg algebra.
1155
J. Math. Phys. 32 (5), May 1991
The Hopf algebra H( 1) 4 just defined is clearly different
from the algebra of the q-deformed creation and annihilation
operators used in the Jordan-Schwinger map of SU (2) 4;4 as
it has been shown in Ref. 5 the right quantum structure for
these q-deformed operators is B( O( 1) 9. This fact is related to
the following properties of the classical a, a’; they close an
algebra [a,at ] = 1, quantized as our H( 1) 4, but also with
H = 2at a + 1 a super algebra {a,~+ ) = H, [ H,a] = - 2a,
[ H,at ] = 2a’, quantized as B( 011 )9 in Ref. 5, that gives rise
to the Jordan-Schwinger approach to SU(2),.
The purpose of this work is to gain a deeper insight into
the structure of H ( 1) ~. We thus derive the universal R-matrix for H( 1 )4 by a contraction on the R-matrix of the quantum group SU(2),.“*’ We find that the leading R-matrix
term emerging from the procedure turns out to be singular.
This singularity can be discussed and removed and we shall
see that the finite contribution to R gives rise to an operator
Nanalogous to the occupation number operator of the usual
Lie algebra of the harmonic oscillator and indeed satisfies
the same commutation laws with A and A +. However, we see
that, as in the classical harmonic oscillator enveloping algebra, N is not a primitive element, and it is reminiscent of its
quantum structure at the coproduct level: this fact provides
a concrete and instructive example of the very stringent requirements imposed by the Hopf algebra structure. Moreover, by using the explicit form of AN, it is easy to see that the
quasitriangular property of Hopf algebras6 fails to be satisfied. The universal and some parameter-dependent solutions
of the QYBE and CYBE (quantum and classical Yang-Baxter equations) are provided.
The R-matrix so far determined will be used to make an
explicit connection with the formalism of matrix quantum
pseudogroups due to Woronowicz’ and to define the Hopf
algebra of the representative elements. In this framework,
we also determine the infinitesimal generators of the matrix
pseudogroup and we calculate both commutation relations
and coproducts. As a matter of fact, the former turn out to be
coincident with the commutators defining the Heisenberg
Lie algebra h ( 1) : It is again the coproducts that give a clear
indication of the underlying quantum structure.
0022-2488191 I051 155-04$03.00
@ 1991 American Institute of Physics
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ILTHER-MATRIX
FOR H(l), AND ITSCLASSICAL
LIMIT
The universal R-matrix of a quantum group Q is defined by its property of being a sort of intertwining operator
on the coproduct A, namely,
RA(u)R
-’ = ah(u),
u&P,
tions of the coproduct, counit, and antipode of the element
N, that will be of use later on.
Proposition(2.7): In U, [h ( 1) ] we have
A.(N) ewuI 2
(2.1)
wherea(amb)
= b@a.
For SU( 2)y the R-matrix has the well-known expression7
R = /JeJJ
c
(l
-e-‘)k
X (erkJ”’b-J+lk8e
- rkJ.s/2(J-
At=~1’2JJ_,
+e(w’4F’HeH
o
Fob
P
(w’/2e
- &/4~
,“I,
(2.2)
fi=2EJ,,
w=&-‘2,
(&/2&0H/4AIk
.
f)k
(2.4)
nevertheless the E-+O limit of the first exponential is singular, and its meaning must be properly clarified.
A first observation can be immediately made from the
defining equation (2.1) of the R-matrix. Indeed, we see that
the multiplication of R by any central factor leaves (2.1)
invariant. When taking the limit of (2.4), we shall thus neglect any possibly emerging central factor, even a singular
one: This is just the case of the leading term of order E - ’ in
the exponent of exp [ (w/4&) He H] .
A second observation arises then naturally from the previous point. Since the leading singular term of the above
exponential will be neglected because it is central, the finite
contribution to the limit of (2.4) is coming from the next
order of the expansion in E. Setting J3 = H/( 2~) - N, this
contribution
results
in
exp[ - (w/2)
finite
x (Hs N + N@ H) ] as can be deduced by looking at the
limiting procedure yielding, using the relations (2.3), the
representations of H( 1 )4:’
J-
[E-$,I’)
= ([%‘],[&-‘&‘-I’+
J+
/E-&J’)
= ([Y+
1]s)“2/&->,V-
l].,[E-‘p-y]s)1’21E--Ip,~+
I),
I),
(2.5)
J3 IE - ‘p,Y)
= (E -
‘p/2 - Y) I& - ‘p,V),
where s = e’“, p = 2&j, Y = j - m, and Nis the diagonal operator with eigenvalues Y. The explicit expression of N in
terms of A, A *, H is given by the series
N=;A+A
-I
[ 1
sh?
=wAfA
2
,-Ck+l/ZWH,
tA
ciD e -
kwH
(k+3/4)wHAtQpe-(k+I/4)WHA
-Fe-
J. Math. Phys., Vol. 32, No. 5, May 1991
e(N)=O,
(k +3/4,&A
o e - (k + b’4)WH~
t)
(2.6)
computa-
y(N)=
--N-l.
Prooj Indeed, using the definitions, some lengthy but
otherwise straightforward computations show that the Hopf
algebra identities m(idsy)A(N)
= e(N), (e@id)A(N)
=N,and(A@id)A(N)
= (ideA)A(N)
(Ref,9),aresatisfied.
Remark: We observe that U4 [h ( 1) ] is an example of
nonquasitriangular Hopf algebra,’ since the defining relations of the latter,
= R,,R,,
(Aeid)R
= R,,R,,
and (ideA)R
are not satisfied, as can be easily checked using ( 2.7) and
(2.10).
We also observe that A, A +, H, N satisfy the commutation relations
[A,A +] = (2/w) sh(wH/2),
[NJ
] = -A,
[%A+] =A+,
(2.8)
with Hcentral, which constitute a quantum deformation ofa
four-dimensional Lie algebra, that could be also obtained as
contraction of an SU(21, with central extension. In this
point of view N is a primitive
generator, with
A(N)=N691fIsN,~(N)=O,andy(N)=
-N,and
the Hopf algebra becomes quasitriangular.
Proposition (2.9): (i) The nonsingular part of the
contraction of (2.2) leads to the element REU, [h( 1 f ]
Q Uq [h( 1) 1 given by
R
_
e - (to/2)(HsN+
NeH)
W1/2ewH/4A)
kzo$
8
(wl/2e-
W11/4A
- (w/Z)CHeN+
=e
k
(
f)k
NGM)~B~B+
(2.10)
3 t = w1/2
,
3 = WI” exp[wH/4]
and
A
where
xexp[ - wH/4lA +.
(ii) The equation (2.1) is satisfied in U, [h ( 1) I.
(iii) The R-matrix (2.10) solves the QYBE:
(2.11)
&R13&3
= R,&,R,z.
Proof: The point (i) has already been discussed. (ii) and
(iii) are again proved by direct calculations. As far as the
QYBE is concerned, we may observe that
R 12 =e-
R 13 =e
k=O
Equation (2.6) also permits straightforward
1156
(k + I/‘lfwH~
and
(2.3)
and then letting ~-0. Therefore, by a straightforward substitution and neglecting higher orders in E, we find that
R
e -
-l-e-
with ez = q.
We shall determine the R-matrix of H( 1 )9 by contracting (2.2) with the rules that produce H( 1 I9 from SU(2),
(see Ref. I), i.e., by defining
A = &“ZJ +,
+
e-zkck-It/4
[k I!
k>O
(e-‘k+ItluN~e-(k+I/2’~HAtA
k=O
Rz3
=
e-
t~/2’(NeH~l’~-
WZ)WC+N~~~~BIB+~I
(w/Z)(NheleHie-
~ZmfsleN~~BellepB+
f
,
~~/2~(I~N~,H~~-truff~~leH~N~~leBoS’
The problem is thus reduced to proving the equality
Celeghini etal,
1156
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eBaB’ol
eBe
=
exp[wH/Z)eB
exp[ -
eleBeB+eBe
The same formula (2.20) then gives a trigonometric
of the CYBE for H(l),,
where now
eleBeB+
wH/21eBteBeB+el
,
s=;(aea+-a+ea),
which indeed holds true, due to the commutation law
[B,B+ ] = 2 sh(wH/2).
From the universal R-matrix, by standard techniques,”
we can readily deduce a new R-matrix depending on a parameter x. In fact, defining the operator TX by its action on
the generators
T,A=x.4,
TxAt=A+/.x,
t=;(aea++a+@a--hhn-nnh).
As in the SU ( 2)4 case, t /u is the rational solution of (2.19).
III. THE MATRIX QUANTUM GROUP H(l),
T,H=H,
We shall now present the quantum group H( 1)4 as a
matrix quantum group b la Woronowicz.’ We therefore
consider a 3 x 3 matrix of the form
we can define
R(x)
= (T@l)R
=e-(w’2)(HoN+NeH)exBeBt
(2.12)
1
a P
1 S,
(3.1)
0
0
1
(
1
where the matrix elements a, 8, and S generate a C *-algebra
d whose relations are to be determined. This will be done by
first giving a 3 x 3 representation of the generators A, A +, H,
N of U, [h ( 1) 1, with commutation relations (2.8). As a consequence, the R-matrix (2.10) will be represented on the
tensor product and the defining equation”
and again by a direct calculation we have:
Proposition (2.13): The matrix R(x) defined in (2.12)
satisfies the QYBE
R,2 (x)R,,
WV&3
(Y) = A,, (yM13 W)R,,
T=O
(xl.
(2.14)
Corollary (2.15): The classical r-matrix corresponding
to (2.10) is
r=asa+-t(h@n+n@h),
(2.16)
where a, at, h, n are defined to be the classical limit ofA, A +,
H, N, and satisfy the commutation rules
[a,a+J = h,
Moreover
[n,a] = -a,
RT,T, = T,T,R,
(3.2)
with T, = TB 1 and T2 = 1 o T, will provide the required
relations.
Proposition (3.3): A representation of (2.8) is given by
[ n,a+] = a+.
(2.16) solves the CYBE
(2.1 7)
+
[r122r23]
$- [r13yr23]
= O.
[r 129r13]
Similarly, the classical limit of (2.12), with x = e”, is
r(u) = e”asa+ -$(hen+neh)
and solves the parameter-dependent
p(A)=!
9
3,
p(A+)=[
;
;)
p(H)=[
;
i),
p(N) =[
8
3.
(2.1 8)
CYBE
[r,2 (u),r,3 (u + VI]
+ [r,2 (u),r23 to)]
+ [r,3 (u f u),r23 (VI] = 0.
Correspondingly,
(2.19)
13
ProoJ Recalling that the classical r-matrix is obtained
p(R) =
by a first-order expansion in w of the quantum R-matrix, i.e.,
R = 181 + wr,itisimmediatetoverifythat
(2.16)~(2.18)
are the classical counterparts of (2.10)-(2.12)
and (2.14),
respectively.
Remark: The contraction technique can again be used
to produce trigonometric and rational solutions of (2.19).
Indeed it is well known that for SU( 2), such a solution is of
the form
r(u) =s-t+2t/(l
-e’),
0
( 0
(2.10) is represented by the 9 X 9 matrix
w/M
+I
- (d2)pW)
13 - (w/2)&H)
0
0
13
,
1
1, being the 3 X 3 identity matrix.
Prooj Immediate.
In the following we shall suppress the explicit indication
of the representationp and we shall simply denote the representatives of the generators by the generators themselves.
Proposition (3.4): The relations between the generators
of LZ?are the following:
(2.20)
a@ - pa = (w/2)a, aS - Sa = 0,
ps - c5p= - (w/2)&
Pro08 Indeed the explicit form of (3.2) reads
where
s=J,
t=J,
solution
eJ_ -JBJ,,
@J, +J2@J2 +J38J3.
I
T
0
0
1157
aTf wA+T
T- (w/2)HT
0
PT-t wA+6T- (w/2)NT
6T- (w/2)HST
=
T
I(
J. Math. Phys., Vol. 32, No. 5, May 1991
T
0
0
wTA++ Ta- (w/2)TaH
T- (w/2)TH
0
- (w/2)TN+
TS
T
Tfi
Celeghini
etal.
1157
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We shall now use the tensor product of representative elements of type (3.1) in order to define a Hopf algebra structure on
d. Since
lel
0
TeT=
1e/9-t-pe1+CYe
1@8fG@l
181
Isa+aal
lel
0
i
(3.5)
3
and
-a
a&--/3
1
-6 ) ,
1
0
we see that a, ,B, and y with the relations specified in (3.4) have coproduct, counit, and antipode given by
T-lEio
A(a) = 1 @a + a@ 1,
y(a) = -a,
c(a) = 0,
A(B) = leB+Bsl
r(P) = -B+aS,
E(P) =O,
y(S) = - 4
E(S) = 0.
+aeS,
A(S) = 1~6+Sel,
(3.6)
(3.7)
I
The verification of the Hopf algebra identities’ is
straightforward.
Let us recall that in the Woronowicz approach,’ three
infinitesimal generators A,, A, , A,, are defined for SU ( 2) 4,
and they satisfy the relations
q,‘2A,A,
qA,A,
- ( l/q,‘2)AoA2
-
(l/q)A,A,
= A,,
= (1 -kq)A,,
(3.8)
= (1 -I- q)A,.
@,A, - (l/qM,A,
We shall now find the analogous generators and relations for
H(l), by using again the contraction technique. However,
in order to do that, we need an explicit expression ofA,, A,,
A, in termsofJ+ , J- , J3,12
l/4
A, =q
A,
=
_
- 5,,/2
q’/4-
J +,
A, =q,‘2--‘[2J&,
Jd2~-
,
(3.9)
where, as usual, [ 2J3 ] = sh ( zJA ) /sh (z/2 ) and z = log q.
Proposition (3.10): The infinitesimal generators of the
quantum matrix pseudogroup H( 1 )9 obtained by contracting the generators A,, A, , A, of SU (2) I are, respectively,
B. =e- wH’4A, B, = (l/w)(l--e-“H),
& = -e-wH’4At,
(3.11)
and they satisfy the relations
BIB0 - BOB, = 0,
B,&
-BOB2
B, B2 - B,B, = 0,
=B,.
(3.12)
Prooj It is a straightforward evaluation of the contraction limit.
Remark: From (3.12) it appears that the intinitesimal
generators B,, B, , B, satisfy the very same commutation
relations as their classical analogs. Of course the quantum
structure has not disappeared and can be recognized in the
1158
J. Math. Phys.. Vol. 32, No. 5, May 1991
coproduct, where the deformation parameter w enters. In
fact, from ( 3.11)) we have
AB, = (1--~~3,)“~s3~
+B,sl,
+B, ol-wB,
eB,,
A4 =lsB,
AB, = (1-wB,),‘2@BB2
t&c31
and we see that, while the classical generators (w = 0) are
primitive elements in the corresponding Hopf algebra, the
quantum generators are not such.
ACKNOWLEDGMENTS
One of us (R.G. ) would like to acknowledge Professor
E. Taft for a critical reading of the manuscript.
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