J. Phys. A: Math. Gen., 1993, v. 26, L5–L8.
The matrix quantum unitary Cayley-Klein groups.
N.A.Gromov
Komi Scientific Centre, Russian Academy of Sciences,
Syktyvkar 167000, Russia
The purpose of this report is to generalize the R-matrix theory of the quantum groups [1] to the case of all quantum matrix unitary Cayley-Klein groups
which also include the quantum groups of the degenerate hermitic forms.
Let Rq matrix is as in the case of An (or su(n + 1)) [1] , i.e.
Rq = q
n
X
n
X
ekk ⊗ ekk +
ekk ⊗ emm + (q − q −1 )
(ekm )ij = δik δjm ,
ekm ⊗ emk ,
k,m=0
k>m
k,m=0
k6=m
k=0
n
X
k, m, i, j, = 0, 1, . . . , n.
(1)
Let us define the algebra Aq (j) as the associative C-algebra generated by
the noncommutative elements
k ≥ m;
(T (j))km = tkm ,
Jkm = 1, k ≥ m,
Jkm =
m
Y
jr , k < m,
2
(T (j))km = tkm Jkm
,
k<m
jr = 1, ιr , i, r = 1, 2, . . . , n, (2)
r=k+1
where ιr are the dual numbers, which are nilpotent ι2r = 0 and obey the
commutative laws of multiplication ιr ιm = ιr ιm 6= 0, m 6= r, [2],[3], and
factorized by the following relationships:
Rq T1 (j)T2 (j) = T2 (j)T1 (j)Rq ,
(3)
where T1 (j) = T (j) ⊗ I, T2 (j) = I ⊗ T (j). The algebra Aq (j) is called the
algebra of functions on the quantum group GLq (n + 1; j). The matrix T (j)
(2) is a multiplicative matrix whose entries generate Aq (j) and therefore
one has here the matrix quantum group GLq (n + 1; j). If one introduce the
quantum determinant detq T (j) as usual
det T (j) =
q
X
2
2
(−q)l(σ) J0σ
t J 2 t . . . Jnσ
t ,
n nσn
0 0σ0 1σ1 1σ1
σ∈Symm(n+1)
1
(4)
where l(σ) is the parity of the substitution σ , and factorized Aq (j) by the
relationship detq T (j) = 1, than one obtain the algebra F un(SLq (n + 1; j))
of functions on the quantum group SLq (n + 1; j) and the matrix quantum
group SLq (n + 1; j) .
The associative algebra generated by n+1 generators x̃0 , x̃1 , . . . , x̃n devided by the ideal corresponding to the relations x̃k x̃m = qx̃m x̃k , 0 ≤ k <
m ≤ n is called the (n+1)-dimensional quantum vector space Cq,n+1 . The
map
ψ : Cq,n+1 7−→ Cq,n+1 (j)
x = Ψ(j)x̃,
Ψ(j) = diag(1, J01 , . . . , J0n ),
(5)
is in accordance with the following transformations of the quantum vector
Cayley-Klein space Cq,n+1 (j) by the matrix quantum group SLq (n + 1; j)
˙
δ(x) = T (j)⊗x,
δ(xk ) =
k
X
n
X
tkm ⊗ xm +
m=0
2
Jkm
tkm ⊗ xm .
(6)
m=k+1
These transformations represent the homomorphisms of quantum spaces.
Let deformation parameter q ∈ R. The matrix quantum unitary CayleyKlein group SUq (n + 1; j) is defined with help of the involution ∗ :
T t (j)Ψ2 (j)T ∗ (j) = Ψ2 (j),
(7)
where T t is the transpose matrix and transform the quantum hermitic
Cayley-Klein space
Uq,n+1 (j) = {x0 , . . . , xn , y0 , . . . , yn |xk xm = qxm xk ,
yk ym = q −1 ym yk , k < m, yk∗ = xk }
(8)
due to the following equations:
˙
δ(x) = T (j)⊗x,
˙
δ(y) = S(T (j))t ⊗y,
(9)
where S is antipode
2 t
k−m t̃
S(Jkm
km ) = (−q)
mk (j),
t̃mk (j) =
P
σ∈Symm(n)
2 t
2
2 t
(−q)l(σ) J0σ
· · · Jm−1,σ
t
J2
t
· · · Jnσ
,
n nσn
0 0σ0
m−1 m−1,σm−1 m+1,σm+1 m+1,σm+1
σ = (σ0 , σ1 , . . . , σm−1 , σm+1 , . . . , σn ) = σ(0, 1, . . . , k − 1, k + 1, . . . , n).
This antipode matrix S(T (j)) obey the usual property
T (j)S(T (j)) = S(T (j))T (j) = I.
2
(11)
(10)
The transformations (9) keep invariant the quadratic form
∗t
2
x Ψ (j)x =
x∗0 x0
+
n
X
2 ∗
J0k
xk xk .
(12)
k=1
The algebra F un(SLq (n + 1; j)) with the involution (7) is denoted by
F un(SUq (n + 1; j)) and is called the algebra of functions on the quantum
matrix unitary Cayley-Klein group SUq (n + 1; j) . If one introduce on the
algebra F un(SUq (n + 1; j)) the coproduct ∆ and counit ε as follows:
˙ (j),
∆(T (j)) = T (j)⊗T
2
Jkm
∆(tkm ) =
n
X
∆(I) = I ⊗ I,
2 2
Jkp
Jpm tkp ⊗ tpm ,
ε(T (j)) = I,
(13)
p=0
then with the antipode (10) we obtaine the Hopf algebra structure.
If all parameters j are equal to real unit j1 = · · · = jn = 1, then we
are in one to one correspondence with R-matrix theory of the quantum
unitary groups [1]. The pseudohermitic cases jk = 1, i, k = 1, . . . , n with
different signatures have been also regarded by Faddeev at all [1]. But when
some parameters j take the dual values, then the matrix Ψ2 (j) and hermitic
form x∗t Ψ2 (j)x are degenerate. Such cases have been not regarded in the
literature [1],[4]. The corresponding quantum matrix unitary Cayley-Klein
group SUq (n + 1; j) is obtained from the classical quantum matrix unitary
group SUq (n + 1) by (multidimensional) contractions.
For the simplest two dimensional case the Rq matrix is in the form




Rq = 
q
0
0
1
0 q − q −1
0
0
0
0
1
0

0
0
0
q




(14)
and do not depend on the parameter j1 . This Rq matrix is the same for
all three quantum matrix Cayley-Klein groups SUq (2; j1 ), j1 = 1, ι1 , i. (The
corresponding quantum algebras have been regarded in [5] ). We also do
not transform the quantum parameter q under contractions.
The generating matrix T (j1 ) with noncommuting elements is as follows:
T (j1 ) =
t00 j12 t01
t10
t11
3
!
.
(15)
The commutation relations for their matrix elements follow from Eqs. (3)
in the form
t10 t11 = qt11 t10 , j12 t00 t01 = j12 qt01 t00 , j12 t01 t11 = j12 qt11 t01 , j12 t01 t10 = j12 t10 t01 ,
t00 t11 − j12 qt01 t10 = t11 t00 − j12 q −1 t01 t10 ≡ det T (j1 ) = 1.(16)
t00 t10 = qt10 t00 ,
q
The antipode of T (j1 ) is easily obtained and is given by the matrix
t11 −j12 q −1 t01
−qt10
t00
S(T (j1 )) =
!
.
(17)
For real q the Eqs. (7) with T (j1 ) as in Eqs. (15) and the matrix Ψ2 (j1 ) =
diag(1, j12 ) are now in the explicit form
t00 t∗00 + j12 t10 t∗10
j12 (t00 t∗01 + t10 t∗11 )
2
∗
∗
j1 (t01 t00 + t11 t10 ) j12 (t11 t∗11 + j12 t01 t∗01 )
!
=
1 0
0 j12
!
.
(18)
These equation define the involutive matrix T ∗ (j1 ) , which for j1 6= ι1 is as
follows:
!
t11 −j12 q −1 t10
∗
T (j1 ) =
.
(19)
−qt01
t00
Let us regard the contraction j1 = ι1 . Then we have the matrix
T (ι1 ) =
t00 0
t10 t11
!
,
(20)
with the following commutation relations for their entries: t00 t10 = qt10 t00 , t10 t11 =
qt11 t10 , t00 t11 = t11 t00 . From detq T (ι1 ) = t00 t11 = 1 follows t11 = t−1
00
and from the matrix equation (18) we obtaine t00 t∗00 = 1, t11 t∗11 = 1 or
−1
∗
∗
t∗00 = t−1
00 , t11 = t11 = t00 . Therefore we can write done t00 = exp iϕ0 , t00 =
exp (−iϕ0 ), t11 = exp (−iϕ0 ), t∗11 = exp iϕ0 . The generator t∗10 is not expressed by the generators tkm and therefore is independent generator.
The matrix T (ι1 ), the antipode S(T (ι1 )) and the involutive matrix T ∗ (ι1 )
of the contracted quantum matrix unitary group SUq (2; ι1 ) take now the
final forms
T (ι1 ) =
eiϕ0
t10
0
e−iϕ0
!
,
S(T (ι1 )) =
∗
T (ι1 ) =
4
e−iϕ0 0
−qt10 eiϕ0
e−iϕ0
t∗10
0
eiϕ0
!
,
!
,
(21)
with the commutation relations exp (iϕ0 )t10 = qt10 exp (iϕ0 ), t∗10 exp (iϕ0 ) =
q exp (iϕ0 )t∗10 . It will be noted that the involutive matrix T ∗ (ι1 ) (21) is not
obtained from the involutive matrix T ∗ (j1 ) (19) for j1 = ι1 , because of the
Eqs. (19) is valid only for j1 6= ι1 .
We have generalized the R-matrix theory of the quantum unitary groups
[1] to the case of the matrix quantum unitary Caley-Klein groups by defining
the generating T (j) , the antipode S(T (j)) and the involutive T ∗ (j) matrices.
The main feature of the developed approach is that the all quantum CayleyKlein groups have the same Rq matrix, i.e. the quantum (or deformation)
parameter q is not transformed under transitions from one group to another.
As far as contractions correspond to the dual values of the parameters j, we
conclude that the dual numbers are the real tool to regard the contractions
of groups not only in the classical [2],[3], but also in the quantum case (see
also [5], where the contractions of the quantum algebras SUq (2) and SOq (3)
was investigated with help of the dual units).
References
[1] N.Yu.Reshetikhin, L.A.Takhtajan, L.D.Faddeev. Algebra i Analiz,
1989, 1, pp.178–206 (in Russian).
[2] N.A.Gromov. Contractions and analytical continuations of the classical
groups. Unified approach. Syktyvkar, 1990, 220 p. (in Russian).
[3] N.A.Gromov. J.Math.Phys., 1991, 32, N 4, pp.837–844
[4] M.Dubois-Violette, G.Launer. Phys.Lett, 1990, 245, N 2, pp.175–177
[5] N.A.Gromov, V.I.Man’ko. J.Math.Phys., 1992, 33, N 4, pp.1374–1378.
5