Multiparameter Quantum Deformations of Jordanian Type for Lie Superalgebras

0.14 MB·English

by  V. N. Tolstoy

Multiparameter Quantum Deformations of Jordanian
Type for Lie Superalgebras ∗
V.N. Tolstoy
7
Institute of Nuclear Physics, Moscow State University,
0
0 119992 Moscow, Russia†
2

n
a February 2, 2008
J

3

]
A
Abstract
Q
. WediscussquantumdeformationsofJordaniantypeforLiesuperalgebras. These
h
t deformations are described by twisting functions with support from Borel subalge-
a
m brasandtheyaremultiparameter inthegeneralcase. Thetotaltwists arepresented
in explicit form for the Lie superalgebras sl(m|n) and osp(1|2n). We show also that
[
the classical r-matrix for a light-cone deformation of D = 4 super-Poincare algebra
1
is of Jordanian type and a corresponding twist is given in explicit form.
v
9
7
0 1 Introduction
1
0
7 The Drinfeld’s quantum group theory roughly includes two classes of Hopf algebras: qua-
0
sitriangular and triangular. The (standard) q-deformation of simple Lie algebras belongs
/
h
to the first class. The simplest example of the triangular (non-standard) deformation is
t
a
the Jordanian deformation of sl(2). In the case of simple Lie algebras of rank ≥ 2 some
m
non-standard deformations were constructed by Kulish, Lyakhovsky et al.[1]–[4]. These
:
v deformations are described by twisting functions (which are extensions of the Jordanian
i
X twist) with support from Borel subalgebras, and they are multiparameter in the general
r case. We call their as the deformations of Jordanian type. Total twists of Jordanian type
a
were constructed for all complex Lie algebras of the classical series A , B , C and D .
n n n n
InthispaperwediscussquantumdeformationsofJordaniantypeforLiesuperalgebras.
The total twists are presented in explicit form for the Lie superalgebras sl(m|n) and
osp(1|2n). We show also that the classical r-matrix for a light-cone deformation of D = 4
super-Poincare algebra is of Jordanian type and a corresponding twist is given in explicit
form.
∗This work is supported by the grants RFBR-05-01-01086and ANR grant NT05-241455GIPM.
†Email:[email protected]
1
2 Classical r-matrices of Jordanian type
Let g be any finite-dimensional complex simple Lie superalgebra then g = n ⊕h⊕n ,
− +
wheren aremaximalnilpotentsubalgebrasandhisaCartansubalgebra. Thesubalgebra
±
n (n ) is generated by the positive (negative) root vectors e (e ) for all β ∈ △ (g).
+ − β −β +
The symbol b will denote the Borel subalgebra of g, b := h⊕n . Let θ be a maximal
+ + +
root of g, and let a Cartan element h ∈ h and a root vector e ∈ n satisfies the relation
θ θ +
[h , e ] = e . (1)
θ θ θ
The elements h and e are homogeneous, i.e.
θ θ
deg(h ) = 0, deg(e ) = 0, or 1 . (2)
θ θ
Moreover, lethomogeneouselementse indexedbythesymbolsiand−i,(i = 1,2,...,N),
±i
satisfy the relations
[h , e ] = t e , [h , e ] = (1−t )e (t ∈ C),
θ −i i −i θ i i i i
(3)
[e , e ] = δ e , [e , e ] = 0 , [e , e ] = 0 ,
i −j ij θ ±i ±j ±i θ
provided that
deg(e ) = deg(e )+deg(e ) (mod2). (4)
θ i −i
For the Lie superalgebra g the brackets [·, ·] always denote the super-commutator:
[x,y] := xy −(−1)deg(x)deg(y)yx (5)
for any homogeneous elements x and y.
Consider the even skew-symmetric two-tensor
N
r (ξ) = ξ h ∧e + (−1)deg(ei)deg(e−i)e ∧e (6)
θ,N θ θ i −i
(cid:16) Xi=1 (cid:17)
where
deg(ξ) = deg(e ) = deg(e )+deg(e ) (mod2), (7)
θ i −i
and we assume that the operation ”∧” in (6) is graded:
e ∧e := e ⊗e −(−1)deg(ei)deg(e−i)e ⊗e . (8)
i −i i −i −i i
It is not hard to check that the element (6) satisfies the classical Yang-Baxter equation
(CYBE),
[r12 (ξ),r13 (ξ)+r23 (ξ)]+[r13 (ξ),r23 (ξ)] = 0 , (9)
θ,N θ,N θ,N θ,N θ,N
and it is called the extended Jordanian r-matrix of N-order. Let N be maximal order, i.e.
we assume that another elements e ∈ n , j > N, which satisfy the relations (3), do not
±j +
exist. Such element (6) is called the extended Jordanian r-matrix of maximal order[5].
Consider a maximal subalgebra b′ ∈ b which co-commutes with the maximal ex-
+ +
tended Jordanian r-matrix (6), b′ := Kerδ ∈ b :
+ +
ξδ(x) = [x⊗1+1⊗x, r (ξ)] = [∆(x), r (ξ)] = 0 (10)
θ,N θ,N
2
for ∀x ∈ b′ . Let r (ξ ) ∈ b′ ⊗b′ is also a extended Jordanian r-matrix of the form
+ θ1,N1 1 + +
(5) with a maximal root θ ∈ h′ and maximal order N . Then the sum
1 1
r (ξ,ξ ) := r (ξ)+r (ξ ) (11)
θ,N;θ1,N1 1 θ,N θ1,N1 1
is also a classical r-matrix.
Further, we consider a maximal subalgebra b′′ ∈ b′ which co-commutes with the
+ +
maximal extended Jordanian r-matrix r (ξ ) and we construct a extended Jordanian
θ1,N1 1
r-matrix of maximal order, r (ξ ). Continuing this process as result we obtain a
θ2,N2 2
canonical chain of subalgebras
b ⊃ b′ ⊃ b′′ ··· ⊃ b(k) (12)
+ + + +
and the resulting r-matrix
r (ξ,ξ ,··· ,ξ ) = r (ξ)+r (ξ )+···+r (ξ ). (13)
θ,N;...;θk,Nk 1 k θ,N θ1,N1 1 θk,Nk k
If the chain (12) is maximal, i.e. it is constructed in corresponding with the maximal
orders N,N ,...N , then the r-matrix (13) is called the maximal classical r-matrix of
1 k
Jordanian type for the Lie superalgebra g.
3 Multiparameter twists of Jordanian type
The twisting two-tensor F (ξ) corresponding to the r-matrix (6) has the form
θ,N
F (ξ) = F (ξ)F (σ ), (14)
θ,N N J θ
where the two-tensor F is the Jordanian twist and F is extension of the Jordanian twist
J N
(see[5]). These two-tensors are given by the formulas
F (σ ) = exp(2h ⊗σ ) , (15)
J θ θ θ
N′
FN(ξ) = exp ξ(−1)deg(ei)deg(e−i)ei ⊗e−i e−2tiσθ Fs(σθ)
i=1 (16)
(cid:16)Q N(cid:0)′ (cid:1)(cid:17)
= exp ξ (−1)deg(ei)deg(e−i)ei ⊗e−i e−2tiσθ Fs(σθ),
i=1
(cid:16) (cid:17)
P
where
eθ/2 eθ/2 (eσθ +1)⊗(eσθ +1)
F (σ ) = 1−ξ ⊗ , (17)
s θ eσθ +1 eσθ +1 s 2(eσθ ⊗eσθ +1)
(cid:16) (cid:17)
if θ/2 is a root, e2 = e , N′ = N −1, and
θ/2 θ
F (σ ) = 1 , (18)
s θ
if θ/2 is not any root, N′ = N. Moreover
deg(ξ) = deg(e ) = deg(e )+deg(e ) (mod2) , (19)
θ i −i
3
1
σ := ln(1+ξe ). (20)
θ 2 θ
It should be noted that if the root vector e is odd then σ = 1ξe .
θ θ 2 θ
We can check that the twisting two-tensor (14) defined by the formulas (15)–(20)
satisfies the cocycle equation
F12(∆⊗id)(F) = F23(id⊗∆)(F) (21)
and the ”unital” normalization condition
(ǫ⊗id)(F) = (id⊗ǫ)(F) = 1 . (22)
The twisted coproduct ∆ (·) := F (ξ)∆(·)F−1(ξ) and the corresponding antipode
ξ θ,N θ,N
S for elements in (3) are given by the formulas
ξ
∆ξ(e±σθ) = e±σθ ⊗e±σθ, ∆ξ(eθ/2) = eθ/2 ⊗1+eσθ ⊗eθ/2 , (23)
ξ
∆ξ(hθ) = hθ ⊗e−2σθ +1⊗hθ + 4eθ/2e−σθ ⊗eθ/2e−2σθ
N′
−ξ (−1)degeidege−iei ⊗e−ie−2(tγi+1)σθ, (24)
i=1
X
∆ξ(ei) = ei ⊗e−2tiσθ +1⊗ei, (25)
∆ξ(e−i) = e−i ⊗e2tiσθ +e2σθ ⊗e−i, (26)
Sξ(e±σθ) = e∓σθ, Sξ(eθ/2) = −eθ/2e−σθ, (27)
N′
1
Sξ(hθ) = −hθe2σθ + 4 e2σθ −1 −ξ (−1)deg(ei)deg(e−i)eie−i, (28)
i=1
(cid:0) (cid:1) X
Sξ(ei) = −eie2tiσθ, Sξ(e−i) = −e−ie−2(ti+1)σθ. (29)
If θ/2 is not any root, the third term in (24) and the second term in (28) should be
removed.
The twisted deformation of U(g) with the new coproduct ∆ (·) and the antipode S
ξ ξ
is denoted by U (g).
ξ
In order to construct the twist corresponding to the r-matrix (9) we can not apply the
second twist F (ξ ) directly in the form (14)–(17) to the twisted superalgebra U (g)
θ1,N1 1 ξ
because the deformed coproduct for the elements of subalgebra b′ can be not trivial, i.e.
+
∆ (x) = x⊗1+1⊗x+something, x ∈ b′ . (30)
ξ +
However, there exists a similarity automorphism w which trivializes (makes trivial) the
ξ
twisted coproduct ∆ (·) for elements of the subalgebra b′ , i.e.
ξ +
∆ (w xw−1) := w xw−1 ⊗1+1⊗w xw−1, x ∈ b′ . (31)
ξ ξ ξ ξ ξ ξ ξ +
4
The automorphism w is connected with the Hopf ”folding” of the two-tensor (16) and it
ξ
is given by the following formula (see[5]):
N′
−ξσ
w = exp θ (−1)deg(ei)deg(ei)e e w , (32)
ξ e2σθ −1 i −i s
(cid:16) Xi=1 (cid:17)
where w = exp(1σ ) if θ/2 is a root, and w = 1 if θ/2 is not any root.
s 4 θ s
With the help of the automorphism w the total twist chain corresponding to the
ξ
r-matrix (11) can be presented as follows
F (ξ,ξ ) = F (ξ;ξ )F (ξ) , (33)
θ,N;θ1,N1 1 θ1,N1 1 θ,N
where
F (ξ;ξ ) := (w ⊗w )F (ξ )(w−1 ⊗w−1). (34)
θ1,N1 1 ξ ξ θ1,N1 1 ξ ξ
Here the two-tensors F (ξ) and F (ξ ) are given by the formulas of type (14)–(18).
θ,N θ1,N1 1
Iterating the formula (34) we obtain the total twist corresponding to the r-matrix
(13):
F (ξ,ξ ,...,ξ ) = F (ξ,ξ ,...,ξ ;ξ )···
θ,N;θ1,N1;...;θk,Nk 1 k θk,Nk 1 k−1 k
(35)
×F (ξ,ξ ;ξ )F (ξ;ξ )F (ξ) ,
θ2,N2 1 2 θ1,N1 1 θ,N
where (i = 1,...,k)
F (ξ,ξ ,...,ξ ;ξ ) := (w ⊗w )···(w ⊗w )(w ⊗w )
θi,Ni 1 i−1 i ξi−1 ξi−1 ξ1 ξ1 ξ ξ
(36)
× F (ξ )(w−1⊗w−1)(w−1⊗w−1)···(w−1 ⊗w−1 ).
θi,Ni i ξ ξ ξ1 ξ1 ξi−1 ξi−1
Now we consider specifically the multiparameter twists for the classical superalgebras
gl(m|n) and osp(1|2n).
4 Quantum deformation of Jordanian type for gl(m|n)
Lete (i,j = 1,2,...,m+n)bestandard(n+m)×(n+m)-matrices, where(e ) = δ δ .
ij ij kl ik jl
For such matrices we define a supercommutator as follows
[e , e ] := e e −(−1)deg(eij)deg(ekl)e e , (37)
ij kl ij kl kl ij
where deg(e ) = 0 for i,j ≤ n or i,j > n, and deg(e ) = 0 in another cases. It is easy to
ij ij
check that
[e , e ] = δ e −(−1)deg(eij)deg(ekl)δ e . (38)
ij kl jk il il kj
The elements e (i,j = 1,2,...,N := m+ n) with the relations (38) are generated the
ij
Lie superalgebra gl(m|n).
The maximal r-matrix of Jordanian type for the Lie superalgebra gl(m|n) has the
form[5]
r ξ ,...,ξ = r (ξ )+···+r (ξ ), (39)
1,...,[N/2] 1 [N/2] 1 1 [N/2] [N/2]
(cid:0) (cid:1)
5
where (i = 1,2,...,[N/2])
1
r (ξ ) = ξ (e −e )∧e +
i i i 2 ii N+1−i,N+1−i i,N+1−i
(cid:16) N−i (40)
+ (−1)deg(eik)deg(ekN+1−i)eik ∧ekN+1−i .
kX=i+1 (cid:17)
Consider the first twist corresponding to the r-matrix r (ξ )
1 1
F (ξ ) = F (ξ )F (σ ), (41)
1,N−2 1 N−2 1 J 1
where
F (σ ) = e(e11−eNN)⊗σ1, (42)
J 1
N−1
FN−2(ξ1) = exp ξ1 (−1)deg(e1k)deg(ekN)e1k ⊗ekN e−2σ1 , (43)
(cid:16) Xk=2 (cid:17)
1
σ := ln(1+ξ e ). (44)
1 2 1 1N
The corresponding automorphism w is connected with the Hopf ”folding” of the two-
ξ
tensor (43) and is given as follows
N−1
−ξ σ
wξ1 = exp e2σ11−11 (−1)deg(e1k)deg(ekN)e1kekN . (45)
(cid:16) Xk=2 (cid:17)
It is easy to see that
w e w−1 = e (46)
ξ1 ij ξ1 ij
for all i, j satisfying the condition 2 ≤ i, j ≤ N − 2, therefore (see the formula (31))
deformed coproducts ∆ (·) := F (ξ )∆(·)F−1 (ξ ) for these elements are trivial:
ξ1 1,N−2 1 1,N−2 1
∆ (e ) = e ⊗1+1⊗e , 2 ≤ i, j ≤ N−1. (47)
ξ1 ij ij ij
This means that the automorphism w in the formula (34) for the case gl(m|n) acts
ξ1
trivially andtherefore thetotaltwist corresponding tothe r-matrix(39)is given asfollows
F (ξ ,ξ ,...,ξ ) = F (ξ )···
1,N−2;2,N−4;...;k,N−2k 1 2 k k,N−2k k
(48)
×F (ξ )F (ξ ) ,
2,N−4 2 1,N−2 1
where (i = 1,...,[N/2])
N−i
Fi,N−2i(ξi) = exp ξi (−1)deg(eik)deg(ekN−2i)eik ⊗ekN−2ie−2σi
(49)
k=i+1
(cid:16) (cid:17)
P
×exp (e −e )⊗σ .
ii N−2i,N−2i i
(cid:0) (cid:1)
6
5 Quantum deformation of Jordanian type for osp(1|2n)
In order to obtain compact formulas describing the commutation relations for generators
of the orthosymplectic superalgebra C(n) ≃ osp(1|2n) we use embedding of this superal-
gebra in the general linear superalgebra gl(1|2n). Let a (i,j = 0,±1,±2,...,±n) be a
ij
standard basis of the superalgebra gl(1|2n) (see the previous Section 4) with the standard
supercommutation relations
[a , a ] = δ a −(−1)deg(eij)deg(ekl)δ a , (50)
ij kl jk il il kj
where deg(e ) = 1 when one index i or j is equal to 0 and another takes any value
ij
±1,...,±n; deg(e ) = 0 in the remaining cases. The superalgebra osp(1|2n) is embedded
ij
in gl(1|2n) as a linear envelope of the following generators:
(i) the even (boson) generators spanning the symplectic algebra sp(2n):
e := a +sign(ij)a = sign(ij)e (i,j = ±1,±2,...,±n) ; (51)
ij i−j j−i ji
(ii) the odd (fermion) generators extending sp(2n) to osp(1|2n):
e := a +sign(i)a = sign(i)e (i = ±1,±2,...,±n) . (52)
0i 0−i i0 i0
We also set e = 0 and introduce the sign function: signx = 1 if a real number x ≥ 0 and
00
signx = −1 if x < 0. One can check that the elements (51) and (52) satisfy the following
relations:
[e , e ] = δ e +δ sign(kl)e −δ e −δ sign(kl)e , (53)
ij kl j−k il j−l ik i−l kj i−k lj
[e , e ] = δ sign(k)e −δ e , (54)
ij 0k j−k i0 i−k 0j
{e ,e } = sign(i)e (55)
0i 0k ik
for all i,j,k,l = ±1,±2,...,±n, where the bracket {·,·} means anti-commutator.
The elements e (i,j = 0,±1,±2) are not linearly independent (we have for example,
ij
e = −e ) and we can choose from them the Cartan-Weyl basis as follows
1−2 −21
rising generators : e ,e ,e (1 ≤ i < j ≤ n, 1 ≤ k ≤ n); (56)
i±j kk 0k
lowering generators : e ,e ,e (1 ≤ i < j ≤ n, 1 ≤ k ≤ n); (57)
±j−i −k−k −k0
Cartan generators : h := e (1 ≤ k ≤ n). (58)
i k−k
Maximal classical r-matrix of Jordanian type for the Lie superalgebra osp(1|2n) has
the form[5]
r (ξ ,ξ ,··· ,ξ ) = r (ξ )+r (ξ )+···+r (ξ ) . (59)
1,2,...,n 1 2 n 1 1 2 2 n n
where
n
1
r (ξ ) := ξ e ∧e −2e ⊗e + e ∧e , (60)
i i i 2 i−i ii 0i 0i i−k ik
(cid:16) kX=i+1 (cid:17)
The total twist corresponding to the r-matrix (59) is given as follows
F (ξ,ξ ,...,ξ ) = F (ξ ,ξ ,...,ξ ;ξ )···
1,n;2,n−1;...;n,1 1 n n,1 1 2 n−1 n
(61)
×F (ξ ;ξ )F (ξ ) .
2,n−1 1 2 1,n 1
7
Here (i = 1,...,k)
F (ξ ,...,ξ ;ξ ) := (w ⊗w )···w )(w ⊗w )
i,n+1−i 1 i−1 i ξi−1 ξi−1 ξ2 ξ1 ξ1
(62)
× F (ξ )(w−1⊗w−1)···(w−1 ⊗w−1 ).
i,n+1−i i ξ1 ξ1 ξi−1 ξi−1
n+1−i
Fi,n+1−i(ξi) = exp ξi ei−k ⊗ek,n+1−ie−2σi Fs(σi)eei−i⊗σi, (63)
(cid:16) kX=i+1 (cid:17)
where F (σ ) is defined by the formula (17), and
s i
n+1−i
−ξ σ 1
w = exp i i e e , σ := ln(1+ξ e ). (64)
ξi e2σi −1 i−k k,n+1−i i 2 1 i−i
(cid:16) kX=i+1 (cid:17)
6 Light-cone κ-deformation of the super-Poincar´e
algebra P(3,1|1)
The Poincar´e algebra P(3,1) of the 4-dimensional space-time is generated by 10 elements,
M , N , P , P (j = 1,2,3) with the standard commutation relations:
j j j 0
[M , M ] = iǫ M , [M , N ] = iǫ N , [N , N ] = −iǫ M ,
j k jkl l j k jkl l j k jkl l
[M , P ] = iǫ P , [M , P ] = 0, (65)
j k jkl l j 0
[N , P ] = −iδ P , [N , P ] = −iP , [P , P ] = 0 .
j k jk 0 j 0 j µ ν
The super-Poincar´e algebra P(3,1|1) is generated by the algebra P(3,1) and four real
supercharges Q (α = ±1,±2)) with the commutation relations
α
[M ,Q(±)] = −i(σ ) Q(±),
j α 2 j αβ β
(66)
[N ,Q(±)] = ∓i(σ ) Q(±), [P ,Q(±)] = 0,
j α 2 j αβ β µ α
and moreover
{Q(±), Q(±)} = 0, {Q(+), Q(−)] = 2 δ P −(σ ) P , (67)
α β α β αβ 0 j αβ j
(±) (±(cid:0)) (cid:1)
where we use the denotations Q := Q ±iQ , Q := Q ±iQ , and σ (j = 1,2,3)
1 1 2 2 −1 −2 j
are 2×2 σ-matrices. It should be noted that the spinor Q(+) := (Q(+),Q(+)) transformes
1 2
as the left-regular representation and the spinor Q(−) := (Q(−),Q(−)) provides the right-
1 2
regular one with respect to P(3,1).
Using the commutation relations (65) and (66), (67) it is easy to check that the
elements iN , P := P +P , P , i(N +M ), P , i(N −M ), and Q (α = 1,2) satisfy
3 + 0 3 1 1 2 2 2 1 α
the relations (1)–(4), namely, {h ,e } → {iN ,P }, {e ,e } → {P ,i(N + M )},
γ γ 3 + 1 −1 1 1 2
0 0
{e ,e } → {P ,i(N −M )}, e → Q , e → Q . Therefore the two-tensor
2 −2 2 2 1 ±3 1 ±4 2
r = 1 P ∧(N +M )+P ∧(N −M )+P ∧N +
κ 1 1 2 2 2 1 + 3
(68)
(cid:16)
+2 Q ∧Q +Q ∧Q ,
1 1 2 2
(cid:17)
(cid:0) (cid:1)
8
is a classical r-matrix of Jordanian type. It is called the classical r-matrix for light-cone
κ-deformation of D = 4 super-Poincar´e. Specializing the general formula (16) to our case
P(3,1|1) we immediately obtain the twisting two-tensor corresponding to this r-matrix
F (P(3,1|1) := F (Q )F (Q )F (P(3,1)) , (69)
κ κ 2 κ 1 κ
where F (P(3,1)) is the twisting two-tensor of the light-coneκ-deformation of the Poinca-
κ
r´e algebra P(3,1)
F (P(3,1)) := eκi P1⊗(N1+M2)e−2σ+ eκi P2⊗(N2−M1)e−2σ+ e2iN3⊗σ+ (70)
κ
and the super-factors F (Q ) (α = 1,2) are given by the formula
κ α
(1+eσ+)⊗(1+eσ+) 2 Q Q
F (Q ) = 1+ α ⊗ α , (71)
κ α s 2(1+eσ+⊗eσ+) κ 1+eσ+ 1+eσ+
(cid:18) (cid:19)
1 1
σ := ln 1+ P . (72)
+ 2 κ +
(cid:16) (cid:17)
The formulas (69)–(72) were obtained by a suitable contraction of the quantum deforma-
tion of Jordanian type[6].
References
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[2] P.P. Kulish, V.D. Lyakhovsky and M.A. del Olmo, Journ. Phys. A: Math. Gen. 32,
8671 (1999).
[3] V.D. Lyakhovsky, S. Stolin and P.P. Kulish, J. Math. Phys. Gen. 42, 5006 (2000).
[4] D.N. Ananikyan, P.P. Kulish and V.D. Lyakhovsky, St.Petersburg Math. J. 14, 385
(2003).
[5] V.N. Tolstoy, in: Proc. of Internat. Workshop ”Supersymmetries and Quantum Sym-
metries (SQS’03)”, Eds: E. Ivanov and A. Pashnev, publ. JINR, Dubna, (2004), p.
242; arXiv:math.QA/0402433.
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