- Email: [email protected]
Super-Yangian double and its central extension
8 September 1997 PHYSICS
Physics Letters A 234 (1997)
ELSEVIER
LETTERS
A
20-26
Super-Yangian double and its central extension Yao-Zhong Zhang ’ Department of Mathematics,
University of Queensland, Brisbane, Queensland 4072, Australia
Received 22 March 1997; accepted for publication Communicated by A.P. Fordy
2 July 1997
Abstract
1. We give their defining We introduce the super-Yangian double DYr,[gl(mln)] and its central extension DY&mln) relations in terms of current generators and obtain Drinfeld co-multiplication. @ 1997 Elsevier Science B.V.
This paper is concerned
with the Drinfeld current realization
[ 1] of the super-Yangian
double DYh[ gI( mln) ]
and its central extension DYh&$%in)]. The Yangian double [2] DYh(p) of a simple bosonic Lie algebra 6 is a quantum double of the Yangian Yh(G) [ 11. It is a deformation of the entire loop algebra and has important applications in massive integrable models [ 3,4]. The Yangian double with center (or central extension of the Yangian double) D=) for G = gZ( n) , sl( n) were introduced in Refs. [5,6] in terms of Drinfeld current generators. The aim of this paper is to introduce the super-Yangian double DYh[gl( m/n) ] and its central extension DYhG[n) 1. This is achieved by generalizing the Reshetikhin and Semenov-Tian-Shansky (RS) construction [7] to the supersymmetric case. Using this super-RS construction and Gauss decomposition [8], we obtain in terms of super current generators. The computation in this paper the defining relations for DYkG\n)] is parallel to our recent work [9] on the Drinfeld current realization of the quantum affine superalgebra uJ&Tb+v’)l ( see also Ref. [ lo] for the special case of m = n = l), which in some sense is a superization of Ref. [Ill. The graded Yang-Baxter equation (YBE) with spectral-parameter dependence has the form R12(U
where R(u)
-
U)R13(U)R23(U)
= R23(U)R13(U)R12(U
-
0)~
(1)
E End( V @$V) with V being a graded vector space and obeys the weight conservation
# 0 only when [(u’] + [p’] + [a] + [p] = Omod2. The multiplication R(u)$’ is defined for homogeneous elements a, b, c, d of a quantum superalgebra by (a@b)(c@d) ’ Queen Elizabeth 03759601/97/$17.00
= (-l)‘b’[C1(.c@bd), II Fellow. E-mail: [email protected]. @ 1997 Elsevier
PII SO375-9601(97)00560-4
Science B.V. All rights reserved
condition
rule for the tensor product
(2)
Z-Z Zhung/Physics Letters A 234 (1997) 20-26
21
where [a] E Zz denotes the grading of the element a. We introduce the graded permutation operator P on the tensor product module V @IV such that P ( ua 18 up) = ( - 1) la1 lP1 (up @Iun) , Vu,, up E V. In most cases the R-matrix enjoys, among others, the following properties, P12R12(~)P12
(3)
= R21(u),
RI~(u)R~I(-u)
= 1.
(4)
The graded YBE, when written in matrix form, carries extra signs [ 13,121,
In Ref. [9] we generalized the RS construction [7] to supersymmetric cases and obtained the Drinfeld realization of the quantum affine superalgebra Us[gl(m]n)“)]. Here we consider the rational solution R(u) to the graded YBE and give a “rational” super-RS algebra: 1. The rational super-RS
Definition
R(u - u)LF(u)L;(u)
algebra is generated by invertible
= L;(u)Lf(u)R(u
R(u+ - u_)L;(u)L;(u)
L*(U) , satisfying
-u).
=L,(u)L~(u)R(u-
-u+),
(6)
where L?(u) = L*(U) @ 1, L:(U) = 1 @L*(U) and u* = uf ;Fic. For the first formula of (6), the expansion direction of R(u - u) can be chosen in u/u or u/u, but for the second formula, the expansion direction must only be in u/u. In matrix form, (6) carries extra signs due to the graded multiplication R( u - u);;~ p(u)$p(u)$ = L*(u)~L*(#&J P
R(u+ - u_)$~
(1
0 $’
(_~)L~‘lw’1+wl) - ufPa”P” ( - 1) [al([Bl+w7)
~+(u)~~,~-(u)~:(_l)[~‘l’la’l+~~l’
= L-(u);L+(u)$‘R(u_
We introduce
rule of tensor products,
- u+)$;:,
(-l)‘a’(‘f11+‘8’1),
(7)
the matrix 8,
= ( -1)‘““%,,,6ppL
(8)
With the help of this matrix 8, one can cast (7) into the usual matrix equation, R(u - u)Lf(u)eL;(u)e
= eL;(u)eLf(u)R(u
R(~+ - u_)L;(u)eq(u)e
= eL;(u)eL;(u)R(u_
- u), - v+).
Now the multiplications in (9) are simply the usual matrix multiplications. We will take R(u) E End( V 8 V) to be Yang’s rational R-matrix associated R(u)
= --&p
+2fiP),
(9)
with superalgebra
gZ(m]n),
( 10)
22
Y-Z Zhang/Physics
where
Letters A 234 (1997) 20-26
V is a (m + n)-dimensional graded vector space. Let basis vectors {ur, ~2, . . . , u,} . .? urn+,} be odd. Then the R-matrix has the following matrix elements,
be even and
{&fl+1, %l+2,.
= (-pq(u)$‘,
R(u)$’
(11) It is easy to check that the R-matrix super-Yangian
L*(u)
has the following
(3)
and (4).
We will construct
the central
extended
Gauss decomposition,
=
e;+,:,(u)
1 f&(u)
x
satisfies
1.
double DY, E/n)
Theorem 1. L*(u)
R(u)
(. ..
:
.
. . . eZt;n,mtn_l (u)
ffyu) .-. .
.
flf,+,W !) ’ fi+n-l‘m+n(u i i
f,$( u) and k’(u)
(i > j)
xl:(u) = f;+Ju+> - &(u-L whereu+=ufiTic,thenc,
are elements
X,‘(u),
decomposition
in the rational
k:(u),
implies
i=1,2
,...,
m+n-1,
double DYhmln)]
that the elements
eifj( u),
determined by L*(u). In the following, we will denote f;+t The following matrix equations can be deduced from (9), R2,(u - v)8L~(u)~L~(u) R21(u_ - v+>BL;(u)t9L~(v)
super-RS
algebra
and kif (u)
XT(u) = ei+,*,i(u-)- ei,l,i(ut>,
relations of the central extended super-Yangian the super-Yangian double DYh [g1( mln) 1. The Gauss
(12)
.
0
where eifi( u), invertible. Let
1
= r;f=(~)OL;(u)t?R~~(u = L;(u)fIL;(u)6’R21(u+
are
(13)
j=1,2
,...,
m+ngivethedefining
which, when c = 0, reduce to those of
f$( u) (i
>
j)
and k’(u)
are uniquely
(u) , eif+,,i( u) as fif (u) , etf( u), respectively.
- v), - v_),
(14) (15)
eL~(U)-1eL~(v)-1R2,(U -u) =R21(~-v)L~(v)-*eL2f(~)-1e,
(16)
eL,f(U)-1eL,(v)-1R2,(U+ -v_) =Rzl(u_ -v+)~,(v)-~e~zf(~)-~e,
(17)
v)eL2f(U)e=eL2f(U)eR2,(UU)~f(U)-l,
(18)
L;(~)-~R~,(u-
E-Z
Zhang/ Physics Letters A 234 (1997)
20-26
23
qtw’R21(*+ - v_)eL,+(u)e = eL;(U)eR*,(U_ - u+)L,(u)-‘,
(19)
- ~+)e~,(u)e=e~z(u)e~~~(~+-v_)~~(~)-~,
L;(“)-~R~,(L
(20)
where R2’ (u - u) = R( v - u)-‘. As in (9), the multiplications in ( 14)-(20) are usual matrix multiplications. Using (8)) ( II), (9), ( 14)-( 20) and Theorem 1, and by calculations parallel to Ref. [ 91, we obtain Dejinition 2. DYhaln)] is an associative algebra over the ring of formal power series in the variable and withDrinfeldcurrentgenerators: Xi’(u), k:(u), i=1,2,...,m+n-1, j= 1,2,...,m+nandacentral element
c. fi)
are invertible.
The grading
of the generators
is: [X;(u)]
c = 0, DYh [ gl( mln) ] reduces to DYh[ g1( mln) 1. The defining relations
are given by
k’(U)kjf(“)
= k;(“)ki(u),
i # j,
k’(u)k,(o)
= k;(u)k;(u),
i < m,
u+ - v- - 2n u +-vu-
+2n
kif(U)ki(“)
‘* - “’ u*-u,+2fi
u_ - v+ - 2n
=
u_-u++2n
k’(“)-‘kF(u)
=
_ ki (v)k+(u),
” - “* uF-u&+2ri
&(u)kT(“)-‘, J
m j,
k;(u)-‘x;(“)k~(Il)
=xZ:(v),
j-i<
k:(u)-‘x;(“)k;(u)
=x+(u),
j-i<-1,
kf(u)-‘X,-(“)k+)
= xl:(v),
j-i>2,
kf(u)-‘x;(u)k;(u)
= Xi+(u),
j-iZ2,
kf(u)-‘X[:(u)k~(u) = kf(u)-‘Xl:(“)kf(u)
uF-“+2fi UT - v
= “‘,
k;‘(u)-‘X*Wkif,(u)
_ xi (“)7
“r--o
2fixI:(“),
= uF-v-2nx;(“), u 7-u UT-u+2Fi _ = xi (“)V
k~‘(U)-‘X~Wk~,(U)
= 1 and zero otherwise.
-1, or
i < m. m
U r---u
U& k'(u)X;(u)k'(u)-'=
k’(u)Xi+(v)k’W’
-u+2fi ** _u
=
u* -u-2zi u* - u
k~,(U)Xif(“)k;‘(U)-’
=
k~,(U)Xif(U)k;,(U>-’
=
k’(u)-lx,-(u)k’(u)
=
k’(u)X;(u)k’(u)-’
=
Xi+(u),
i < m,
Xi’(“),
m
u*-v-22ri u* -u u* -u+2n U& - v
UT - u+2n UT - v
Xi+(“),
i < m,
X,‘(u),
m
_ X, (“),
i=m,
m+l,
X,+(u),
i=m,
m+l,
u* - u + 2h Uf - v
6
When
r-2 Bang/Physics
24
(u -
u ww’(u)x~(u)
= (u -
(u - u * 2@X,‘(c4)X,r(u)
u
Letters
A 234 (1997) 20-26
i < m,
*22n)x3u)x3u),
m
= (u - u r2h)X3u)XF(u),
{x;(~>,x~(u)}=o, (u - u)Xif(u)X,+,,(u)
= (u - u + 2n)Xi+,,(u)Xif(u),
i < m,
(u - u)Xi+(u)X,‘,,(u,
= (u - u - 2fi)X:,,
m
(u)Xi+(u),
(U-u+22li)X~~(u)X~S,(u)
=(u-u)x~;,(u)x~~(u),
i < m,
(u - u - 2fi)xL~(u)x*<,(u)
= (u - u)xi-,I(u)x;(u),
m
[x’(u),X,‘(U)l
i,j
=-2@ij
(fi(U-
-U+)k,+,l(U+)k+(U+)-’
-6(U+
-U_)k,,(u+)k,(u+)-‘),
# m,
vm+(~)xxu))
=2fq@-
- u+)k;+,(u+)k;(u+)-’
where [X, Y] E XY - YX stands for a commutator S(u - u) = c
-6(u+
(21)
- u_>k,+,(u+)k,(u+)-1))
and {X, Y} E XY + Ix for an anti-commutator
l&l-k-’
and (22)
kc2
Serre and extra Serge [ 14,151 relations,
is a formal series, together with the following WWX~h)Xif+,W + {Ul H
u2)
- 2x,f(w)x~,(u)x~(u*) = 0,
i # 171,
= 0,
i#m-1,
v.$(w;,(~*)x~(u) +
{(w
{lq
*
u2)
((4
-u1
+ {u, {(w
(23)
-2x,=~(~,)xif(u)x~,(u,>
-u2
+x~(u)X~*(u,)X~l(U*)}
(24)
- u2 T 2hi) K3~,)x;(~z)x~_,(u) + {Ul +-+ u2)
+x~,(u)x~(u,)x~(u2)}
- 2x~(u,)x~_,(u)x,f(u*)
(25)
+2~)[x~(~l)x,f(~*)x~+,(u)
-~~~(u,)xmf+,(u)x,f(u;!)
+x~+,(u>x~(u,)x,f(u*)]}
c-) 112) = 0,
(26)
-2X;(u,)X;_,(u,)X;(u;!)X~+,(ua)]
r2~)[X~(u,)X~(u2)X~_,(u,)X~+,(u2)
~f~~~-1(~1~~~(~1~~~(~2)~~+1(~2~
+ (u2 -u1
+ x:-1
+ {Ul *
(Ul w;+,
(~2)X,f(~l)X~(~2)1}
k’(u)kjf(u)
= kif(u)k’(u),
k;(u)k;(u)
= k,(u)k:(uL
u
(u)x;(u,)x~(u2)]}
= 0,
r2fN-2X~_,(u,)X~(u,)X~+,(u2)X~(u2) u2)
Remark. For the special case of m = n = 1, we have
z4
+ x2_,
+ - u_ - 2h + _ u_ + 2fik;(u)k,(u)
‘* -” k;(u)-‘k:(u) u* - UT + 2h
i, j=
1,2,
= ‘- 2’ u_ -_ “+ u+ + 2fik;(u)k:(u)v =
” - ‘* uF-u*+frF
k;(u)k;(u)-‘,
= 0.
(27)
E-Z Zhang/Physics Letters A 234 (1997) 20-26
k?(u)-‘X&l)k~(u) = UTUT
k~(u)X;(u)k;(u)-’
{X~(u>,X,(u)}
=
u+2RX;(,), u -
U& -u+2fi U& - u
=2fi(S(u_
Xj%&
- u+)k:(u+)k~(o+)-’
These are the defining relations of DYbz] super-Yangian double DYh[ gE( 111) ] obtained Theorem 2. The algebra by the following formulae. Coproduct h:
25
DYhE]n)]
- qu,
1) 1, which, in Ref. [ 161.
- u-)k,b+)k,(u+)-l).
when c = 0, reduce
given by Definition
(28)
to those of the centerless
2 has a Hopf algebra structure,
which is given
A(c)=c@l+l@c, A@(u))
= kj’(u zt ;ricz) @ ki’(u F $7q),
A(X;(u))
= Xi+(u) @ 1 + @i(u + $1)
j=
1,2 ,...,
@ X”(u + Tic,), i= 1,2,. ..,m+n-
A(X~(U))=~~X~~(U)+X~~(U+~~C~)~~~(U+~~~C~), where CI = c@ 1, c2 = 1 BC, &(u) Counit l :
E(kf(U)) = 1,
E(C) = 0, Antipode
= k,,(u)k,(u)-’
l(Xif(u>)
1,
(29)
and q&(u) = kL,(u)kt(u)-‘.
= 0.
(30)
S:
S(c) = -c, s(x+(u))
mfn,
S(k;(u)) = -&(u
= k;(u)-‘,
- $c)-‘x;(u
- hc),
S(X*‘(U))
=-X~‘(U-ti)$i(U-
ih)-‘.
(31)
Acknowledgement This work is supported by the Australian Research Council, Staff Research Grant and External Support Enabling Grant.
and in part by a University
of Queensland
New
References ]I] [2] [3] [ 41 [5] [6] (71
V.G. Drinfeld, Sov. Math. Dokl. 36 (1988) 212. S. Khoroshkin and V. Tolstoy, Lett. Math. Phys. 36 (1996) 319. D. Bernard and A. L&lair, Nucl. Phys. B 399 (1993) 709. EA. Smimov, hit. J. Mod. Phys. A 7 suppl. 1B ( 1992) 813. S. Khoroshkin, Central extension of the Yangian double, preprint q-alg/960203 1. K. Iohara and M. Kohno, Lett. Math. Phys. 37 (1996) 319. N.Yu. Reshetikhin and M.A. Semenov-Tian-Shansky, Lett. Math. Phys. 19 (1990) 133. [S] J. Ding and I.B. Frenkel, Commun. Math. Phys. 155 (1993) 277. [9] Y.-Z. Zhang, Comments on Drinfeld realization of quantum affine superalgebra Us[gl(mjn)tt)] preprint q-alg/9703020.
and its Hopf algebra
structure,
26
E-Z Zhang/Physics titters A 234 (1997) 20-26
[ IO] J.-E Cai, SK. Wang, K. Wu and W.-Z. Zhao, Drinfeld realization of quantum affine superalgebra
Uq [ex)],
preprint q-aIg/9703022
[ 111H. Fan. E.-Y. Hou and K.-J. Shi, J. Math. Phys. 38 (1997) 41 I. [ 121 Y.-Z. Zhang, On the graded Yang-Baxter and reflection equations, Commun. Theor. Phys. ( 1996). [ 131 A.J. Bracken, G.W. Delius. M.D. Gould and Y.-Z. Zhang, J. Phys. A 27 (1994) 6551. [ 141 M. Scheunert, Lett. Math. Phys. 24 (1992) 173. [ 151H. Yamane, On defining relations of affine Lie superalgebras and their quantized universal enveloping superalgebras, preprint qalg/9603015. [ 161 J.-E Cai, G.-X. Ju, K. Wu and S.-K. Wang, Super Yangian double DY(gl( 111))and its Gauss decomposition, preprint q-alg/9701011,
Physics Letters A 234 (1997)
ELSEVIER
LETTERS
A
20-26
Super-Yangian double and its central extension Yao-Zhong Zhang ’ Department of Mathematics,
University of Queensland, Brisbane, Queensland 4072, Australia
Received 22 March 1997; accepted for publication Communicated by A.P. Fordy
2 July 1997
Abstract
1. We give their defining We introduce the super-Yangian double DYr,[gl(mln)] and its central extension DY&mln) relations in terms of current generators and obtain Drinfeld co-multiplication. @ 1997 Elsevier Science B.V.
This paper is concerned
with the Drinfeld current realization
[ 1] of the super-Yangian
double DYh[ gI( mln) ]
and its central extension DYh&$%in)]. The Yangian double [2] DYh(p) of a simple bosonic Lie algebra 6 is a quantum double of the Yangian Yh(G) [ 11. It is a deformation of the entire loop algebra and has important applications in massive integrable models [ 3,4]. The Yangian double with center (or central extension of the Yangian double) D=) for G = gZ( n) , sl( n) were introduced in Refs. [5,6] in terms of Drinfeld current generators. The aim of this paper is to introduce the super-Yangian double DYh[gl( m/n) ] and its central extension DYhG[n) 1. This is achieved by generalizing the Reshetikhin and Semenov-Tian-Shansky (RS) construction [7] to the supersymmetric case. Using this super-RS construction and Gauss decomposition [8], we obtain in terms of super current generators. The computation in this paper the defining relations for DYkG\n)] is parallel to our recent work [9] on the Drinfeld current realization of the quantum affine superalgebra uJ&Tb+v’)l ( see also Ref. [ lo] for the special case of m = n = l), which in some sense is a superization of Ref. [Ill. The graded Yang-Baxter equation (YBE) with spectral-parameter dependence has the form R12(U
where R(u)
-
U)R13(U)R23(U)
= R23(U)R13(U)R12(U
-
0)~
(1)
E End( V @$V) with V being a graded vector space and obeys the weight conservation
# 0 only when [(u’] + [p’] + [a] + [p] = Omod2. The multiplication R(u)$’ is defined for homogeneous elements a, b, c, d of a quantum superalgebra by (a@b)(c@d) ’ Queen Elizabeth 03759601/97/$17.00
= (-l)‘b’[C1(.c@bd), II Fellow. E-mail: [email protected]. @ 1997 Elsevier
PII SO375-9601(97)00560-4
Science B.V. All rights reserved
condition
rule for the tensor product
(2)
Z-Z Zhung/Physics Letters A 234 (1997) 20-26
21
where [a] E Zz denotes the grading of the element a. We introduce the graded permutation operator P on the tensor product module V @IV such that P ( ua 18 up) = ( - 1) la1 lP1 (up @Iun) , Vu,, up E V. In most cases the R-matrix enjoys, among others, the following properties, P12R12(~)P12
(3)
= R21(u),
RI~(u)R~I(-u)
= 1.
(4)
The graded YBE, when written in matrix form, carries extra signs [ 13,121,
In Ref. [9] we generalized the RS construction [7] to supersymmetric cases and obtained the Drinfeld realization of the quantum affine superalgebra Us[gl(m]n)“)]. Here we consider the rational solution R(u) to the graded YBE and give a “rational” super-RS algebra: 1. The rational super-RS
Definition
R(u - u)LF(u)L;(u)
algebra is generated by invertible
= L;(u)Lf(u)R(u
R(u+ - u_)L;(u)L;(u)
L*(U) , satisfying
-u).
=L,(u)L~(u)R(u-
-u+),
(6)
where L?(u) = L*(U) @ 1, L:(U) = 1 @L*(U) and u* = uf ;Fic. For the first formula of (6), the expansion direction of R(u - u) can be chosen in u/u or u/u, but for the second formula, the expansion direction must only be in u/u. In matrix form, (6) carries extra signs due to the graded multiplication R( u - u);;~ p(u)$p(u)$ = L*(u)~L*(#&J P
R(u+ - u_)$~
(1
0 $’
(_~)L~‘lw’1+wl) - ufPa”P” ( - 1) [al([Bl+w7)
~+(u)~~,~-(u)~:(_l)[~‘l’la’l+~~l’
= L-(u);L+(u)$‘R(u_
We introduce
rule of tensor products,
- u+)$;:,
(-l)‘a’(‘f11+‘8’1),
(7)
the matrix 8,
= ( -1)‘““%,,,6ppL
(8)
With the help of this matrix 8, one can cast (7) into the usual matrix equation, R(u - u)Lf(u)eL;(u)e
= eL;(u)eLf(u)R(u
R(~+ - u_)L;(u)eq(u)e
= eL;(u)eL;(u)R(u_
- u), - v+).
Now the multiplications in (9) are simply the usual matrix multiplications. We will take R(u) E End( V 8 V) to be Yang’s rational R-matrix associated R(u)
= --&p
+2fiP),
(9)
with superalgebra
gZ(m]n),
( 10)
22
Y-Z Zhang/Physics
where
Letters A 234 (1997) 20-26
V is a (m + n)-dimensional graded vector space. Let basis vectors {ur, ~2, . . . , u,} . .? urn+,} be odd. Then the R-matrix has the following matrix elements,
be even and
{&fl+1, %l+2,.
= (-pq(u)$‘,
R(u)$’
(11) It is easy to check that the R-matrix super-Yangian
L*(u)
has the following
(3)
and (4).
We will construct
the central
extended
Gauss decomposition,
=
e;+,:,(u)
1 f&(u)
x
satisfies
1.
double DY, E/n)
Theorem 1. L*(u)
R(u)
(. ..
:
.
. . . eZt;n,mtn_l (u)
ffyu) .-. .
.
flf,+,W !) ’ fi+n-l‘m+n(u i i
f,$( u) and k’(u)
(i > j)
xl:(u) = f;+Ju+> - &(u-L whereu+=ufiTic,thenc,
are elements
X,‘(u),
decomposition
in the rational
k:(u),
implies
i=1,2
,...,
m+n-1,
double DYhmln)]
that the elements
eifj( u),
determined by L*(u). In the following, we will denote f;+t The following matrix equations can be deduced from (9), R2,(u - v)8L~(u)~L~(u) R21(u_ - v+>BL;(u)t9L~(v)
super-RS
algebra
and kif (u)
XT(u) = ei+,*,i(u-)- ei,l,i(ut>,
relations of the central extended super-Yangian the super-Yangian double DYh [g1( mln) 1. The Gauss
(12)
.
0
where eifi( u), invertible. Let
1
= r;f=(~)OL;(u)t?R~~(u = L;(u)fIL;(u)6’R21(u+
are
(13)
j=1,2
,...,
m+ngivethedefining
which, when c = 0, reduce to those of
f$( u) (i
>
j)
and k’(u)
are uniquely
(u) , eif+,,i( u) as fif (u) , etf( u), respectively.
- v), - v_),
(14) (15)
eL~(U)-1eL~(v)-1R2,(U -u) =R21(~-v)L~(v)-*eL2f(~)-1e,
(16)
eL,f(U)-1eL,(v)-1R2,(U+ -v_) =Rzl(u_ -v+)~,(v)-~e~zf(~)-~e,
(17)
v)eL2f(U)e=eL2f(U)eR2,(UU)~f(U)-l,
(18)
L;(~)-~R~,(u-
E-Z
Zhang/ Physics Letters A 234 (1997)
20-26
23
qtw’R21(*+ - v_)eL,+(u)e = eL;(U)eR*,(U_ - u+)L,(u)-‘,
(19)
- ~+)e~,(u)e=e~z(u)e~~~(~+-v_)~~(~)-~,
L;(“)-~R~,(L
(20)
where R2’ (u - u) = R( v - u)-‘. As in (9), the multiplications in ( 14)-(20) are usual matrix multiplications. Using (8)) ( II), (9), ( 14)-( 20) and Theorem 1, and by calculations parallel to Ref. [ 91, we obtain Dejinition 2. DYhaln)] is an associative algebra over the ring of formal power series in the variable and withDrinfeldcurrentgenerators: Xi’(u), k:(u), i=1,2,...,m+n-1, j= 1,2,...,m+nandacentral element
c. fi)
are invertible.
The grading
of the generators
is: [X;(u)]
c = 0, DYh [ gl( mln) ] reduces to DYh[ g1( mln) 1. The defining relations
are given by
k’(U)kjf(“)
= k;(“)ki(u),
i # j,
k’(u)k,(o)
= k;(u)k;(u),
i < m,
u+ - v- - 2n u +-vu-
+2n
kif(U)ki(“)
‘* - “’ u*-u,+2fi
u_ - v+ - 2n
=
u_-u++2n
k’(“)-‘kF(u)
=
_ ki (v)k+(u),
” - “* uF-u&+2ri
&(u)kT(“)-‘, J
m
k;(u)-‘x;(“)k~(Il)
=xZ:(v),
j-i<
k:(u)-‘x;(“)k;(u)
=x+(u),
j-i<-1,
kf(u)-‘X,-(“)k+)
= xl:(v),
j-i>2,
kf(u)-‘x;(u)k;(u)
= Xi+(u),
j-iZ2,
kf(u)-‘X[:(u)k~(u) = kf(u)-‘Xl:(“)kf(u)
uF-“+2fi UT - v
= “‘,
k;‘(u)-‘X*Wkif,(u)
_ xi (“)7
“r--o
2fixI:(“),
= uF-v-2nx;(“), u 7-u UT-u+2Fi _ = xi (“)V
k~‘(U)-‘X~Wk~,(U)
= 1 and zero otherwise.
-1, or
i < m. m
U r---u
U& k'(u)X;(u)k'(u)-'=
k’(u)Xi+(v)k’W’
-u+2fi ** _u
=
u* -u-2zi u* - u
k~,(U)Xif(“)k;‘(U)-’
=
k~,(U)Xif(U)k;,(U>-’
=
k’(u)-lx,-(u)k’(u)
=
k’(u)X;(u)k’(u)-’
=
Xi+(u),
i < m,
Xi’(“),
m
u*-v-22ri u* -u u* -u+2n U& - v
UT - u+2n UT - v
Xi+(“),
i < m,
X,‘(u),
m
_ X, (“),
i=m,
m+l,
X,+(u),
i=m,
m+l,
u* - u + 2h Uf - v
6
When
r-2 Bang/Physics
24
(u -
u ww’(u)x~(u)
= (u -
(u - u * 2@X,‘(c4)X,r(u)
u
Letters
A 234 (1997) 20-26
i < m,
*22n)x3u)x3u),
m
= (u - u r2h)X3u)XF(u),
{x;(~>,x~(u)}=o, (u - u)Xif(u)X,+,,(u)
= (u - u + 2n)Xi+,,(u)Xif(u),
i < m,
(u - u)Xi+(u)X,‘,,(u,
= (u - u - 2fi)X:,,
m
(u)Xi+(u),
(U-u+22li)X~~(u)X~S,(u)
=(u-u)x~;,(u)x~~(u),
i < m,
(u - u - 2fi)xL~(u)x*<,(u)
= (u - u)xi-,I(u)x;(u),
m
[x’(u),X,‘(U)l
i,j
=-2@ij
(fi(U-
-U+)k,+,l(U+)k+(U+)-’
-6(U+
-U_)k,,(u+)k,(u+)-‘),
# m,
vm+(~)xxu))
=2fq@-
- u+)k;+,(u+)k;(u+)-’
where [X, Y] E XY - YX stands for a commutator S(u - u) = c
-6(u+
(21)
- u_>k,+,(u+)k,(u+)-1))
and {X, Y} E XY + Ix for an anti-commutator
l&l-k-’
and (22)
kc2
Serre and extra Serge [ 14,151 relations,
is a formal series, together with the following WWX~h)Xif+,W + {Ul H
u2)
- 2x,f(w)x~,(u)x~(u*) = 0,
i # 171,
= 0,
i#m-1,
v.$(w;,(~*)x~(u) +
{(w
{lq
*
u2)
((4
-u1
+ {u, {(w
(23)
-2x,=~(~,)xif(u)x~,(u,>
-u2
+x~(u)X~*(u,)X~l(U*)}
(24)
- u2 T 2hi) K3~,)x;(~z)x~_,(u) + {Ul +-+ u2)
+x~,(u)x~(u,)x~(u2)}
- 2x~(u,)x~_,(u)x,f(u*)
(25)
+2~)[x~(~l)x,f(~*)x~+,(u)
-~~~(u,)xmf+,(u)x,f(u;!)
+x~+,(u>x~(u,)x,f(u*)]}
c-) 112) = 0,
(26)
-2X;(u,)X;_,(u,)X;(u;!)X~+,(ua)]
r2~)[X~(u,)X~(u2)X~_,(u,)X~+,(u2)
~f~~~-1(~1~~~(~1~~~(~2)~~+1(~2~
+ (u2 -u1
+ x:-1
+ {Ul *
(Ul w;+,
(~2)X,f(~l)X~(~2)1}
k’(u)kjf(u)
= kif(u)k’(u),
k;(u)k;(u)
= k,(u)k:(uL
u
(u)x;(u,)x~(u2)]}
= 0,
r2fN-2X~_,(u,)X~(u,)X~+,(u2)X~(u2) u2)
Remark. For the special case of m = n = 1, we have
z4
+ x2_,
+ - u_ - 2h + _ u_ + 2fik;(u)k,(u)
‘* -” k;(u)-‘k:(u) u* - UT + 2h
i, j=
1,2,
= ‘- 2’ u_ -_ “+ u+ + 2fik;(u)k:(u)v =
” - ‘* uF-u*+frF
k;(u)k;(u)-‘,
= 0.
(27)
E-Z Zhang/Physics Letters A 234 (1997) 20-26
k?(u)-‘X&l)k~(u) = UTUT
k~(u)X;(u)k;(u)-’
{X~(u>,X,(u)}
=
u+2RX;(,), u -
U& -u+2fi U& - u
=2fi(S(u_
Xj%&
- u+)k:(u+)k~(o+)-’
These are the defining relations of DYbz] super-Yangian double DYh[ gE( 111) ] obtained Theorem 2. The algebra by the following formulae. Coproduct h:
25
DYhE]n)]
- qu,
1) 1, which, in Ref. [ 161.
- u-)k,b+)k,(u+)-l).
when c = 0, reduce
given by Definition
(28)
to those of the centerless
2 has a Hopf algebra structure,
which is given
A(c)=c@l+l@c, A@(u))
= kj’(u zt ;ricz) @ ki’(u F $7q),
A(X;(u))
= Xi+(u) @ 1 + @i(u + $1)
j=
1,2 ,...,
@ X”(u + Tic,), i= 1,2,. ..,m+n-
A(X~(U))=~~X~~(U)+X~~(U+~~C~)~~~(U+~~~C~), where CI = c@ 1, c2 = 1 BC, &(u) Counit l :
E(kf(U)) = 1,
E(C) = 0, Antipode
= k,,(u)k,(u)-’
l(Xif(u>)
1,
(29)
and q&(u) = kL,(u)kt(u)-‘.
= 0.
(30)
S:
S(c) = -c, s(x+(u))
mfn,
S(k;(u)) = -&(u
= k;(u)-‘,
- $c)-‘x;(u
- hc),
S(X*‘(U))
=-X~‘(U-ti)$i(U-
ih)-‘.
(31)
Acknowledgement This work is supported by the Australian Research Council, Staff Research Grant and External Support Enabling Grant.
and in part by a University
of Queensland
New
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