universal weight function plan - nested (off-shell) bethe vectors - borel subalgebras in the quantum...
TRANSCRIPT
![Page 1: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/1.jpg)
Universal Weight FunctionUniversal Weight Function, o ,oS.KHOROSHKIN S.PAKULIAK
PlanPlan
- Nested (off-shell) Bethe vectors
- Borel subalgebras in the quantum affine algebras
- Projections and an Universal weight function
o
*
Institute of Theoretical and Experimental Physics, Moscow Laboratory of Theoretical Physics J oint Institute of Nuclear Research, Dubna
- Weight functions in theoryµ1( )NqU gl
Part IPart I Weight functions and the Hierarchical Bethe ansatzWeight functions and the Hierarchical Bethe ansatz
Part II Part II Universal weight function and Drinfeld’s currents Universal weight function and Drinfeld’s currents
![Page 2: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/2.jpg)
Algebraic Bethe Ansatz ( 2gl case)Let be a -operator of some quantum integrable
Due to the RTT relation
is the generating series of quantum integrals of motion.
the transfer matrixtr( ) ( ) ( ) ( )t T A D u u u u
( )L u 2 2 Lmodel associated with
2 2( )gl sl
2 1
( ) ( )( ) ( ) ( ) ( )
( ) ( )M
A BT L L L
C D
L
u uu u u u
u u
1 2 2 1( , ) ( ) ( ) ( ) ( ) ( , )R T T T T Ru v u v v u u v
the problem of finding eigenfunctions for
If we can find a vector
in the form
is reduced to the Bethe equations for parameters
vac such that( ) 0 ( ) ( ) ( ) ( )C A D d vac vac vac vac vacu u a u u u
( )t u1 1
1 1 1
,..., ( ) ( )
( ) ,..., ( ; ,..., ) ,...,
n n
n n n
B B
t
u u u u
u u u u u u u u
L vac
.ku
Part I Weight functions and the Hierarchical Bethe ansatzPart I Weight functions and the Hierarchical Bethe ansatz
![Page 3: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/3.jpg)
The Hierarchical Bethe AnsatzP. Kulish, N. Reshetikhin. Diagonalization of ( )GL N invariant transfer matrices and
quantum N-wave system (Lee model) J.Phys. A: Math. Gen. 16 (1983) L591-L596
(short review)
The Hierarhical Bethe Ansatz starts from the decomposition ( )T uinto blocks
( 1)( ) ( )
( ) (
))
)
((
NN B
D
TT
C
u
uu
u
u
where ( )D u is a scalar, ( )B u -dimensional column and row, and ( 1) ( )NT u monodromy
and ( )C u are ( 1)N is 1N gl
a matrix.
Let , 1,...,V Mm m be representations of with the highest ( ) ( )1( ,..., ).Nm mh h
and auxiliary spaces respectively. The monodromy is now
( , ) ( )N N
R E E E E E E E E E E
ii ii ii jj jj ii ij ji ji iji i j
u v
The problem is to find eigenstates of the transfer matrix( ) tr ( )Wt Tu u
Let 1N
MV V W LH and £ be quantum
N N
Nglweightweight
![Page 4: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/4.jpg)
Let (0) (0)0 1 MV V LH H
The column
fundamental representation of
11
1 1 111 1
1 1 (1) 1 11 ... 1 1
...
( ) ( ) ( ) ( ; ; )N
N NF t B t B t F t t t t n
n
n
iin i i n n
i i
L K
where1 1
(1).... 0F
ni i H f
of the monodromy
1 11 1
1 10
N N
t t
n
HLC C
by the comultiplication
( 1) ( 1) ( 1)( ) ( ) ( )N N NT T T ij kj ikk
u u u
be such that any vector 0f H satisfies
( ) ( ) , ( ) 0, 1,..., 1D f d f C f N iu u u i
( )B u may be considered as an operator valued
1.N gl We look for eigenvectors
( )t u in the form
These vectors are defined by the elements( 1) ( )NT u acting in the tensor product
1,..., 1N ki
1N gl
![Page 5: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/5.jpg)
Let ( )R P u u
In the fundamental
23 131 22 32( ) ( ) (( ; ) (( ) ( ) ( )) ) ( )iT t T tF t t T B t tT T is s s s vac
be a rational -matrix, where P is the
( ) 1, ( ) ( ) , ( ) ( ) 1. u u u u u u u
representation the monodromy matrix acts by the R-matrix
permutation operator
ExamplesExamples
( ) ( )N N
R E E E E E E E E E E
ii ii ii jj jj ii ij ji ji iji i j
u
N Rgl
( )( ( )) ( ) ( ) ( )T e e e ij k ij k kj iu v u v v v
LetFix 3.N 1 21 1. and n n Denote 11t t and 2
1t s
1 2 1 2 12 2 12 1
2 1
23
13
1
2 2 22 1
1 22 2 13 2 1
( ; , ) ( )( ) ( ) ( )
(
+ ( 1) ( ) ( ) +
+
)
( )
( )( ) ( ) ( )
F t t t T T
t
T t
T T
t
T
T
t
t TT
s s s s s s
s s s
s s s
vac
vac
vac
Let 1 21 2. and n n Denote 11t t and 2 2
1 1 2 2, t t s s
![Page 6: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/6.jpg)
Weight function as specific element of Weight function as specific element of monodromy matrix ( case)monodromy matrix ( case)
A.Varchenko, V.Tarasov. Jackson integrals for the solutions to Knizhnik-Zamolodchikov equation, Algebra and Analysis 2 (1995) no.2, 275-313
µ1( )glNqU
1( ) ( ) ( ),NqL U
End C b$u µ1( ) ( )Nq qU U $b glLet
It satisfies the -relation with1 1
1 1( , ) ( , ) ( , ) 1 ( , ) ( , )q q q q
u vu v u v u v u u v v u v
u v u vDefine an element
(1) ( ) ( ,...,1)1 1 1( , ..., ) ( ) ( ) ( , ..., )K M M
M M ML L u u u u u uT R
( ,...,1) ( )1
1
( ,..., ) ( , )MM
M
R
jij i
i j
u u u uR
Let 1, , NKn n be nonnegative integers such that
Rename the variables
µ1( ).NqU gl
1 .N M Ln n
where
is an -operator realization of Borel subalgebra of
RLL
iu
1 2
1 1 2 21 1 1 1{ ,..., } { ,..., ; ,..., ;......; , , }
N
N NM t t t t t t Kn n nu u
11 ( ) ( )( ,..., ) N M
M qU
u u bCT End
L
![Page 7: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/7.jpg)
Example:
1 2 2 1 23 12 2 12 1
2 1 13 12 2 22 1
2 1 13 22 2 12 1
( ; , ) ( , ) ( , ) ( ) ( ) ( )
( , ) ( , ) ( ) ( ) ( ) +
+ ( , ) ( , ) ( ) ( ) ( )
t t t L t L L
t t L t L L
t t L t L L
s s s s s s
s s s s
s s s s
B
1 2 21 2 1 1 1 2 22, 3, 1, 2, , , N M t t t t n n s s
Define an element
1
1
1 11 1 21 1,(( ) )( ( ,..., ; ; ,..., ) 1)N
N
M N NN Nt t t t E E K L nn
n nTtr id
1
1 1
( ) ( , )N
N
t t t
a aj i
a i j
B
( )( ) qUt $B b
1 2 1 2 23 12 2 12 1
2 13 12 2 22 1
1 13 22 2 12 1
( ; , ) ( )( ) ( ) ( ) ( )
( 1) ( ) ( ) ( ) +
+ ( ) ( ) ( ) ( )
F t t t T t T T
t T t T T
t T t T T
s s s s s s
s s s
s s s
vac
vac
vac
In the case of the rational In the case of the rational RR-matrix-matrix
![Page 8: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/8.jpg)
a vector-valued weight function of the weight
associated with the vector
Let be the set of indices of the simple roots
for the Lie algebra
A -multiset is a collection of indexes together with a
map , where and We
associate a formal variable to the index
Weight function
.v1 , )N N N n n n
Let be -module. A vector is called a weight
singular vector with respect to the action of if
for and If we call
( )qU b$V Vv( ),qU b$ ( ) 0L ij u v
1 1N j i ( ) ( ) .L ii iu v u v ,Vv
( ) ( )V t tw B v
1( ,...,N Ln n
{1,..., }N
1{ ,..., }N 1.Ngl
( , )a i
.I i ¢: ,I ( )i atai
a
( , ).a i
![Page 9: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/9.jpg)
that is, is a formal power series over the variables
2 1 3 2 1, ,..., , 1t t t t t t t
n n ni i i i i i i
For any -multiset we choose the formal series where
I
32
1 1 1
1 2 1
1 1 1,, ,,{ ,..., } ( )[ , ,..., , ]q
tt tU t t U t t t t
t t t t
n
n n n
n n
ii ii i i i i i
i i i i
b1 1
( ,..., ) { ,..., }, ,W t t U t t I n ni i i i ki
1( ,..., )W t t
ni i
with coefficients in the ring of polynomials :1 1
1 1( )[ , , , , ]qU t t t t
n ni i i ib
• For any representation with singular weight vector V v
1 1( ,..., ) ( ,..., )V t t W t tw
n ni i i i v
converges to a meromorphic -valued weight function.V
• If then and0I 1W .V w v• If are two weight singular vectors, then is a
weight singular vector in the tensor product and for
any -multiset the weight function satisfies the
recurrent relation
1 2, v v 1 2 v v
1 2V V I
1 2( )V V tw
![Page 10: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/10.jpg)
1 2 1 1 2 2
1 2
({ | }) ({ | }) ({ | })V V I V I V II I I
t t t
w w w
i i i i i i
1 2 2, 1 1, 2
1 1(2) (1)
11 , ,
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )N
I I I I I I
q t qt qt q tt t
t t t t
a a a ai j i ja a
a i a i a a a aa i i i j i j i j i ji j i j
i a i a i j a i j a
2, 1 2, 1
1 1 1
1 1 1 , ,
( ) , ( ) 1 ( ) 1, ( )j I I I I
qt q t t t
t t q t qt
a a a ai j i j
a a a ai i j i j i ji j i ji a j a i a j a
(1) (2)1 1 2 2( ) ( )L t L t aa a aa av v v v
We call the element theµ1 1( ,..., ) ( ) ( )Nq qW t t U U $
ni i b gl
Universal Weight Function
![Page 11: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/11.jpg)
Chevalley description: 1, , , 0,1,...,e f
i i ik i r
1
11
, , etc.k
f k q f e fq q
iij i
i i jj i j
ai ij
i i
kk
Standard Hopf structures:
1) ( 1 , etc.f f f
i i ii i i i
k k k k
Current (Drinfeld’s «new») realization of ( ), 1,..., :qU i rg$
0
[ ]( ) [ ] ( ) ( ) [ ]e e f f
m
i i i i i i
k k k
k k k
z k z z k z z k zy yZ Z
( , ) ( , )( ) ( ) ( ) ( ) etc.q e e e e q i j i j
i j j iz w z w w z z w
Current Hopf structure: D D( ) 1 ( ) ( ) ( ) , ( ) ( ) ( ) etc.f f f
i i i i i i i
z z z z z z zy y y y
Different realizations of the QAA $gU Uq= ( )
Part II Universal weight function and Drinfeld’s current
![Page 12: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/12.jpg)
Here are the operators of the adjoint action:
1 1( ) [0] [0 ] ( ) [0] [0]e e f f
i i i i i i i ii iS x x k xk S x x k xk
iS
Relation between the two realizationsRelation between the two realizations
Let be the longest root of the Lie algebra0 1
r
r rin g.
Let be The assignment ge
[0] [0] 1,...,e e e f a ai i i i
i r
0
1
1
a a i
i i i
rn
i
k k k k k
0 1 2 0 1 2( [1]) ( [ 1])e f e e
a L a L
n j n ji i i i i iS S S S S S
establishes the isomorphism of the two realizations.
1 2
[ ,[ ,...[ , ]...]]. i i i jn
e e e e e
![Page 13: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/13.jpg)
Different Borel Subalgebras in Different Borel Subalgebras in ( )qU g$
1 1( ) ( ) ( ) :{ , } ( ) :{ , }q q q qU U U e U e
$ $ $ $i i i ik kb g b b
( ) :{ } ( ) :{ } 0,1,...,q qU e U e $ $i i
i rn n
( ( )) ( ) ( ) ( ( )) ( ) ( )q q q q q qU U U U U U n b n n n b
We call the STANDARD Borel subalgebras of( )qU b$ ( )qU g$
1:{ [ ], ; , [ ], 0; 1,..., } ( )F qU f U $
i i in n k n n i r gy¢
1:{ [ ], ; , [ ], 0; 1,..., } ( )E qU e U $
i i in n k n n i r gy¢
We call and CURRENT Borel subalgebras ofFU ( )qU g$EU
:{ [ ], } :{ [ ], }f F e EU f U U e U i in n n n¢ ¢
( ) ( )( ) ( )D Df f F e E eU U U U U U
( ) ( ) ( )f F q F q F F qU U U U U U U U - - +b n bI I I
( ) ( ) ( )e E q E q E E qU U U U U U U U + + -b n bI I I
![Page 14: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/14.jpg)
of linear spaces;
(i) The algebra admits a decomposition that is, theA 1 2,A A Amultiplication map establishes an isomorphism1 2: A A A
Let be a bialgebra with unit 1 and counit We say that itsA .subalgebras and determine an orthogonal decomposition 1A 2A
,Aof if
Orthogonal decompositions of Hopf algebrasOrthogonal decompositions of Hopf algebras
1 1 1 2 1 2 2 2 1 2 1 2 1 1 2 2: ( ) ( ) : ( ) ( ) P P P P aa a a aa a a a aA A(1) (2) (1) (2)
1 2( ) ( ) ( )=P P a a a a a afor
: ( ) ( ) ( ) ( )
: ( ) ( ) ( ) ( )
F F F f
F F f F
U U U U P f f f f P f f f f
U U U U P f f f f P f f f f
A
A
ýý
curves. Israel J.Math 112 (1999) 61-108 B. Enriquez, V. Rubtsov. Quasi-Hopf algebras associated with and complex2sl
(ii) is a left coideal, is a right coideal:1A 2A
1 1 2 2( ) ( ) A A A A A A
![Page 15: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/15.jpg)
a system of simple roots of a Lie algebra The Universal The Universal
1 1 1( ) ( )( ,..., ) ( ( ) ( ))t t P f t f t Ln n ni i i i i iW
given by the projectionweight function weight function for the quantum affine algebra is ( )qU
$g.g
1{ ,..., }I ni iLet be an ordered -multiset, where is
Projections and the weight functionProjections and the weight function
[ ], , J U e n n Z
elements where[ ],e n , n Z
( )( ( )) ( ) ( ) mod DP f P P f U J
Theorem. Theorem. Let be a left ideal of generated by theJ ( ),qU$g
Let be a singular weight vector in a highest weight vrepresentation of then V ( ),qU g$
1 1( ,..., ) ( ,..., )t t t t
n nV i i i iw vW
is a meromorphic -valued weight functionV ( )qU$g
![Page 16: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/16.jpg)
Coproduct property of the weight functionCoproduct property of the weight function
representations with singular vectors They are eigen-
( )( ) ( ) 1,..., 1,2t t i
kk i kv v i r ky
vectors of the Cartan currents
1 2, .v v1 2V V V Let be the tensor product of highest weight
1 2 2
1 2
({ | } ) = ({ | }) ({ | })I I V II I I
t t t
1V i i V i i i iw w w
( ), ( )
( ), ( )
2 2, 1
( )
(1)( )
1 , 1 , ( ) ( ) , ( )
( )i j
i j
i j
I b j I I i jj b
q t tt
t q t
r r
a ia i a i i j
i a i a
( )qU$g
( , )D 1
( , )( ) 1 ( ) ( ) ( ) ( ) ( )( ( )) ( )q
f f f f fq
i j
i i i i i j i ji j
z wz z z z z z z z
z wy y y
![Page 17: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/17.jpg)
µ1( )NqU gl weight function as a projectionweight function as a projection
representation of µ1( ).NqU glV
vLet be a singular weight vector in a highest weight
1
1 11 1 1 1( ) ( ) ( ) ( ) ( )
N
N NN Nt P F t F t F t F t L L LV n nw v
1 1
1 1 2 1
( ) ( ) ( ) ( )N N
V
q t qtt t t t
t t
w a
a
a a ni j a
V a ia aa i j n a ii j
v w vB v
Then the weight function is equal toµ1( )NqU gl
( ) ( )L t taa av v
![Page 18: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/18.jpg)
Ding-Frenkel isomorphismDing-Frenkel isomorphism
affine algebra Comm.Math. Phys. 156 (1993) 277-300
J. Ding, I. Frenkel. Isomorphism of two realizations of quantum µ
1( ).NqU gl
2,1
1,1 1,
1
( ) 0( )
1
( ) ( ) 1
q
N N N
eU
e e
O $M O
L
bz
z z
1, 1, , 1 , 1( ) ( ) ( ) ( ) ( ) ( )E e e F f f i i i i i i i i i iz z z z z z
11,2 1, 1
, 1
1
(1 ( (
01( )
00 (
(1
N
N N
N
f f
Lf
k z)z) z)
zz)
k z)
, 1 , 1( ( )) ( ) ( ) ( ( )) ( ) ( )q qP F f U P F f U i i i i i iz z z zb b$ $
![Page 19: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/19.jpg)
Composed currents and projectionsComposed currents and projections
, 1, , 1( ) res ( ) ( )d
F F F i j s j i sw=z
wz z w
w
11 1( ) ( ) ( ) ( ) ( )( )q F F F F i i i iz-qw z w w z z-w
, 1( ) ( )F t F t i i i
1 2, [0] [0] [0] 1( ( )) ( ( ))F F FP F t F t
L
i i ji j jS S S
1 1, ,( ( )) ( ) ( )P F t q q f t j i
i j i jv v
Define the screening operators ( )B A AB qBA S
1, 1 1,
0
( ) [ ] ( ) ,q q F F
ki s s j
k
k z z
1, , 1 , 1 1, ,( ) [0]) ( )( [0]F F F qF F s j i s i si j s j zz z
1
1, , 1 1 ,, 1,( ) ( ) ( ) ( () ) ( )q q
F F F F F
s j i s i s s j i j
z wz w w z z w
1- wz
z
, 1,..., 2. s i i j
µ1( )q NU gl
![Page 20: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/20.jpg)
Calculation of the projectionsCalculation of the projections
, , , , , , ,0 0
( ) ( ) ( ), ( ) [ ] , ( ) [ ] F F F F F F F
n ni j i j i j i j i j i j i j
n n
z z z z n z z n z
1
1, , 1 , 1 1,( ) ( ) ( ) ( )q q
F F F F
s j i s i s s j
z wz w w z
z-w1
, 1 1, ,
( )( ) ( ) ( )
q qF F F
i s s j i j
z zw z z
z w z w
First step 22 1 3 2
, 1 2,3 1,2 , 1 3,4 1,3 2 1( ( ) ( )) ( ( )) ( ( ) ( ) ( )) N N
N N N N
t
t tP F t F t P F t P F t F t F t
L L
Let us calculate using2 1, 1 2,3 1,2( ( ) ( ) ( ))N
N NP F t F t F t L
1 2 1 2 1 2( ( )) ( ) ( ) ( ( ) ) 0P FP F P F P F P P F F and
2 2 1 2 2 21,3 1,3 1,2 2,3 1,3( ) ( ( )) ( )( ( ) ( )) ( )F t P F t q q F t F t F t
1, 1 1,2( ( ) ( ))N
N NP F t F t L121
1, 1 , 1 1, 1
2 1
( ( ) ( )) ( ( ))N
NN N
tP F t F t P F t
t t
jmm m
m m m j jm j
After -steps:N
![Page 21: Universal Weight Function Plan - Nested (off-shell) Bethe vectors - Borel subalgebras in the quantum affine algebras - Projections and an Universal weight](https://reader036.vdocument.in/reader036/viewer/2022062422/56649e735503460f94b7285f/html5/thumbnails/21.jpg)
1 1,2 2 1, 1 1
2
, 1 1
1
( ) ( ) ( ) ( ) ( )
0 ( )( )
0 ( ) ( )
( )
N N
N N N
N
t f t t f t t
tL t
f t t
t
L
O M
O
k k k
kv v
k
k
Taking into account
we conclude that
1 1, 1 1 1
1
( ,..., ) ( ( ) ( )) ( )N
N NN Nt t P F t F t t
L jj
j
v k vB
The element for a collection of times satisfies:( )tB 1{ ,..., }Nt t
1111 1 1 1
1 , 12 2
( )( ,..., ) ( ,..., ) ( ) ( )
NN N q q t
t t t t L t L tt t
jmm m j
m j j j jm j
v vB B