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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$ $

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

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

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