[PPT] - Linear ODEs from an algebraic point of view Inna Scherbak (joint with PowerPoint Presentation

SLIDE 1

Linear ODEs from an algebraic point of view

Inna Scherbak (joint with Letterio Gatto) Conference Legacy of Vladimir Arnold

Fields Institute, Toronto November 24 – 28, 2014

SLIDE 2

We solve the generic order linear ODE

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

     

r

page 1

SLIDE 3

We solve the generic order linear ODE indeterminate constant coefficients,

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

     

r

1,..., r

e e

1

( ) [[ ]], [ ,..., ].

r r r

u t B t B e e  

page 1

SLIDE 4

Theorem 1: Let be given by Then is a fundamental system of solutions.

, ,

j r

h B j  

2 1 2

1 . 1 ... ( 1)

j j r r j r

h z e z e z e z



     



( ) , 0 1, !

n j n j n j

t u t h j r n

  

   



We solve the generic order linear ODE indeterminate constant coefficients,

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

     

r

1,..., r

e e

1

( ) [[ ]], [ ,..., ].

r r r

u t B t B e e  

page 1

SLIDE 5

Theorem 1: Let be given by Then is a fundamental system of solutions.

, ,

j r

h B j  

2 1 2

1 . 1 ... ( 1)

j j r r j r

h z e z e z e z



     



( ) , 0 1, !

n j n j n j

t u t h j r n

  

   



We solve the generic order linear ODE indeterminate constant coefficients,

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

     

r

1,..., r

e e

1

( ) [[ ]], [ ,..., ].

r r r

u t B t B e e  

generates this fundamental system:

.

1 ( ) r

u t



( 1 ) 1

( ) ( )

r j j r

u t u t

   



page 1

SLIDE 6

Theorem 1: Let be given by Then is a fundamental system of solutions.

, ,

j r

h B j  

2 1 2

1 . 1 ... ( 1)

j j r r j r

h z e z e z e z



     



( ) , 0 1, !

n j n j n j

t u t h j r n

  

   



We solve the generic order linear ODE indeterminate constant coefficients,

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

     

r

1,..., r

e e

1

( ) [[ ]], [ ,..., ].

r r r

u t B t B e e  

the elementary sym functions in the complete sym functions:

2 1 1 2 1 2

1, , ,... h h e h e e    

1,

,

r

 

1,..., r

e e

generates this fundamental system:

.

1 ( ) r

u t



( 1 ) 1

( ) ( )

r j j r

u t u t

   



page 1

, ,

j

h j  0 ( 0),

j

h j  

SLIDE 7

'

r

X M X 

1

( ) ( )

r

x t X x t



          

( )

( ) ( )

j j

x t u t 

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

     

page 2

SLIDE 8

'

r

X M X 

1 2 3 1 1 2

( 1) ( 1) ( 1)

1 1 1

r r r r r r

r e e e e

M

    

  

                  

1

( ) ( )

r

x t X x t



          

( )

( ) ( )

j j

x t u t 

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

     

page 2

SLIDE 9

'

r

X M X 

1 2 3 1 1 2

( 1) ( 1) ( 1)

1 1 1

r r r r r r

r e e e e

M

    

  

                  

1

( ) ( )

r

x t X x t



          

( )

( ) ( )

j j

x t u t 

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

       

1

( ) exp , (0)

r

c c

X t M t C C X



             

page 2

SLIDE 10

'

r

X M X 

1 2 3 1 1 2

( 1) ( 1) ( 1)

1 1 1

r r r r r r

r e e e e

M

    

  

                  

1

( ) ( )

r

x t X x t



          

( )

( ) ( )

j j

x t u t 

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

       

1

( ) exp , (0)

r

c c

X t M t C C X



             

( )

( ) ( ), (0)

j j

u t x t u c  

page 2

SLIDE 11

Remark: is the Wronski matrix of the standard fundamental system of the solutions to the ODE, , that is (standard initial conditions).

'

r

X M X 

1 2 3 1 1 2

( 1) ( 1) ( 1)

1 1 1

r r r r r r

r e e e e

M

    

  

                  

1

( ) ( )

r

x t X x t



          

( )

( ) ( )

j j

x t u t 

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( )

r r r r r

u t e u t e u t e u t

 

       

1

( ) exp , (0)

r

c c

X t M t C C X



             

( )

( ) ( ), (0)

j j

u t x t u c  

 

exp

r

M t

1

( ), , ( )

r

v t v t

 ( )

, 1

(0) ,

j i ij

i j r

v 

  



page 2

SLIDE 12

Theorem 2: The last column of is our universal fundamental system:

 

1 2 ' ' ' 1 1 ( 1) ( 1) ( 1) 1 1 1 1

exp

r r r r r r r r r

v v v u u v v v t v u M v v

       

                   

exp

r

M t

page 3

1 1 1

, ... , ( ) ( )

k k

k k r k k r r

t t

u t u h t h

     

 

 

! k

SLIDE 13

The universal and the standard fundamental systems are related as follows, and

1 1 2 1 1 2 1 1

1 1 0 1 1

r r

v v h h h h v u u u r r h

  

                                   

          

2 1 1 2 1 1 1 1 1

1 ( 1) 1 ( 1) 1 1

r r r r

u u u r v v r e e e e v e

   

                                       

          

page 4

SLIDE 14

The universal and the standard fundamental systems are related as follows, and

H E HE=I

1 1 2 1 1 2 1 1

1 1 0 1 1

r r

v v h h h h v u u u r r h

  

                                   

          

2 1 1 2 1 1 1 1 1

1 ( 1) 1 ( 1) 1 1

r r r r

u u u r v v r e e e e v e

   

                                       

          

page 4

SLIDE 15

The universal and the standard fundamental systems are related as follows, and

H E HE=I

Generating functions of the complete and the elementary symmetric functions, and

1 1 2 1 1 2 1 1

1 1 0 1 1

r r

v v h h h h v u u u r r h

  

                                   

          

2 1 1 2 1 1 1 1 1

1 ( 1) 1 ( 1) 1 1

r r r r

u u u r v v r e e e e v e

   

                                       

          

page 4 1 1

( ) (1 )

k k k i i

H z h z z 

  

    

1

( ) (1 )

r k r k k i i

E z e z z 

 

    

SLIDE 16

The universal and the standard fundamental systems are related as follows, and

H E HE=I

Generating functions of the complete and the elementary symmetric functions, and , satisfy

1 1 2 1 1 2 1 1

1 1 0 1 1

r r

v v h h h h v u u u r r h

  

                                   

          

2 1 1 2 1 1 1 1 1

1 ( 1) 1 ( 1) 1 1

r r r r

u u u r v v r e e e e v e

   

                                       

          

page 4 1 1

( ) (1 )

k k k i i

H z h z z 

  

    

1

( ) (1 )

r k r k k i i

E z e z z 

 

    

( ) ( ) 1 H z E z  

SLIDE 17

Corollary: The (unique) solution to the Cauchy problem , Solution in the non-homogeneous case, : , where the the particular solution with vanishing i. c

 

2 1 1 1 1

1 1

1

1 ( 1) 1 ( 1) . 1 1

( )

r r r r C

e e e e e r

u u u r

u t c c

 

                      

          

( ) ( 1) ( 2) ( ) 1 2

... ( 1) 0, (0) ,

r r r r j j

r r

u e u e u e u u c B

 

       

page 5

SLIDE 18

Corollary: The (unique) solution to the Cauchy problem , Solution in the non-homogeneous case, : , where the particular solution with vanishing initial conditions.

 

2 1 1 1 1

1 1

1

1 ( 1) 1 ( 1) . 1 1

( )

r r r r C

e e e e e r

u u u r

u t c c

 

                      

          

( ) ( 1) ( 2) ( ) 1 2

... ( 1) 0, (0) ,

r r r r j j

r r

u e u e u e u u c B

 

       

page 5

( ) ( )

C

u t p t 

( ) p t

( ) [[ ]] !

k

r

k k

t g t b B t k



 



SLIDE 19

Corollary: The (unique) solution to the Cauchy problem , Solution in the non-homogeneous case, : , where the particular solution with vanishing initial conditions. Theorem 3: We have where are given by

 

2 1 1 1 1

1 1

1

1 ( 1) 1 ( 1) . 1 1

( )

r r r r C

e e e e e r

u u u r

u t c c

 

                      

          

( ) ( 1) ( 2) ( ) 1 2

... ( 1) 0, (0) ,

r r r r j j

r r

u e u e u e u u c B

 

       

page 5

( ) ( )

C

u t p t 

( ) p t

( ) [[ ]] !

k

r

k k

t g t b B t k



 



( ) , !

k

k k r

t p t p k



 

, ,

k r

p B k r  

2 1 2

. 1 ... ( 1)

k k k

k

k r r r r

k r

b z p z e z e z e z







      



SLIDE 20

Schur function: where a partition of of length , Young diagram of variables.

page 6

1 1

( ... 0) ,

r r

P           

j

    r  { }

i i



 1 r

 



1



, 1

( ) det( ,

)

j i j

i j r

S



 

    





SLIDE 21

Schur function: where a partition of of length , Young diagram of variables.

Examples: The coefficients of the ODE and the coefficients

f the universal solution are related by the Giambelli formula

page 6

1 1

( ... 0) ,

r r

P           

j

    r  { }

i i



 1 r

 



1



, 1

( ) det( ,

)

j i j

i j r

S



 

    





(1 )

k

 

1,..., r

e e { }

j j

h h





(1 )( ).

k

h

e S 

SLIDE 22

Schur function: where a partition of of length , Young diagram of variables.

Examples: The coefficients of the ODE and the coefficients

f the universal solution are related by the Giambelli formula

As the elementary and the complete symmetric functions in , , , where

page 6

1 1

( ... 0) ,

r r

P           

j

    r  { }

i i



 1 r

 



1



, 1

( ) det( ,

)

j i j

i j r

S



 

    





(1 )( ).

k

h

e S 

(1 )

k

 

1,..., r

e e { }

j j

h h





(1 )( )

k

e S  

...

( ) k  

{ } , 1, 0, 0, .

j j j

j j r    



    

( )( ) k k

h S  

1 2

, ,...,

r

  

SLIDE 23

For denote . The Wronskian of : Motivation: The role Wronskians play in Schubert calculus on Grassmannian [papers of L. Goldberg; A. Eremenko and A. Gabrielov; B. and M. Shapiro;

E. Mukhin, V. Tarasov, and A. Varchenko; …]

Generalized Wronskian of corresponding to :

page 7

( ) , 1

[ ]( ) det( ) .

i j i j r

W f t f

  

 ( ) [[ ]] (0 1),

j j r

f f t B t j r     

1 1

( , ,..., )

r

f f f f  

f

SLIDE 24

For denote . The Wronskian of : Motivation: The role Wronskians play in Schubert calculus on Grassmannian [papers of L. Goldberg; A. Eremenko and A. Gabrielov; B. and M. Shapiro;

E. Mukhin, V. Tarasov, and A. Varchenko; …]

page 7

( ) , 1

[ ]( ) det( ) .

i j i j r

W f t f

  

 ( ) [[ ]] (0 1),

j j r

f f t B t j r     

1 1

( , ,..., )

r

f f f f  

f

SLIDE 25

For denote . The Wronskian of : Motivation: The role Wronskians play in Schubert calculus on Grassmannian [papers of L. Goldberg; A. Eremenko and A. Gabrielov; B. and M. Shapiro;

E. Mukhin, V. Tarasov, and A. Varchenko; …]

Generalized Wronskian of corresponding to :

page 7

1 1

( ... 0)

r

         

1

( ) , 1

[ , ]( ) det( ) .

r i

i j r

i j

W f t f  

 

  





( ) , 1

[ ]( ) det( ) .

i j i j r

W f t f

  

 ( ) [[ ]] (0 1),

j j r

f f t B t j r     

1 1

( , ,..., )

r

f f f f  

f f

SLIDE 26

Theorem 4 (Derivatives of the Wronskian in terms of the generalized Wronskians): where .

page 8

( ) | |

[ ] ( ) [ , ]( ),

k k

W f t c W f t

 







1 | |

| |! c k k

 

 

SLIDE 27

Theorem 4 (Derivatives of the Wronskian in terms of the generalized Wronskians): where . (hook formula)

page 8

( ) | |

[ ] ( ) [ , ]( ),

k k

W f t c W f t

 







1 | |

| |! c k k

 

 

4

j

k 

j

SLIDE 28

Theorem 4 (Derivatives of the Wronskian in terms of the generalized Wronskians): where . (hook formula) Let now be a fundamental system of the ODE

page 8

( ) | |

[ ] ( ) [ , ]( ),

k k

W f t c W f t

 







1 | |

| |! c k k

 

 

4

j

k 

j

1 1

( , ,..., )

r

f f f f  

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( ) 0.

r r r r r

u t e u t e u t e u t

 

     

SLIDE 29

Theorem 4 (Derivatives of the Wronskian in terms of the generalized Wronskians): where . (hook formula) Let now be a fundamental system of the ODE Theorem 5: We have [Giambelli’s formula],

page 8

( ) | |

[ ] ( ) [ , ]( ),

k k

W f t c W f t

 







1 | |

| |! c k k

 

 

4

j

k 

j

1 1

( , ,..., )

r

f f f f  

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( ) 0.

r r r r r

u t e u t e u t e u t

 

      [ , ] ( ) [ ] W f S h W f



 

SLIDE 30

Theorem 4 (Derivatives of the Wronskian in terms of the generalized Wronskians): where . (hook formula) Let now be a fundamental system of the ODE Theorem 5: We have [Giambelli’s formula], [Pieri’s formula], the sum over all partitions satisfying

page 8

( ) | |

[ ] ( ) [ , ]( ),

k k

W f t c W f t

 







1 | |

| |! c k k

 

 

4

j

k 

j

1 1

( , ,..., )

r

f f f f  

( ) ( 1) ( 2) 1 2

( ) ( ) ( ) ... ( 1) ( ) 0.

r r r r r

u t e u t e u t e u t

 

      [ , ] ( ) [ ] W f S h W f



 

[ , ] [ , ]

k

h W f W f



  

1 1 1 1

| | | |, ... .

r r

k        

 

      

r

P  

SLIDE 31

Denote the free -module generated by , were a fundamental system of the ODE.

page 8

r

W

{ [ , ], }

r

W f P   

f

SLIDE 32

Denote the free -module generated by , were a fundamental system of the ODE. Let be the Grassmannian, Schubert classes, the Poincare duals.

page 8

r

W

( , ) G G r





*( , )

H G



 

*( , )

H G



 

{ [ , ], }

r

W f P   

f

SLIDE 33

Denote the free -module generated by , were a fundamental system of the ODE. Let be the Grassmannian, Schubert classes, the Poincare duals. Mapping defines a surjection , and its kernel gives relations on generators ( the ring of symmetric functions).

page 8 *( , ) r

B H G 

r

W

( , ) G G r





*( , )

H G



 

*( , )

H G



 

( ) j j

h 

{ [ , ], }

r

W f P   

1

[ ,..., ]

r r

B e e 

f

SLIDE 34

Denote the free -module generated by , were a fundamental system of the ODE. Let be the Grassmannian, Schubert classes, the Poincare duals. Mapping defines a surjection , and its kernel gives relations on generators ( the ring of symmetric functions). Well-known Giambelli’s and Pieri’s formulae of Schubert calculus: where special Schubert classes.

page 8 *( , ) r

B H G 

r

W

1

[ ,..., ]

r r

B e e 

( , ) G G r





*( , )

H G



 

*( , )

H G



 

( ) j j

h 

{ [ , ], }

r

W f P   

( )

( ), ,

k

S

    

       

(1) (2) ( )

(1, , ,...),

j

    

f

SLIDE 35

Denote the free -module generated by , were a fundamental system of the ODE. Let be the Grassmannian, Schubert classes, the Poincare duals. Mapping defines a surjection , and its kernel gives relations on generators ( the ring of symmetric functions). Well-known Giambelli’s and Pieri’s formulae of Schubert calculus: where special Schubert classes. Corollary: The -module isomorphism defined by is an isomorphism of -modules.

page 8 *( , ) r

B H G 

r

W

( , ) G G r





*( , )

H G



 

*( , )

H G



 

( ) j j

h 

{ [ , ], }

r

W f P   

( )

( ), ,

k

S

    

       

(1) (2) ( )

(1, , ,...),

j

    

*( , ) r

H G  W [ , ] W f



 

*( , )

H G

1

[ ,..., ]

r r

B e e 

f

SLIDE 36

From and to bosonic and fermionic spaces :

page 9

r

B

r

W

SLIDE 37

From and to bosonic and fermionic spaces :

We write ;

page 9

( ) , !

n j n j n

t u t h j n

 

 



r

B

r

W

SLIDE 38

From and to bosonic and fermionic spaces :

We write ;

the universal fundamental system of the ODE;
the sub-module of all the solutions of the ODE;

page 9

( ) , !

n j n j n

t u t h j n

 

 



1 1

( , , ... )

r

u u u u

 



1 1

{ ( ), 1} { , ,..., } [[ ]]

r

r j B r r

K Span u t j r Span u u u B t

 

     

r

B

r

W

SLIDE 39

From and to bosonic and fermionic spaces :

We write ;

the universal fundamental system of the ODE;
the sub-module of all the solutions of the ODE;

Theorem 6: We have , where . .

page 9

( ) , !

n j n j n

t u t h j n

 

 



1 1

( , , ... )

r

u u u u

 



1 1

{ ( ), 1} { , ,..., } [[ ]]

r

r j B r r

K Span u t j r Span u u u B t

 

     

1 1

{ : ... , };

r r

r r B r r

K Span u u u u P

   





  

       ( )

O

u S h u

 



1 1

...

O r

u u u u

 

   

r

B

r

W

SLIDE 40

From and to bosonic and fermionic spaces :

We write ;

the universal fundamental system of the ODE;
the sub-module of all the solutions of the ODE;

Theorem 6: We have , where . Corollary: The correspondence gives the -module isomorphism .

page 9

( ) , !

n j n j n

t u t h j n

 

 



1 1

( , , ... )

r

u u u u

 



1 1

{ ( ), 1} { , ,..., } [[ ]]

r

r j B r r

K Span u t j r Span u u u B t

 

     

1 1

{ : ... , };

r r

r r B r r

K Span u u u u P

   





  

       ( )

O

u S h u

 



1 1

...

O r

u u u u

 

   

~

r r r

K  W

r

B [ , ] u W u



 

r

B

r

W

SLIDE 41

We call the r-th Bosonic space; the r-th Fermionic space (of zero total charge)

page 10

1

[ ,..., ]

r r

B e e 

1

: { ..., }

r

r B r r r

F Span u u u P



  

    

SLIDE 42

We call the r-th Bosonic space; the r-th Fermionic space (of zero total charge) We have .

page 10

1

[ ,..., ]

r r

B e e 

1

: { ..., }

r

r B r r r

F Span u u u P



  

    

0 ~

~

r r r r

F K  W

SLIDE 43

We call the r-th Bosonic space; the r-th Fermionic space (of zero total charge) We have . Remark: is well-defined for whereas does not.

page 10

1

[ ,..., ]

r r

B e e 

1

: { ..., }

r

r B r r r

F Span u u u P



  

    

0 ~

~

r r r r

F K  W

r  

r

F

r

W

SLIDE 44

We call the r-th Bosonic space; the r-th Fermionic space (of zero total charge). We have . Remark: is well-defined for whereas does not. If we have ; all are solutions to the “ODE of infinite order”, where are defined in the same way as for ,

page 10

1

[ ,..., ]

r r

B e e 

1

: { ..., }

r

r B r r r

F Span u u u P



  

    

0 ~

~

r r r r

F K  W

r  

r

F

r

W

1 2 1

[ , ,...], { , ,...}

B

B e e K Span u u



  

 

r  

( ) , , !

n j n j n

t u t h j n

 

 



{ }

i i

h

 1

( ) 1, ( ) 1 ( 1) .

n j j n j n j

E t h t E t e t

   

   

 

r  

SLIDE 45

The Boson-Fermion correspondence [this terminology is standard if , see V. Kac, A.K. Raina] .

page 11

SLIDE 46

The Boson-Fermion correspondence is the -module isomorphism defined by [this terminology is standard if , see V. Kac, A.K. Raina] .

page 11

1

... ( )

r r

u u u S h

    

  

r

B

:

r r r

c F B 

r  

SLIDE 47

The Boson-Fermion correspondence is the -module isomorphism defined by [this terminology is standard if , see V. Kac, A.K. Raina] . Denote the elements of defined by the generating function In terms of symmetric functions, .

page 11

1

... ( )

r r

u u u S h

    

  

r

B

:

r r r

c F B 

r

B

2 1 2 1

log(1 ... ( 1) ).

j r r j r j x z

e z e z e z



      



r  

1

...

j j j r

jx     

1

{ }

j j

x



SLIDE 48

The Boson-Fermion correspondence is the -module isomorphism defined by [this terminology is standard if , see V. Kac, A.K. Raina] . Denote the elements of defined by the generating function In terms of symmetric functions, . We have . Thus all are polynomials in

page 11

1

... ( )

r r

u u u S h

    

  

r

B

:

r r r

c F B 

r

B

2 1 2 1

log(1 ... ( 1) ).

j r r j r j x z

e z e z e z



      



1

[ ,..., ]

r r

B x x 

1,...,

, ( 1)

r i

e e h i 

1,...,

.

r

x x r  

1

...

j j j r

jx     

1

{ }

j j

x



SLIDE 49

The Boson-Fermion correspondence is the -module isomorphism defined by [this terminology is standard if , see V. Kac, A.K. Raina] . Denote the elements of defined by the generating function In terms of symmetric functions, . We have . Thus all are polynomials in If , then

page 11

1

... ( )

r r

u u u S h

    

  

r

B

:

r r r

c F B 

r

B

2 1 2 1

log(1 ... ( 1) ).

j r r j r j x z

e z e z e z



      



1

[ ,..., ]

r r

B x x 

1,...,

, ( 1)

r i

e e h i 

1,...,

.

r

x x r  

1

...

j j j r

jx     

1

{ }

j j

x

 1 1

log ( ), ( ) 1 ( 1) .

n j j n j n j

x t E t E t e t

   

    

 

r  

SLIDE 50

We write now

page 12 1

( ,..., ; ), . !

n j n j j r n

t u h u x x t j n

 

  



SLIDE 51

We write now Operator induces a -linear map

page 12

/ :

i i r r

x B B     

: ( 1,..., ).

i r r

K K i r   

1

( ,..., ; ), . !

n j n j j r n

t u h u x x t j n

 

  



SLIDE 52

We write now Operator induces a -linear map Operator acts on by shift,

page 12

/ :

i i r r

x B B     

{ }

j j

u



, 0.

k j k j

D u u k



 

: ( 1,..., ).

i r r

K K i r   

1

( ,..., ; ), . !

n j n j j r n

t u h u x x t j n

 

  



:

k k k r r

d dt

D B B  

SLIDE 53

We write now Operator induces a -linear map Operator acts on by shift, They both can be naturally extended first to operators in , and then in . We denote them and , resp. (recall: ).

page 12

/ :

i i r r

x B B      :

k k k r r

d dt

D B B  

{ }

j j

u



, 0.

k j k j

D u u k



 

: ( 1,..., ).

i r r

K K i r   

1 1

{ , ,..., } { , 1}

r

r B r j

K Span u u u Span u j r



    

r

F

r r

K 

1

( ,..., ; ), . !

n j n j j r n

t u h u x x t j n

 

  



ˆ

i



ˆ k D

SLIDE 54

Theorem 7: For the operators , we have

page 13

ˆ ˆ (1 , )

k r r i

D F F i k r      

SLIDE 55

Theorem 7: For the operators , we have

page 13 1 1

ˆ ( ) ˆ ( ) , ˆ ˆ [ , ] [ , ] .

r r i i r k r k k i i k ik

c c x c D c kx D x kx i

 

            

ˆ ˆ (1 , )

k r r i

D F F i k r      

SLIDE 56

Theorem 7: For the operators , we have Denote the Lie algebra over generated by such that

page 13 1 1

ˆ ( ) ˆ ( ) , ˆ ˆ [ , ] [ , ] .

r r i i r k r k k i i k ik

c c x c D c kx D x kx i

 

            

r

H

{ , }

i r i r

p

   ,

[ , ] , [ , ] 0.

i k i k i

p p i p     ˆ ˆ (1 , )

k r r i

D F F i k r      

dim 2 2.

r

r   H

SLIDE 57

Theorem 7: For the operators , we have Denote the Lie algebra over generated by such that is the oscillator Heisenberg Algebra generated over by such that

page 13 1 1

ˆ ( ) ˆ ( ) , ˆ ˆ [ , ] [ , ] .

r r i i r k r k k i i k ik

c c x c D c kx D x kx i

 

            

r

H

{ , }

i r i r

p

   ,

[ , ] , [ , ] 0.

i k i k i

p p i p     ˆ ˆ (1 , )

k r r i

D F F i k r      



H

{ , }

i i

p



,

[ , ] , [ , ] 0.

i k i k i

p p i p    

dim 2 2.

r

r   H

SLIDE 58

We obtain two one-parametric families of representations of the Lie algebra :

page 14

r

H

SLIDE 59

We obtain two one-parametric families of representations of the Lie algebra : Bosonic representations ,

page 14

:

m r r

B B  

/ , ( ) , ( ) , ( ) . ,

j m m m j j

P x j p P mP P P p P jx j   



           

r

H

:

r

P B  

SLIDE 60

We obtain two one-parametric families of representations of the Lie algebra : Bosonic representations , Fermionic representations ,

page 14

:

m r r

B B  

/ , ( ) , ( ) , ( ) . ,

j m m m j j

P x j p P mP P P p P jx j   



           

r

H

:

r r m F

F  

ˆ , ( ) , ( ) , ( ) . ˆ ,

r j r r r r r m m m j j r

j p m p D j

      

  



                

:

r

P B  

1

... :

r r r r

u u u F

    

     

SLIDE 61

We obtain two one-parametric families of representations of the Lie algebra : Bosonic representations , Fermionic representations , On this way, finite-dimensional counterparts of vertex operators appear [L. Gatto and P. Salehyan, arXiv 13.10.5132]

page 14

:

m r r

B B  

/ , ( ) , ( ) , ( ) . ,

j m m m j j

P x j p P mP P P p P jx j   



           

r

H

:

r r m F

F  

ˆ , ( ) , ( ) , ( ) . ˆ ,

r j r r r r r m m m j j r

j p m p D j

      

  



                

:

r

P B  

1

... :

r r r r

u u u F

    

     