## Infinite Analysis Seminar Tokyo

Seminar information archive ～09/13｜Next seminar｜Future seminars 09/14～

Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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### 2011/10/22

13:30-16:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Department of Mathematics) 13:30-14:30

Quantization of Quasimaps' Spaces (joint work with M. Finkelberg) (ENGLISH)

Instituteof Biochemical Physics) 15:00-16:00

Quantum integrable models with elliptic R-matrices

and elliptic hypergeometric series (ENGLISH)

**Leonid Rybnikov**(IITP, and State University Higher School of Economics,Department of Mathematics) 13:30-14:30

Quantization of Quasimaps' Spaces (joint work with M. Finkelberg) (ENGLISH)

[ Abstract ]

Quasimaps' space Z_d (also known as Drinfeld's Zastava space) is a

remarkable compactification of the space of based degree d maps from

the projective line to the flag variety of type A. The space Z_d has a

natural Poisson structure,

which goes back to Atiyah and Hitchin. We describe

the Quasimaps' space as some quiver variety, and define the

Atiyah-Hitchin Poisson structure in quiver terms.

This gives a natural way to quantize this Poisson structure.

The quantization of the coordinate ring of the Quasimaps' space turns

to be some natural subquotient of the Yangian of type A.

I will also discuss some generalization of this result to the BCD types.

Quasimaps' space Z_d (also known as Drinfeld's Zastava space) is a

remarkable compactification of the space of based degree d maps from

the projective line to the flag variety of type A. The space Z_d has a

natural Poisson structure,

which goes back to Atiyah and Hitchin. We describe

the Quasimaps' space as some quiver variety, and define the

Atiyah-Hitchin Poisson structure in quiver terms.

This gives a natural way to quantize this Poisson structure.

The quantization of the coordinate ring of the Quasimaps' space turns

to be some natural subquotient of the Yangian of type A.

I will also discuss some generalization of this result to the BCD types.

**Anton Zabrodin**(Instituteof Biochemical Physics) 15:00-16:00

Quantum integrable models with elliptic R-matrices

and elliptic hypergeometric series (ENGLISH)

[ Abstract ]

Intertwining operators for infinite-dimensional representations of the

Sklyanin algebra with spins l and -l-1 are constructed using the technique of

intertwining vectors for elliptic L-operator. They are expressed in

terms of

elliptic hypergeometric series with operator argument. The intertwining

operators obtained (W-operators) serve as building blocks for the

elliptic R-matrix

which intertwines tensor product of two L-operators taken in

infinite-dimensional

representations of the Sklyanin algebra with arbitrary spin. The

Yang-Baxter equation

for this R-matrix follows from simpler equations of the star-triangle

type for the

W-operators. A natural graphic representation of the objects and

equations involved

in the construction is used.

Intertwining operators for infinite-dimensional representations of the

Sklyanin algebra with spins l and -l-1 are constructed using the technique of

intertwining vectors for elliptic L-operator. They are expressed in

terms of

elliptic hypergeometric series with operator argument. The intertwining

operators obtained (W-operators) serve as building blocks for the

elliptic R-matrix

which intertwines tensor product of two L-operators taken in

infinite-dimensional

representations of the Sklyanin algebra with arbitrary spin. The

Yang-Baxter equation

for this R-matrix follows from simpler equations of the star-triangle

type for the

W-operators. A natural graphic representation of the objects and

equations involved

in the construction is used.