Last updated Oct 11

KIAS

International Conference on

Lie algebras and related topics

# Oct 20-23, KIAS International Conference room

## Invited speakers (alphabetical order)

Henning Haahr Andersen  (Aarhus U.)

Susumu Ariki (RIMS)

Georgia Benkart (U. Wisconsin)

Michio Jimbo (U. Tokyo)

Seok-Jin Kang (KIAS)

Jae-Hoon Kwon (U. Seoul)

Kyu-Hwan Lee (U. Toronto)

Hyeonmi Lee (KIAS)

Olivier Mathieu (U. Lyon)

Tetsuji Miwa (U. Kyoto)

Young-Tak Oh (KIAS)

Arun Ram (U. Wisconsin)

Toshiyuki Tanisaki (Osaka City U.)

Efim Zelmanov (KIAS, U. C. San Diego)

Titles and abstracts

Henning Haahr Andersen

Hochschild cohomology for the plus part of a quantum group.

We present some computations for the derived functors of the fixed point functor for the plus part of a quantum group.

In the generic case these results are easy to obtain but at roots of unity the situation is much more complicated. We shall give some general methods as well as some precise results for special cases. There are connections to the cohomology of line bundles on the cotangent bundle for flag varieties.

Susumu Ariki

Recent advances in modular representation theory of Hecke algebras

I will explain advances in our field after my book was published.

I will explain two results on the classification of simple modules, one by J. Hu and the other by N. Jacon.

I also explain classification of representation type, a result by the speaker, and generalization of Gyoja's theorem by Leclerc and Miyachi.

Georgia Benkart (U. Wisconsin)

A Look at locally finite Lie algebras

Tits construction and Freudenthal's Magic Square

The well-known Tits construction provides models of the exceptional simple Lie algebras using, as ingredients, a unital

composition algebra and a degree three central simple Jordan algebra. By varying both ingredients, Freudenthal's Magic Square is obtained.

This construction will be reviewed and extended to get models of the exceptional simple classical Lie superalgebras.

This is joint work with Georgia Benkart.

On the other hand, since degree three Jordan algebras are obtained as 3x3 matrices over unital composition algebras, Tits

construction can be interpreted as a construction based on two unital composition algebras.

But, even though the construction is not symmetric on the two algebras, the outcome (the Magic Square) is.

Several more symmetric construction have been proposed, based on the triality phenomenon.

Here a new construction of the Magic Square will be presented, based on a pair of the so called

symmetric composition algebras'', which provide very simple formulas for triality.

Michio Jimbo (U. Tokyo)

Form factors and representations of $U_{\sqrt{-1}}(\widehat{sl}_2)$

Local fields in massive integrable quantum field theory are described by a tower of meromorphic functions called form factors. One of the basic issues is to classify all local operators in the theory and establish a one-to-one correspondence with those in the limiting conformal field theory.

We report some recent progress on this topic, based on representations of the quantum affine algebra $U_q(\widehat{sl}_2)$ at $q=\sqrt{-1}$. This is a joint work with T.Miwa, Y.Takeyama, E.Mukhin, B.Feigin and M.Kashiwara.

Seok-Jin Kang (KIAS)

Crystal Bases for Quantum Generalized Kac-Moody Algebras

We develop the crystal basis theory for quantum generalized Kac-Moody algebras.

For a quantum generalized Kac-Moody algebra $U_q$, we first introduce the category $O_{int}$ of $U_q$-modules and prove its semisimplicity. Next, we define the notion of crystal bases for $U_q$-modules in the category $O_{int}$ and for the subalgebra $U^{-1}_q$. We then prove the tensor product rule and the existence theorem for crystal bases.

Finally, we construct the global bases for $U_q$-modules in the category $O_{int}$ and for the subalgebra $U^{-1}_q$.

Jae-Hoon Kwon (U. Seoul)

Crystal bases of Fock space representations and combinatorics of partitions

We generalize the classical method of abacus in the theory of partitions to the case of the crystal bases of Fock space representations of quantum affine algebras. We establish a natural bijection between the crystal basis of a Fock space and a certain product of partitions sets, and hence compute a string function of the corresponding basic representation in a combinatorial way.

Hyeonmi Lee (KIAS)

Crystal graphs and Young walls

We introduce new realizations of higher level perfect crystals and crystal graphs of quantum affine algebras, using combinatorics of Young walls. The notions of slice and splitting of blocks play crucial roles in the constructions.

Kyu-Hwan Lee (U. Toronto)

Spherical Hecke algebras of algebraic groups over 2-dimensional local fields

The theory of higher local fields was launched by K. Kato and A. N. Parshin independently in the attempt of generalizing the class field theory. An example of algebraic groups over 2-dimensional local fields is a p-adic loop group, whose central extension is a Kac-Moody group. We will construct spherical Hecke algebras of algebraic groups over 2-dimensional local fields and prove the Satake isomorphism of the spherical Hecke algebras. This is a joint work with Henry Kim.

Olivier Mathieu (U. Lyon)

Tetsuji Miwa (U. Kyoto)

A space of coinvariants and Kostka polynomials

Young-Tak Oh (KIAS)

R-analogue of the Burnside ring of profinite groups and free Lie (super)algebras

The classical construction of Witt vectors can be understood as a special case of a construction

which can be defined relative to any given profinite group in terms of  its (completed) Burnside ring,

due to A. Dress and C. Siebeneicher

In this talk we will generalize their works by constructing the R-analogue of the Burnside ring of profinite groups,

where R is any finite dimensional torsion-free special $\lambda$-ring.
In particular we remark that the (Grothendieck) Lie-module denominator identity of free Lie algebras

is closely related to the canonical isomorphism between the $R$-analogue and Grothendieck's ring of formal power series
with coefficients in $R$ and constant term 1.

Arun Ram (U. Wisconsin)

Affine Hecke algebras, Fock space and translation functors

This talk describes a point of view on the relationship between Fock space and representations of the affine Hecke algebra of type A which provides a natural generalization to other Lie types.  Applying this point of view in one case in type B provides of the representation theory of affine BMW (Birman-Murakami-Wenzl) algebras in terms of the Kazhdan-Lusztig polynomials for the affine Hecke algebra of type B.

Toshiyuki Tanisaki (Osaka City U.)

D-modules on quantized flag manifold

Theory of the quantized flag manifold as a quasi-scheme (non-commutative scheme) has been developed by Lunts-Rosenberg.
They have formulated an analogue of the Beilinson-Bernstein correspondence using the q-differential operators introduced in their earlier paper. In this talk we shall give a version of the Beilinson-Bernstein correspondence for the quantized flag manifold using a class of q-differential operators, which is (possibly) smaller than the one used by Lunts-Rosenberg.

Efim Zelmanov (KIAS, U. C. San Diego)

TBA

## Schedules

 Oct .20 (Mon) Oct .21 (Tue) Oct .22 (Wed) Oct .23 (Thur) Opening remark Breakfast Breakfast Breakfast 9:30-10:30 Zelmanov Tanisaki Ram Mathieu 10:45-11:45 Jimbo Miwa Kwon Kang Lunch Photo Session/ Lunch Lunch Lunch 1:30-2:30 H. Lee K. Lee Oh Benkart Tea break Tea break Tea break 3:00-4:00 Ariki Andersen Elduque Reception Banquet

## Organizing Committee

Efim Zelamnov (KIAS, U. C. San Diego)

Hyo Chul Myung (KIAS)

Seok-Jin Kang (KIAS)

Jae-Hoon Kwon (U. Seoul)

## Photos

H.C. Myung (Opening remark)

E. Zelmanov

M. Jimbo

H.M. Lee

S. Ariki

T. Tanisaki

T. Miwa

K.H. Lee

H.H. Andersen

A. Ram

J.H. Kwon

Y.T. Oh

A. Elduque

O. Mathieu

S.J. Kang

G. Benkart

Banquet