Japanese

KODERA Ryosuke

Research Fellow of the Japan Society for the Promotion of Science (PD)

Research Institute for Mathematical Sciences, Kyoto University
mail address

Research Interests

Representation theory

Papers

  1. A generalization of adjoint crystals for the quantized affine algebras of type An(1), Cn(1) and Dn+1(2), Journal of Algebraic Combinatorics 30 (2009), no. 4, 491--514.

    Benkart-Frenkel-Kang-Lee gave a uniform construction of so-called adjoint crystals for all quantized affine algebras and proved that they are level-one perfect. According to their work, it turned out that the adjoint crystals have nice symmetry.

    In this paper I propose to generalize them for higher-level cases (conjecturally they are perfect) and describe their structures for type An(1), Cn(1) and Dn+1(2). The generalization forms a family of crystals indexed by nonnegative integers (the 0th member in the family is the trivial crystal and the 1st coincides with the adjoint crystal) and my result says that certain inductive structure appears in the family.

    For other types, the same statement in the paper does not hold in general. However I still expect that they have some (more complicated) inductive structures and one can describe them in a uniform manner as in the case of level-one.

  2. Extensions between finite-dimensional simple modules over a generalized current Lie algebra, Transformation Groups 15 (2010), no. 2, 371--388.

    Representation theory of loop Lie algebras has been developed for a long time. Since the category of its finite-dimensional modules is not semisimple, it is important to investigate its homological properties. The first extension groups for finite-dimensional simple modules were identified with certain Hom spaces for modules over the underlying simple Lie algebra by Fialowski-Malikov for a special class of modules and by Chari-Greenstein for general simple modules, while the blocks of the category were determined by Chari-Moura.

    The present paper are concerned with representations of a generalized current Lie algebra, which is a generalization of the loop Lie algebra. It is defined as the tensor product of a finite-dimensional semisimple Lie algebra and a finitely generated commutative algebra, both over the field of complex numbers. I calculate the first extension groups for its finite-dimensional simple modules. To say in more detail, they are described in terms of the same Hom spaces as in the works of Fialowski-Malikov and Chari-Greenstein, together with the space of derivations of the commutative algebra. I also determine the blocks of the category of finite-dimensional modules, which generalizes the result of Chari-Moura.

  3. Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties (with Katsuyuki Naoi), Publications of the Research Institute for Mathematical Sciences 48 (2012), no. 3, 477--500.

    Weyl modules for current Lie algebras are defined as the universal finite-dimensional highest weight modules. When the current Lie algebra is associated with a simple Lie algebra of type ADE, they are known to be isomorphic to the level-one Demazure modules for the affine Lie algebras, and the standard modules defined as the homology groups of the Lagrangian quiver varieties.

    In this joint work with Katsuyuki Naoi, we study the graded module structures of Weyl modules for the current Lie algebra associated with a simple Lie algebra of type ADE. It also contains some applications to the corresponding quiver varieties.

    One of the main results is rigidity of Weyl modules, that is, each Weyl module has a unique Loewy series. This is concluded by showing that the radical series, the socle series, and the grading filtration for a Weyl module all coincide. Further we use this result to show that the gradings on a Weyl module and a standard module coincide under the isomorphism mentioned above. However, it should be remarked that this fact, coincidence of the gradings, itself can be proved in a more direct way. It seems to be known to specialists, but not in the literature.

    Combining the coincidence of the gradings with known results, we obtain the following applications to quiver varieties.

Proceedings (non-refereed)

  1. Ext1 for simple modules over Uq(Lsl2), 14th Conference on Representation Theory of Algebraic Groups and Quantum Groups. [PDF]

    I calculate the first extension groups for some (far from all) finite-dimensional simple modules over the quantum loop algebra Uq(Lsl2). In particular the finite-dimensional simple modules that admit non-trivial extensions with the trivial module are determined. In the proof, I mainly use a result of Chari-Pressley on tensor products of evaluation modules and of Chari-Moura on blocks of the category of finite-dimensional modules as well as the well-known adjointness between the functor tensoring with a finite-dimensional module and that tensoring with its dual, which was used also in my paper [2] for generalized current Lie algebras.

Talks

  1. On Kirillov-Reshetikhin crystals for type A, 5th Kinosaki Shinjin Seminar, Hyogo, February 19th 2008.
  2. A generalization of adjoint crystals for the quantized affine algebras of type An(1), Cn(1) and Dn+1(2), 11th Conference on Representation Theory of Algebraic Groups and Quantum Groups, Okayama, May 28th 2008.
  3. A generalization of adjoint crystals for the quantized affine algebras of type An(1), Cn(1) and Dn+1(2), RAQ Seminar, University of Tokyo, June 26th 2008.
  4. A generalization of adjoint crystals for the quantized affine algebras of type An(1), Cn(1) and Dn+1(2), Workshop ``Crystals and Tropical Combinatorics'', Kyoto, August 28th 2008.
  5. A generalization of adjoint crystals for the quantized affine algebras of type An(1), Cn(1) and Dn+1(2), Mathematical Society of Japan Autumn Meeting 2008, Tokyo Institute of Technology, September 25th 2008.
  6. A generalization of adjoint crystals for the quantized affine algebras of type An(1), Cn(1) and Dn+1(2), Russia-Japan School of Young Mathematicians, Kyoto University, January 29th 2009.
  7. Extensions between finite-dimensional simple modules over a generalized current Lie algebra, poster session in Infinite Analysis 09, Kyoto University, July 29th 2009.
  8. Extensions between finite-dimensional simple modules over a generalized current Lie algebra, Workshop ``Algebras, Groups and Geometries 2009 in Tambara'', Tambara Institute of Mathematical Sciences, August 22nd 2009.
  9. Extensions between finite-dimensional simple modules over a generalized current Lie algebra, Mathematical Society of Japan Autumn Meeting 2009, Osaka University, September 24th 2009.
  10. Extensions between finite-dimensional simple modules over a generalized current Lie algebra, Lie Groups and Representation Theory Seminar, University of Tokyo, October 13th, 2009.
  11. Extensions between finite-dimensional simple modules over a generalized current Lie algebra, RIMS workshop ``Algebraic Combinatorics and related groups and algebras'', Shinshu University, November 17th 2009.
  12. Extensions between finite-dimensional simple modules over a generalized current Lie algebra, Tokyo-Seoul Conference in Mathematics: Representation Theory, University of Tokyo, December 5th 2009.
  13. Extensions between finite-dimensional simple modules over a generalized current Lie algebra, Representation Theory Seminar, Nagoya University, May 11th 2010.
  14. Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties, RAQ Seminar, Sophia University, May 13th 2011.
  15. Ext1 for simple modules over Uq(Lsl2), 14th Conference on Representation Theory of Algebraic Groups and Quantum Groups, Kagawa, June 3rd 2011.
  16. Ext1 for simple modules over Uq(Lsl2), poster session in Infinite Analysis 11, University of Tokyo, July 27th 2011.
  17. Self-extensions and prime factorizations for simple Uq(Lsl2)-modules, Representation Theory Seminar, Research Institute for Mathematical Sciences, February 10th 2012.
  18. Loewy series of Weyl modules for current Lie algebras, Geometric/categorical aspects of representation theory, Hokkaido University, February 21st 2012.
  19. Ext1 for simple modules over Uq(Lsl2), Séminaire d'Algèbre, Institut Henri Poincaré, March 19th 2012.
  20. Quiver varieties and one-dimensional sums, Mathematical Society of Japan Annual Meeting 2012, Tokyo University of Science, March 29th 2012.
  21. Self-extensions and prime factorizations for simple Uq(Lsl2)-modules, Mathematical Society of Japan Autumn Meeting 2012, Kyushu University, September 19th 2012. Slide [PDF]
  22. Loewy series of Weyl modules and the Poincaré polynomials of quiver varieties, Shanghai Workshop on Representation Theory: Special session at Osaka, Osaka University, December 15th 2012.
  23. Representation theory of loop algebras and quantum loop algebras, GCOE tea time, Research Institute for Mathematical Sciences, January 30th 2013.
  24. Homological properties of current algebras and Yangians, Algebra Seminar, Osaka City University, February 5th and 7th 2013.
  25. Ext1 for simple modules over Uq(Lsl2), poster session in The 9th RIMS-Kyoto University and SNU joint symposium on mathematics, Seoul National University, February 18th 2013.
  26. Kostka systems for current Lie algebras, Shanghai Workshop on Representation Theory, East China Normal University, December 5th 2013.

Curriculum Vitae

Last update: December 6th 2013