Department of Mathematics and Informatics, Graduate School of Science, Chiba University
Representation theory
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 A_{n}^{(1)}, C_{n}^{(1)} and D_{n+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.
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.
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.
Combining the coincidence of the gradings with known results, we obtain the following applications to quiver varieties.
Uglov constructed an action of the Yangian of type A on the level one Fock space and calculated eigenvalues of the action of the Gelfand-Zetlin subalgebra on a distinguished basis of the Fock space (He called them Jack(gl_N) symmetric functions). I obtain an explicit formula for the action of the Drinfeld generator of the Yangian on the Uglov's basis in this paper.
Then I compare the formula with one for the action of the affine Yangian of type A on the torus fixed point basis of the localized equivariant homology group of the quiver variety associated with the cyclic quiver. This shows that the Yangian action due to Uglov extends to an action of the affine Yangian on the Fock space.
I construct an action of the affine Yangian of type A on the Fock space with an arbitrary positive integer level. This work is a degenerate analog of a result by Takemura-Uglov who constructed an action of the quantum toroidal algebra on the higher level q-deformed Fock space.
I calculate the first extension groups for some (far from all) finite-dimensional simple modules over the quantum loop algebra U_{q}(Lsl_{2}). 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.
In the main body of the paper, Braverman-Finkelberg-Nakajima identify the Coulomb branches associated with framed quiver gauge theories of type ADE with generalized slices in the affine Grassmannian. The dimension vectors of a given framed quiver representation give two coweights lambda and mu, where lambda is always dominant. Under the assumption that mu is dominant, the generalized slice is a genuine transversal slice.
Before Braverman-Finkelberg-Nakajima proposed a mathematical definition of Coulomb branches, Kamnitzer-Webster-Weekes-Yacobi studied quantizations of the slices in the affine Grasmannian via shifted Yangians. They showed certain quotients of shifted Yangians give quantizations modulo nilpotents and conjectured that they give quantizations.
In this appendices we show that the quantized Coulomb branch is isomorphic to the quotient of the shifted Yangian under the condition that the coweight mu is dominant. This result gives an affirmative answer to the conjecture by Kamnitzer-Webster-Weekes-Yacobi.
I construct a filtration of a level one integrable irreducible representation of the affine Lie algebra of type ADE whose associated graded pieces are quotients of global Weyl modules of the current Lie algebra. By comparing it with the character identity proved by Cherednik-Feigin, each surjective map from the global Weyl module turns out to be an isomorphism. This gives a concrete construction of the filtration which is proved to exist abstractly by Kato-Loktev for more general situation in the main body of the paper.