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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.
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. 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.
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  for generalized current Lie algebras.