Deniz Kus

Local Weyl modules for equivariant map algebras with free abelian group actions

Suppose a finite group acts on a scheme X and a finite-dimensional Lie algebra g. The associated equivariant map algebra is the Lie algebra of equivariant regular maps from X to g. Examples include generalized current algebras and (twisted) multiloop algebras. Local Weyl modules play an important role in the theory of finite-dimensional represen- tations of loop algebras and quantum affine algebras. In the current paper, we extend the definition of local Weyl modules (previously defined only for generalized current algebras and twisted loop algebras) to the setting of equivariant map algebras where g is semisimple, X is affine of finite type, and the group is abelian and acts freely on X. We do so by defin- ing twisting and untwisting functors, which are isomorphisms between certain categories of representations of equivariant map algebras and their untwisted analogues. We also show that other properties of local Weyl modules (e.g. their characterization by homological prop- erties and a tensor product property) extend to the more general setting considered in the current paper.

Demazure modules and Weyl modules: The twisted current case

We study finite-dimensional respresentations of twisted current algebras and show that any graded twisted Weyl module is isomorphic to level one Demazure modules for the twisted affine Kac-Moody algebra. Using the tensor product property of Demazure modules, we obtain, by analyzing the fundamental Weyl modules, dimension and character formulas. Moreover, we prove that graded twisted Weyl modules can be obtained by taking the associated graded modules of Weyl modules for the loop algebra, which implies that its dimension and classical character are independent of the support and depend only on its classical highest weight. These results were previously known for untwisted current algebras and are new for all twisted types.

Weyl Modules for the Hyperspecial Current Algebra

We develop the theory of global and local Weyl modules for the hyperspecial maximal parabolic subalgebra of type

Fusion products and toroidal algebras

A_{2n}^{(2)}

Twisted Demazure modules, fusion product decomposition and twisted -systems

. We prove that the dimension of a local Weyl module depends only on its highest weight, thus establishing a freeness result for global Weyl modules. Furthermore, we show that the graded local Weyl modules are level one Demazure modules for the corresponding affine Lie algebra. In the last section we derive the same results for the special maximal parabolic subalgebras of the twisted affine Lie algebras not of type

Crystal Bases as Tuples of Integer Sequences

A_{2n}^{(2)}

Borel–de Siebenthal pairs, global Weyl modules and Stanley–Reisner rings

.

Realization of affine type A Kirillov-Reshetikhin crystals via polytopes

We study the category of finite--dimensional bi--graded representations of toroidal current algebras associated to finite--dimensional complex simple Lie algebras. Using the theory of graded representations for current algebras, we construct in different ways objects in that category and prove them to be isomorphic. As a consequence we obtain generators and relations for certain types of fusion products including the

The PBW filtration and convex polytopes in type B

N

Kirillov---Reshetikhin crystals, energy function and the combinatorial R-matrix

--fold fusion product of