Jacob Greenstein

Current algebras, highest weight categories and quivers

Abstract We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametrized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on the affine weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the injective cogenerator of the subcategory associated to an interval closed finite subset of the weight lattice. Finally, we prove that there is a large number of interesting quivers of finite, affine and star-shaped type, as well as tame quasi-hereditary algebras, that arise from our study.

A family of Koszul algebras arising from finite-dimensional representations of simple Lie algebras

Abstract Let g be a finite-dimensional simple Lie algebra and let S g be the locally finite part of the algebra of invariants ( End C V ⊗ S ( g ) ) g where V is the direct sum of all simple finite-dimensional modules for g and S ( g ) is the symmetric algebra of g . Given an integral weight ξ, let Ψ = Ψ ( ξ ) be the subset of roots which have maximal scalar product with ξ. Given a dominant integral weight λ and ξ such that Ψ is a subset of the positive roots we construct a finite-dimensional subalgebra S Ψ g ( ⩽ Ψ λ ) of S g and prove that the algebra is Koszul of global dimension at most the cardinality of Ψ. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of Ψ. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.

Minimal affinizations as projective objects

We prove that the specialization to q = 1 of a Kirillov-Reshetikhin module for an untwisted quantum affi ne algebra of classical type is projective in a suitable cate gory. This yields a uniform character formula for the Kirillov-Reshetikhin modules. We conjecture that these results holds for specializations of minimal affi nization with some restriction on the corresponding highest weight. We discuss the connection with the conjecture of Nakai and Nakanishi on q-characters of minimal affi nizations. We establish this conjecture in some special cases. This also leads us to conjecture an alternating sum formula for Jacobi-Trudi determinants.

Quantum loop modules

We classify the simple infinite dimensional integrable modules with finite dimensional weight spaces over the quantized enveloping algebra of an untwisted affine algebra. We prove that these are either highest (lowest) weight integrable modules or simple submodules of a loop module of a finite-dimensional simple integrable module and describe the latter class. Their characters and crystal bases theory are discussed in a special case.

On homomorphisms between global Weyl modules

Global Weyl modules for generalized loop algebras

GRADED LEVEL ZERO INTEGRABLE REPRESENTATIONS OF AFFINE LIE ALGEBRAS

\lie g\tensor A

Path model for quantum loop modules of fundamental type

, where

Characters of Simple Bounded Modules over an Untwisted Affine Lie Algebra

\lie g

Littelmann's path crystal and combinatorics of certain integrable slℓ+1 modules of level zero☆

is a simple finite dimensional Lie algebra and A is a commutative associative algebra were defined, for any dominant integral weight

Double canonical bases

\lambda