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

Vyjayanthi Chari, Jacob Greenstein

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