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

Let g be a finite-dimensional simple Lie algebra and let S be the locally finite part of the algebra of invariants (End(C) V circle times S(g))(g) where V is the direct sum of all simple finite-dirnensional modules for g and S(g) is the symmetric algebra of g. Given an integral weight xi, let Psi = Psi(xi) be the subset of roots which have maximal scalar product with xi. Given a dominant integral weight lambda and xi such that Psi is a subset of the positive roots we construct a finite-dimensional subalgebra S-Psi(g)(<=Psi lambda) of S-g and prove that the algebra is Koszul of global dimension at most the cardinality of Psi. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of Psi. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras. (C) 2008 Elsevier Inc. All rights reserved,