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

摘要

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 §, let ψ = WO be the subset of roots which have maximal scalar product with § . Given a dominant integral weight and such that ψ is a . subset of the positive roots we construct a finite-dimensional subalgebra S_ψ~g ,(~ф X) of S and prove that the algebra is Koszul of global dimension at most the cardinality of J'. Using this we construct naturally an infinite-dimensional non-commutative Koszul algebra of global dimension equal to the cardinality of W. The results and the methods are motivated by the study of the category of finite-dimensional representations of the affine and quantum affine algebras.