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

• Autores: Vyjayanthi Chari, Jacob Greenstein
• Localización: Advances in mathematics, ISSN 0001-8708, Vol. 220, Nº 4, 2009, págs. 1193-1221
• Idioma: inglés
• DOI: 10.1016/j.aim.2008.11.007
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• Resumen

  • 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