Modules which are direct sum of weight spaces. - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T11:15:04Z http://mathoverflow.net/feeds/question/93589 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/93589/modules-which-are-direct-sum-of-weight-spaces Modules which are direct sum of weight spaces. Jianrong Li 2012-04-09T18:38:15Z 2012-04-09T23:36:13Z <p>For a semisimple Lie algebra $\mathfrak{g}$, a highest weight module $V(\lambda)$ with highest weight weight $\lambda$ has the property that every submodule $W$ of $V(\lambda)$ is a direct sum of the weight spaces of $W$. </p> <p>Now consider quantum affine algebra $U_q(\hat{\mathfrak{g}})$. Let $V(\lambda)$ be the highest weight $U_q(\hat{\mathfrak{g}})$-module with highest weight weight $\lambda$. Do we still have the property that every submodule $W$ of $V(\lambda)$ is a direct sum of the weight spaces of $W$?</p> http://mathoverflow.net/questions/93589/modules-which-are-direct-sum-of-weight-spaces/93614#93614 Answer by Christopher Drupieski for Modules which are direct sum of weight spaces. Christopher Drupieski 2012-04-09T23:36:13Z 2012-04-09T23:36:13Z <p>Yes. See Proposition 3.2.1 in Jin Hong and Seok-Jin Kang's book <em>Introduction to Quantum Groups and Crystal Bases</em>.</p>