Archives mensuelles : avril 2013

More New Modules from Old

The Unapologetic Mathematician

There are a few constructions we can make, starting with the ones from last time and applying them in certain special cases.

First off, if $latex V$ and $latex W$ are two finite-dimensional $latex L$-modules, then I say we can put an $latex L$-module structure on the space $latex \hom(V,W)$ of linear maps from $latex V$ to $latex W$. Indeed, we can identify $latex \hom(V,W)$ with $latex V^*\otimes W$: if $latex \{e_i\}$ is a basis for $latex V$ and $latex \{f_j\}$ is a basis for $latex W$, then we can set up the dual basis $latex \{\epsilon^i\}$ of $latex V^*$, such that $latex \epsilon^i(e_j)=\delta^i_j$. Then the elements $latex \{\epsilon^i\otimes f_j\}$ form a basis for $latex V^*\otimes W$, and each one can be identified with the linear map sending $latex e_i$ to $latex f_j$ and all the other basis elements of $latex V$ to $latex 0$. Thus we have an inclusion…

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