Categorification means the construction of tensor categories with given Grothendieck groups. In this paper the authors obtain two categorifications of the \(U(\mathfrak{sl}_2)\)-action in \(V^{\otimes n}_1\) and of its commutant, the Temperley-Lieb algebra by functors between representation categories of \(\mathfrak{gl}_n\).
Starting with a collection of preliminary facts and definitions in section 2: Lie algebra \(\mathfrak{sl}_2\) and categories of highest weight modules, section 3: Singular categories, is devoted to the first type of categorification. In this case \(V^{\otimes n}_1\) is realized as a Grothendieck group of a category defined as a direct sum of singular blocks in the category of highest weight \(\mathfrak{gl}_n\)-modules. The categorification of the \(U(\mathfrak{sl}_2)\)-action is given by projective functors, while Zuckerman functors are used to categorify the action of the Temperley-Lieb algebra on \(V^{\otimes n}_1\).
Another categorification is constructed in section 4: Parabolic categories. In that case \(V^{\otimes n}_1\) is realized by a direct sum of parabolic subcategories of regular blocks of \(\mathfrak{gl}_n\)-modules. The role of projective and Zuckerman functors is reversed, the categorification of the Temperley-Lieb algebra action is given by projective functors, while the action of \(U(\mathfrak{sl}_2)\) is achieved via Zuckerman functors. There is a conjecture, that the Koszul duality functor exchanges the categorifications constructed in section 3 and section 4, and more generally, even exchanges projective and Zuckerman functors.