Categorical Schur's Lemma

Question

In attempt to prove (and compute) a formula for the dimensions of the HOMFLY homology of the (p,q)-torus knot one could try to follow original proof by Jones of a formula for HOMFLY polynomial of the torus knot. One of the key moments of the Jones proof is Schur-Weyl duality and Schur's Lemma. Then the question:

What is a categorical analogues of the Schur's Lemma and Schur-Weyl duality. In general what are the obstacles for finding formula for the HOMLFLY homology of the (p,q)-torus knot. Any refs and suggestions would be greatly appreciated.

Answer 1

One categorical analogue of Schur's lemma is the definition of a simple object in, say, an abelian category. This is essentially an object $V$ for which a generalization of Schur's lemma holds. In particular, all morphisms $f \colon V \to V$ are either isomorphisms or equal zero.

If the category is enriched over vector spaces over some field $k$, it follows that the vector space $\mathrm{hom}(V,V)$ becomes a division algebra over $k$. If $k$ is algebraically complete, it then follows that $\mathrm{hom}(V,V) \cong k$.