# Emmy-Noether-Seminar: The non-semisimple Kazhdan-Lusztig category for affine sl_2 at admissible levels

**The non-semisimple Kazhdan-Lusztig category for affine sl_2 at
admissible levels**

Robert McRae (Beijing)

Abstract: The Kazhdan-Lusztig category KL^k(sl_2) is the category of finite-length modules for affine sl_2 at level k whose composition factors are irreducible highest weight modules whose highest weights are dominant integral for the finite -dimensional subalgebra sl_2. In this talk, we show that for admissible levels

k = −2 + p/q, where p>1 and q>0 are relatively prime integers, KL^k(sl_2) admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang, but that it is not rigid, that is, not every object has a dual. Instead, an object of KL^k(sl_2) is rigid if and only if it is projective and, moreover, KL^k(sl_2) has enough projectives. Most of the indecomposable projective objects are logarithmic modules, which means that the Virasoro L_0 operator acts non-semisimply. We show also that the monoidal subcategory of rigid and projective objects is tensor equivalent to the category of tilting modules for quantum sl_2 at the root of unity e^{pi i/(k+2)}. This leads to a universal property for KL^k(sl_2), which allows us to construct an essentially surjective (but not fully faithful) exact tensor functor from KL^k(sl2) to the category of finite -dimensional weight modules for quantum sl_2 at e^{pi i/(k+2)}. This is joint work with Jinwei Yang.