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Lie Group Quantum Mathematics Seminar

Non-rigid tensor categories for affine sl_2 at admissible levels

Robert McRae, Yau Mathematical Sciences Center at Tsinghua University

Location:  Hill 705
Date & time: Friday, 29 September 2023 at 12:10PM - 1:10PM

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 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.

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