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Over a field of characteristic p>0, the higher Verlinde categories Ver䂞 are obtained by taking the abelian envelope of a quotient of the category of tilting modules for the algebraic group SL₂. These symmetric tensor categories have been introduced in arXiv: 2003. 10499 & arXiv: 2003. 10105, and their properties have been extensively studied in the former reference. In arXiv: 2105. 07724, the above construction for SL₂ has been generalized to Lusztig's quantum group for sl₂ and root of unity, which produces the mixed higher Verlinde categories Ver^^ (₍). Inspired by the results of arXiv: 2003. 10499, we study the properties of these braided tensor categories in detail. In particular, we establish a Steinberg tensor product formula for the simple objects of Ver^^ (₍), construct a braided embedding Ver䂞 Ver^^ (₍+₁), compute the symmetric center of Ver^^ (₍), and identify its Grothendieck ring.
Thibault D. Décoppet (Mon,) studied this question.