Trace decategorification of categorified quantum $$\mathfrak {sl}_2$$ sl 2

Abstract

Abstract. The trace or the 0th Hochschild-Mitchell homology of a linear category C may be regarded as a kind of decategorification of C. We compute the traces of the two versionsU and U * of categorified quantum sl 2 introduced by the third author. The trace of U is isomorphic to the split Grothendieck group K 0 (U ), and the higher Hochschild-Mitchell homology ofU is zero. The trace of U * is isomorphic to the idempotented integral form of the current algebra U(sl 2 [t]).

Some remarks on the unrolled quantum group of sl(2)

“…In the paper [3], the concepts of Coend, trace and the Hochschild-Mitchell homology in a linear category are related. In [4], a graded TQFT is defined for manifolds equipped with a 1-cohomology class with value in C/2Z.…”

Sutured annular Khovanov-Rozansky homology

“…In this section, we'll discuss trace decategorifications of linear 2-categories, in particular the vertical and horizontal traces of categorified quantum groups and foam 2-categories. See [4,3,5] for a detailed study of the vertical trace decategorification of U Q (sl m ).…”

“…Using Rickard complexes [14] in U Q (gl m ), we assign a complex C(L) in this category to any link L ⊆ A × [0, 1]. This complex lifts 1 to a complex C(L) in vTr(2Foam), which in turn lifts to a complex D(L) in vTr(U Q (gl m ) 0≤2 ), the universal annular sl 2 invariant of L. Results of Beliakova, Guliyev, Habiro, Lauda, Webster, andZivković [4,3,5] show that vTr(U Q (sl m )) is isomorphic toU(sl m [t]), the idempotented form of the current algebra of sl m . This pairs with skew Howe duality on ( 2 ⊗ m ) to give a functorU (sl m [t]) to D(L) gives a complex whose chain groups are sl 2 representations and whose differentials are sl 2 intertwiners.…”

“…Objects in the 2-category 2Foam are sequences of the symbols 1 and 2, which we view as labeling boundary points on {0} × Ê and {1} × Ê, together 3 We use the negative of the grading convention from [2], to match with the categorified quantum group convention that a dot has degree 2. 4 Here, and throughout, we've ✿✿✿✿✿✿✿✿ underlined the term of a complex in homological degree zero. Additionally, we use the grading shifts for complexes assigned to crossings from [34] throughout.…”

The elliptic Hall algebra and the deformed Khovanov Heisenberg category

“…There is a natural Chern character map K 0 (C) → Tr(C) which sends an object to the class of its identity morphism. In many cases the Chern character is an isomorphism, but recently several examples have been given where Tr(C) is a much larger, more interesting algebra than K 0 (C) [BGHL,BHLW14,BHLZ14,EL16,CLLS15].…”