Morphism spaces in stable categories of Frobenius algebras

We present an explicit formula computing morphism spaces in the stable category of a Frobenius algebra.Comment: V3 contains minor corrections. 11 pages. Comments are welcom

Hopfological Algebra

We develop some basic homological theory of hopfological algebra as defined by Khovanov. Several homological properties in hopfological algebra analogous to those of usual homological theory of DG algebras are obtained.Comment: Minor corrections and format changin

A Categorification of the Burau representation at prime roots of unity

We construct a p-DG structure on an algebra Koszul dual to a zigzag algebra used by Khovanov and Seidel to construct a categorical braid group action. We show there is a braid group action in this p-DG setting.Comment: 33 pages, many PSTricks figures, color viewing not essential, v2 contains minor corrections, comments welcom

Categorifying Hecke algebras at prime roots of unity, part I

We equip the type $A$ diagrammatic Hecke category with a special derivation, so that after specialization to characteristic $p$ it becomes a $p$-dg category. We prove that the defining relations of the Hecke algebra are satisfied in the $p$-dg Grothendieck group. We conjecture that the $p$-dg Grothendieck group is isomorphic to the Iwahori-Hecke algebra, equipping it with a basis which may differ from both the Kazhdan-Lusztig basis and the $p$-canonical basis. More precise conjectures will be found in the sequel. Here are some other results contained in this paper. We provide an incomplete proof of the classification of all degree $+2$ derivations on the diagrammatic Hecke category, and a complete proof of the classification of those derivations for which the defining relations of the Hecke algebra are satisfied in the $p$-dg Grothendieck group. In particular, our special derivation is unique up to duality and equivalence. We prove that no such derivation exists in simply-laced types outside of finite and affine type $A$. We also examine a particular Bott-Samelson bimodule in type $A_7$, which is indecomposable in characteristic $2$ but decomposable in all other characteristics. We prove that this Bott-Samelson bimodule admits no nontrivial fantastic filtrations in any characteristic, which is the analogue in the $p$-dg setting of being indecomposable.Comment: 44 pages, many figures, color viewing essential. V2 contains corrections from referee reports. To appear in Transactions of the AM

An approach to categorification of some small quantum groups II

We categorify an idempotented form of quantum sl2 and some of its simple representations at a prime root of unity.Comment: 59 pages, many PSTricks figures, v2 contains minor corrections and updated reference

A categorification of cyclotomic rings

For any natural number $n \geq 2$, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the ring of cyclotomic integers $\mathbb{O}_n$.Comment: 28 pages. Comments welcome! v2, v3: minor correction

The differential graded odd nilHecke algebra

We equip the odd nilHecke algebra and its associated thick calculus category with digrammatically local differentials. The resulting differential graded Grothendieck groups are isomorphic to two different forms of the positive part of quantum sl(2) at a fourth root of unity.Comment: 53 page