arXiv math/05090832005-09math.QA

Hopfological algebra and categorification at a root of unity: the first steps

Mikhail Khovanov

备注：: 30 pages, latex, this version contains very minor corrections

英文摘要：

Any finite-dimensional Hopf algebra H is Frobenius and the stable category of H-modules is triangulated monoidal. To H-comodule algebras we assign triangulated module-categories over the stable category of H-modules. These module-categories are generalizations of homotopy and derived categories of modules over a differential graded algebra. We expect that, for suitable H, our construction could be a starting point in the program of categorifying quantum invariants of 3-manifolds.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Categorification at prime roots of unity and hopfological finiteness

We survey some recent results in hopfological algebra and the program of categorification at prime roots of unity. A categorical Jones-Wenzl projector at prime roots of unity is studied, and it is shown that this projector is hopfologically finite in special characteristics, while generically it is infinite.

2017-03-04math.KTmath.QA

A categorification of quantum sl(2) at prime roots of unity

We categorify the Beilinson-Lusztig-MacPherson integral form of quantum sl(2) specialized at a prime root of unity.

2016-07-01math.QAmath.RT

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.

2015-03-25math.QAmath.RT

A categorification of the colored Jones polynomial at a root of unity

There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.

2021-11-25math.GTmath.QAmath.RT

Cell 2-Representations and Categorification at Prime Roots of Unity

Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of…

2019-12-11math.CTmath.QAmath.RT

Derived equivalences and sl_2-categorifications for U_q(gl_n)

We give a construction of sl_2-categorifications (in the sense of Chuang-Rouquier) for representations of U_q(gl_n), for generic q and for q a root of unity.

2011-01-16math.QAmath.RT

更多相关论文收起更多论文

A categorification of a quantum Frobenius map

A quantum Frobenius map a la Lusztig for $\mathfrak{sl}_2$ is categorified at a prime root of unity.

2017-07-24math.KTmath.QAmath.RT

An approach to categorification of some small quantum groups

We categorify one half of the small quantum sl(2) at a prime root of unity. An extension of this construction to an arbitrary simply-laced case is proposed.

2013-06-17math.KTmath.QAmath.RT

素数单位根处的有色 Khovanov 同调理论

Colored Khovanov homology theories at prime roots of unity

本文为有色 Khovanov 同调引入次数 2 导子，在特征 p>2 域上得到 p-幂零微分，实现 2p 次单位根处 gl(2) 同调的范畴化。

2026-10-02math.GTmath.QA

Handle Slides and Localizations of Categories

We propose a means by which some categorifications can be evaluated at a root of unity. This is implemented using a suitable localization in the context of prior work by the authors on categorification of the Jones-Wenzl projectors. Within this construction we define objects, invariant under handle slides, which decategorify to the SU(2) quantum invariants at low levels.

2011-10-11math.GTmath.QA

广义簇代数的幺半范畴化与Fraser和Gleitz猜想

Monoidal categorification of generalized cluster algebras and conjectures of Fraser and Gleitz

School of Mathematics and Statistics, Lanzhou University（兰州大学数学与统计学院）；School of Information Engineering and Artificial Intelligence, Lanzhou University of Finance and Economics（兰州财经大学信息工程与人工智能学院）；Faculty of Mathematics, University of Vienna（维也纳大学数学系）

本文定义了广义簇代数的幺半范畴化概念，并证明了对于k=3和任意ℓ≥2，限制量子环代数模块范畴的子范畴的Grothendieck环同构于秩为2ℓ-2的广义簇代数，从而证明了Gleitz猜想的第一部分。

2026-07-08math.QAmath.RT

Bost-Connes systems, Categorification, Quantum statistical mechanics, and Weil numbers

In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unit are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unit, Weil restriction, algebraic numbers, Weil numbers, CM fields, germs, completion of Weil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unit is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck's catego…

2014-11-12math-phmath.AGmath.AT

关于若干$p$-微分分次链环同调 II

On some $p$-differential graded link homologies II

本研究拓展了链环同调的范畴化工作，提出一族将Cautis微分特化为N=kp+2形式的链环同调，证明k为偶数时它们均范畴化2p次单位根处的琼斯多项式且互不同构，丰富了相关不变量体系。

2021-08-24math.GTmath.QA

来自量子一般线性群的 $G(d,1,n)$ 的傅里叶矩阵

Fourier matrices for $G(d,1,n)$ from quantum general linear groups

本研究针对特殊复反射群 $G(d,1,n)$ 的幺正特征模数据，基于循环群特征标表外幂的克罗内克积分解，利用量子一般线性李代数表示实现范畴化，并在去范畴化层面证明了Cuntz的部分正性猜想。

2020-11-23math.QAmath.RT