nLab anti-Yetter-Drinfeld module

Idea

Stable anti-Yetter–Drinfeld modules are natural coefficients for Hopf-algebraic cyclic homology.

Their definition is a slight variant of that of Yetter-Drinfeld modules.

Definition

Given a commutative ring $k$ and a Hopf algebra $H$ over $k$ with an invertible antipode $S$, a $k$-module $M$ is called a left-left anti-Yetter–Drinfeld module if

(i) $M$ is a left $H$-module

(ii) $M$ is a left $H$-comodule via a coaction $m\mapsto \sum m_{(-1)}\otimes m_{(0)}$

(iii) The anti-Yetter–Drinfeld compatibility condition between the $H$-module and $H$-comodule structure is satisfied

$$\sum (h m)_{(-1)} \otimes (h m)_{(0)} = \sum h_{(1)}m_{(-1)} S^{-1}(h_{(3)})\otimes h_{(2)}m_{(0)}$$

Properties

The category of AYD modules is not monoidal but the tensor product of an AYD module with a YD module results in an AYD module.

By the work of Rangipour–Sutlu one knows that there is such category over Lie algebras and there is a one-to-one correspondence between AYD modules over a Lie algebra and those over the universal enveloping algebra of the Lie algebra. This correspondence is extended by the same authors for bicrossed product Hopf algebras.

The true meaning of the AYD modules in noncommutative geometry is not known yet. There are some attempts by Kaygun & Khalkhali 2006 to relate them to the curvature of flat connections similar to the work of T. Brzeziński on YD modules, however their identification are not restricted to AYD and works for a wide variety of YD type modules.

Related entries

• Yetter-Drinfeld module

• Hopf module

• cyclic homology

Literature

AYD modules appeared for the first time in different name in B. Rangipour’s PhD thesis under supervision of M. Khalkhali.

The notion was generalized in:

• P. M. Hajac, M. Khalkhali, B. Rangipour, Y. Sommerhaeuser: Hopf-cyclic homology and cohomology with coefficients, C. R. Math. Acad. Sci. Paris 338(9), 667–672 (2004) math.KT/0306288;

• P. M. Hajac, M. Khalkhali, B. Rangipour, Y. Sommerhaeuser: Stable anti-Yetter–Drinfeld modules. C. R. Math Acad. Sci. Paris 338(8), 587–590 (2004)

See also:

• Gabriella Böhm, Dragos Stefan, (Co)cyclic (co)homology of bialgebroids: An approach via (co)monads, Comm. Math. Phys. 282 (2008), no.1, 239–286, arxiv/0705.3190; A categorical approach to cyclic duality, J. Noncommutative Geometry 6 (2012), no. 3, 481–538, arxiv/0910.4622

• Atabey Kaygun, Masoud Khalkhali, Hopf modules and noncommutative differential geometry, Lett. Math. Phys. 76 (2006) 77–91 doi

We define a new algebra of noncommutative differential forms for any Hopf algebra with an invertible antipode. We prove that there is a one-to-one correspondence between anti-Yetter–Drinfeld modules, which serve as coefficients for the Hopf cyclic (co)homology, and modules which admit a flat connection with respect to our differential calculus. Thus, we show that these coefficient modules can be regarded as “flat bundles” in the sense of Connes’ noncommutative differential geometry.

• B. Rangipour, Serkan Sütlü, Characteristic classes of foliations via SAYD-twisted cocycles, arXiv:1210.5969; SAYD modules over Lie–Hopf algebras, arXiv:1108.6101; Cyclic cohomology of Lie algebras, arXiv:1108.2806

• Florin Panaite, Mihai D. Staic, Generalized (anti) Yetter–Drinfeld modules as components of a braided T-category, arXiv:math.QA/0503413

category: algebra