Topological Hopf algebras and their Hopf-cyclic cohomology

Abstract

A natural extension of the Hopf-cyclic cohomology, with coefficients, is introduced to encompass topological Hopf algebras. The topological theory allows to work with infinite dimensional Lie algebras. Furthermore, the category of coefficients (AYD modules) over a topological Lie algebra and those over its universal enveloping (Hopf) algebra are isomorphic. For topological Hopf algebras, the category of coefficients is identified with the representation category of a topological algebra called the anti-Drinfeld double. Finally, a topological van Est type isomorphism is detailed, connecting the Hopf-cyclic cohomology to the relative Lie algebra cohomology with respect to a maximal compact subalgebra.

Keywords:

• Hopf-cyclic cohomology
• infinite dimensional Lie algebras
• topological Hopf algebras

2010 Mathematics Subject Classification:

• 19D55
• 16S40
• 57T05

Acknowledgements

B.R. would like to thank the Hausdorff Institute in Bonn for its hospitality and support during the time this work was in progress.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

Notes

1 Strong dual, [34, Sect. II.2.3].

2 Any element of V is contained in a finite dimensional (differentiable) G-module, [14, Sect. 1.2].

3 Follows from $\text{exp}(tY)⊳X=X+t\mu (X,Y,t),\hspace{1em}\mu (X,Y,0)=Y⊳X,$ for any $X\in {\mathfrak{g}}_1$, and any $Y\in {\mathfrak{g}}_2$, which, in turn, follows from the action of G₂ on G₁ being analytical.

4 A semi-norm $\rho :A\to [0,\infty )$ is said to be submultiplicative if $\rho (ab)\le \rho (a)\rho (b)$ for any $a,b\in A$.

5 Convex and balanced, where a subset $\mathcal{U}$ of a l.c. t.v.s. W is called balanced if $\lambda w\in \mathcal{U}$ for any $w\in \mathcal{U}$ and any $\lambda \in \mathbb{C}$ with $|\lambda |\le 1$.

6 E, F, G being topological spaces; a map $E\times F\to G$ is called “jointly continuous” if it is continuous with respect to the product topology on E × F.