Amenability of Lie, Group and Hopf algebras

Abstract

We propose to define amenability of a Lie algebra by the existence of almost-invariant finite-dimensional subcoalgebras of its universal enveloping algebra. More generally, a module coalgebra of a cocommutative Hopf algebra is amenable if it admits almost-invariant finite-dimensional subcoalgebras. We prove a coalgebraic rounding theorem: almost-invariant finite-dimensional subspaces can be replaced, without loss in the Følner constant, by almost-invariant finite-dimensional subcoalgebras. For Lie algebras this implies that our definition is equivalent to Elek's amenability of the left regular module of the universal enveloping algebra seen merely as an associative algebra. For groups this recovers the result that a group is amenable if and only if its group ring is algebraically amenable. It furthermore shows that, for every amenable group, all its nonzero modules are amenable, thus proving an assertion by Gromov. We prove that amenable Hopf algebras are closed under taking subalgebras, quotients, cleft extensions, and directed unions, and that every Hopf algebra locally of subexponential growth is amenable. We give examples of amenable Lie algebras which are not elementarily amenable. Finally, we show that amenability passes to the associated graded Hopf-module coalgebra.

On amenability of group algebras, I
Laurent Bartholdi
Isr. J. Math.3 Nov 2026Peer-reviewedLicence not statedFull text18 citedAlgebra & number theoryHAL
Exponents of factorized groups and Kashina's conjecture for group-theoretical Hopf algebras
Ningyi Li, Ningyi Li
arXiv8 Oct 2026PreprintarXiv licenceFull textAlgebra & number theoryarXiv
Equivalence of Pseudo- and Approximate Amenability in Banach Algebras
Fereidoun Ghahramani, Maedeh Soroushmehr, Yong Zhang
arXiv15 Sep 2026PreprintCC0Full textAnalysis & PDEsarXiv
On extensive amenability of algebras
Laurent Bartholdi
arXiv1 Oct 2026PreprintCC BY 4.0Full textAlgebra & number theoryarXiv
Character cyclic amenability of Banach algebras
Hamid Sadeghi Nahrekhalaji
arXiv1 Oct 2026PreprintarXiv licenceFull textAnalysis & PDEsarXiv

Cite

BibTeX RIS

BibTeX

@misc{bartholdi2026amenability,
  title = {{Amenability of Lie, Group and Hopf algebras}},
  author = {Laurent Bartholdi},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.10373},
  eprint = {2609.10373},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.10373},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Amenability of Lie, Group and Hopf algebras
AU  - Laurent Bartholdi
PY  - 2026
DA  - 2026/09/09
PB  - arXiv
DO  - 10.48550/arxiv.2609.10373
UR  - https://arxiv.org/abs/2609.10373
AB  - We propose to define amenability of a Lie algebra by the existence of almost-invariant finite-dimensional subcoalgebras of its universal enveloping algebra. More generally, a module coalgebra of a cocommutative Hopf algebra is amenable if it admits almost-invariant finite-dimensional subcoalgebras. We prove a coalgebraic rounding theorem: almost-invariant finite-dimensional subspaces can be replaced, without loss in the Følner constant, by almost-invariant finite-dimensional subcoalgebras. For Lie algebras this implies that our definition is equivalent to Elek's amenability of the left regular module of the universal enveloping algebra seen merely as an associative algebra. For groups this recovers the result that a group is amenable if and only if its group ring is algebraically amenable. It furthermore shows that, for every amenable group, all its nonzero modules are amenable, thus proving an assertion by Gromov. We prove that amenable Hopf algebras are closed under taking subalgebras, quotients, cleft extensions, and directed unions, and that every Hopf algebra locally of subexponential growth is amenable. We give examples of amenable Lie algebras which are not elementarily amenable. Finally, we show that amenability passes to the associated graded Hopf-module coalgebra.
ER  -

Abstract

Related

Cite