Lie、群与Hopf代数的可均性

中文摘要 AI 辅助

我们提出通过其泛包络代数中几乎不变的有限维子余代数的存在性来定义Lie代数的可均性。更一般地，一个余交换Hopf代数的模余代数如果允许几乎不变的有限维子余代数，则称其为可均的。我们证明了一个余代数舍入定理：几乎不变的有限维子空间可以在不损失Følner常数的情况下被几乎不变的有限维子余代数所替代。对于Lie代数，这意味着我们的定义等价于Elek关于泛包络代数（仅视为结合代数）的左正则模的可均性。对于群，这恢复了如下结果：一个群是可均的当且仅当其群环是代数可均的。此外，它还表明，对于每个可均群，其所有非零模都是可均的，从而证明了Gromov的一个断言。我们证明了可均Hopf代数在取子代数、商、裂扩张和定向并下封闭，并且每个局部次指数增长的Hopf代数都是可均的。我们给出了不是初等可均的可均Lie代数的例子。最后，我们证明了可均性传递到相关的分次Hopf模余代数。

英文摘要

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.

发表机构

• Université de Lyon 1（里昂第一大学）
• Université de Genève（日内瓦大学）

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