The amenability and non-amenability of skew fields

关于代数的广泛可均性

On extensive amenability of algebras

本文定义了余交换Hopf代数上模的广泛可均性，证明了其与可均性及短正合列的关系，并讨论了置换模与外代数情形。

Amenability of monomial algebras, minimal subshifts and free subalgebras

We give a combinatorial characterization of amenability of monomial algebras and prove the existence of monomial Folner sequences, answering a question due to Ceccherini-Silberstein and Samet-Vaillant. We then use our characterization to prove that over projectively simple monomial algebras, every module is exhaustively amenable; we conclude that convolution algebras of minimal subshifts admit the same property. We deduce that any minimal subshift of positive entropy gives rise to a graded algebra which does not satisfy an extension of Vershik's conjecture on amenable groups, proposed by Barth…

Lie、群与Hopf代数的可均性

Amenability of Lie, Group and Hopf algebras

本文通过泛包络代数中几乎不变的有限维子余代数定义Lie代数可均性，证明其与Elek定义等价，并推广至Hopf代数，验证Gromov猜想及多种封闭性质。

Infinite dimensional representations of finite dimensional algebras and amenability

We present a novel approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra $R$ over a finite field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on $R$-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types.…

Skew Meadows

A skew meadow is a non-commutative ring with an inverse operator satisfying two special equations and in which the inverse of zero is zero. All skew fields and products of skew fields can be viewed as skew meadows. Conversely, we give an embedding of non-trivial skew meadows into products of skew fields, from which a completeness result for the equational logic of skew fields is derived. The relationship between regularity conditions on rings and skew meadows is investigated.

On Rota-Baxter operators of non-zero weight induced by non skew-symmetric solutions of the classical Yang-baxter equation on simple anticommutative algebras

Let $L$ be a simple anti-commutative algebra. In this paper we prove that a non skew-symmetric solution of the classical Yang-Baxter equation on $L$ with $L$-invariant symmetric part induces on $L$ a Rota-Baxter operator of a non-zero weight.