A note in algebras with Laurent polynomial identity

Algebras with Laurent polynomial identity

In this article we shows some results about algebra with the group of units having special polynomial identity.

Group algebras whose units satisfy a Laurent Polynomial Identity

Let $KG$ be the group algebra of a torsion group $G$ over a field $K$. We show that if the units of $KG$ satisfy a Laurent polynomial identity which is not satisfied by the units of the relative free algebra $K[α,β: α^2=β^2=0]$ then $KG$ satisfies a polynomial identity. This extends Hartley Conjecture which states that if the units of $KG$ satisfies a group identity then $KG$ satisfies a polynomial identity. As an application of our results we prove that if the units of $KG$ satisfies a Laurent polynomial identity with a support of cardinality at most 3 then $KG$ satisfies a polynomial identit…

Nil-generated algebras and group algebras whose units satify a Laurent polynomial identity

Let A be an algebra whose group of units U(A) satisfies a Laurent polynomial identity (LPI). We establish conditions on these polynomials in such a way that nil-generated algebras and group algebras with torsion groups over infinite fields in characteristic p > 0 have nonmatrix identities. We also determine, in the determine in the context of group algebras with arbitrary LPI for the group of units, the existence of polynomial identities.

A1-homotopy invariants of corner skew Laurent polynomial algebras

In this note we prove some structural properties of all the A1-homotopy invariants of corner skew Laurent polynomial algebras. As an application, we compute de mod-l algebraic K-theory of Leavitt path algebras using solely the kernel/cokernel of the incidence matrix. This leads naturally to some vanishing and divisibility properties of the algebraic K-theory of these algebras.

二维仿射霍普夫代数的微分光滑性

Differential smoothness of affine Hopf algebras of Gelfand-Kirillov dimension two

构造了单变量多项式环和洛朗多项式环的Ore扩张类的二维可积微分演算。从而得出结论：所有非多项式单位环的Gelfand-Kirillov维2的仿射点Hopf域都是微分光滑的.

$A_1^{(1)}$型簇代数的半正则基生成元

Semicanonical basis generators of the cluster algebra of type $A_1^{(1)}$

研究了仿射型无系数簇代数$A_1^{(1)}$的簇变量和半正则基的虚元。P.Caldero和作者在math.RT/0604054中得到了这些元素的Laurent展开式的一个闭公式。作为副产品，给出了所讨论的洛朗多项式的组合解释，等价于G.Musiker和J.Propp在Math.CO/0602408中获得的解释。math.RT/0604054中的参数使用了P.Caldero和F.Chapoton（math.RT/0410184）对Laurent多项式的几何解释。本文给出了同样结果的一个快速、完备和完全初等的替代证明。