Classification of quantum tori with involution

具有分级反演的代数结构分类及二次余维增长

A structural classification of algebras with graded involution and quadratic codimension growth

本文研究了具有分级反演的代数结构，通过分类具有二次余维增长的代数，揭示了变种生成与最小变种分解之间的对应关系。

On biquaternion algebras with involution of orthogonal type

We investigate the pfaffians of decomposable biquaternion algebras with involution of orthogonal type. In characteristic two, a classification of these algebras in terms of their pfaffians and some other related invariants is studied. Also, in arbitrary characteristic, a criterion is obtained for an orthogonal involution on a biquaternion algebra to be metabolic.

具有分级反演的代数的最小品种与二次增长

Minimal varieties of algebras with graded involution and quadratic growth

本文研究了具有有限群 $G$ 和分级反演的代数的最小品种分类，重点在于二次增长的代数结构。

关于矩阵代数上的同构反演

On homogeneous involutions on matrix algebras

本文研究了在特定分级下的矩阵代数中的同构反演分类及其在不同分级下的性质。

On the Classification of Algebras

We classify (possibly non commutative) algebras of low rank over a domain R. We first review results for algebras of rank 2 and for finite-dimensional division algebras over the real numbers. These results motivate us to consider which algebras possess a standard involution. Our main result is that algebras of rank 3 are either commutative or possess a standard involution.

Cayley-Dickson构造中的对合

Involutions in the Cayley-Dickson construction

本文确定了Cayley-Dickson构造中所有扩展原$*$-代数对合的对合，并分类了由此得到的$*$-代数同构类，应用包括证明无零因子的Cayley双代数恰好有一个额外对合，以及分类实数域上至多4维的$*$-代数。

超代数与带对合的代数：三次余维序列的分类

Superalgebras and Algebras with Involution: Classifying Cubic Codimension Sequences

本文对呈现三次多项式增长的φ-余维序列的单位φ-代数完成分类，并确定了φ-余维增长至多二次的单位φ-代数相关T^φ-理想的最低次数多重线性生成元。

Anti-isomorphisms and involutions on the idealization of the incidence space over the finitary incidence algebra

Let $K$ be a field and $X$ a partially ordered set (poset). Let $FI(X,K)$ and $I(X,K)$ be the finitary incidence algebra and the incidence space of $X$ over $K$, respectively, and let $D(X,K)=FI(X,K)(+)I(X,K)$ be the idealization of the $FI(X,K)$-bimodule $I(X,K)$. In the first part of this paper, we show that $D(X,K)$ has an anti-automorphism (involution) if and only if $X$ has an anti-automorphism (involution). We also present a characterization of the anti-automorphisms and involutions on $D(X,K)$. In the second part, we obtain the classification of involutions on $D(X,K)$ to the case when …

On the classification of rational quantum tori and the structure of their automorphism group

An n-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational n-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

具有均质反交换的分级代数与几乎多项式增长的变体

Graded algebras with homogeneous involution and varieties of almost polynomial growth

本文研究了具有均质反交换的G-分级代数中码度序列的行为，通过排除列表中的代数来表征多项式增长的变体，并提供了几乎多项式增长变体的分类。

关于带有格度反演的代数的colength序列

On the colength sequence of algebras with graded involution

本文研究了带有格度反演的代数的colength序列，通过显式描述分解来分类其变体。

Values of Noncommutative Polynomials, Lie Skew-Ideals and the Tracial Nullstellensatz

A subspace of an algebra with involution is called a Lie skew-ideal if it is closed under Lie products with skew-symmetric elements. Lie skew-ideals are classified in central simple algebras with involution (there are eight of them for involutions of the first kind and four for involutions of the second kind) and this classification result is used to characterize noncommutative polynomials via their values in these algebras. As an application, we deduce that a polynomial is a sum of commutators and a polynomial identity of $d\times d$ matrices if and only if all of its values in the algebra of…

Unitary Lie Algebras and Lie Tori of Type BC_r, r \geq 3

A Lie G-torus of type X_r is a Lie algebra with two gradings -- one by an abelian group G and the other by the root lattice of a finite irreducible root system of type X_r. In this paper we construct a centreless Lie G-torus of type BC_r, which we call a unitary Lie G-torus, as it is a special unitary Lie algebra of a nondegenerate G-graded hermitian form of Witt index r over an associative torus with involution. We prove a structure theorem for centreless Lie G-tori of type BC_r, r \geq 3, that states that any such Lie torus is bi-isomorphic to a unitary Lie G-torus, and we determine necessar…

K-theory Classification of Graded Ultramatricial Algebras with Involution

We consider a generalization $K_0^{\operatorname{gr}}(R)$ of the standard Grothendieck group $K_0(R)$ of a graded ring $R$ with involution. If $Γ$ is an abelian group, we show that $K_0^{\operatorname{gr}}$ completely classifies graded ultramatricial $*$-algebras over a $Γ$-graded $*$-field $A$ such that (1) each nontrivial graded component of $A$ has a unitary element in which case we say that $A$ has enough unitaries, and (2) the zero-component $A_0$ is 2-proper (for any $a,b\in A_0,$ $aa^*+bb^*=0$ implies $a=b=0$) and $*$-pythagorean (for any $a,b\in A_0,$ $aa^*+bb^*=cc^*$ for some $c\in A_…

复环面上自守李代数的分类

A classification of automorphic Lie algebras on complex tori

本文分类了复环面到 $\mathfrak{sl}_2(\mathbb{C})$ 的等变映射所对应的自守李代数，给出规范形基，并证明除四种同构于 Onsager 代数的情形外，其余均由 $\mathrm{PSL}_2(\mathbb{Z})$ 的模曲线参数化为两个互不相交的族。