Involutions on graded matrix algebras
Yuri Bahturin, Mikhail Zaicev
本文描述了特征不等于2的任意代数闭域上的有限阿贝尔群赋予群分次的矩阵代数上的有穷级分次自同构和反自同构。
In this paper we describe graded automorphisms and antiautomorphisms of finite order on matrix algebras endowed with a group gradings by a finite abelian group over an arbitrary algebraically closed field of charcteristic different from 2.
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Homogeneous involution on graded division algebras and their polynomial identities
本文研究代数闭域上有限维分次可除代数的齐合对合，计算其带对合的分次多项式恒等式，指出齐合对合在分次多项式恒等式与相容对合研究中自然出现。
Identities of finitely generated graded algebras with involution
考虑特征为零的域上对合被有限阿贝尔群G分次的结合代数.假设对合与分次相容。我们表示条件允许PI-表示这样的代数。特别地，证明了对于任意素数q或q = 4，一个带对合的n阶生成（Z/qZ）-分次结合PI-代数与有限维带对合的（Z/qZ）-分次代数满足完全相同的带对合的分次恒等式.这是一个类似的定理的A.凯梅尔普通身份，和推广的结果，作者的身份与对合。对于分次恒等式，最近也证明了类似的结果。
We classify, up to isomorphism and up to equivalence, involutions on graded-division finite-dimensional simple real (associative) algebras, when the grading group is abelian.
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。
Let $F$ be an infinite field, and let $M_{n}(F)$ be the algebra of $n\times n$ matrices over $F$. Suppose that this algebra is equipped with an elementary grading whose neutral component coincides with the main diagonal. In this paper, we find a basis for the graded polynomial identities of $M_{n}(F)$ with the transpose involution. Our results generalize for infinite fields of arbitrary characteristic previous results in the literature which were obtained for the field of complex numbers and for a particular class of elementary G-gradings.
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。
Graded polynomial identities for matrices with the transpose involution
设$G$是阶为$k$的群。我们考虑$k$的代数$M_k(\mathbb{C})$通过$k$矩阵在复数上，并认为它是一个交叉积关于$G$通过嵌入$G$在对称群$S_k$通过正则表示和嵌入$S_k$在$M_k(\mathbb{C})$在通常的方式。这在$M_k(\mathbb{C})$上产生自然的$G$分级，我们称之为交叉积分级。我们研究了具有这种交叉积分次和转置对合的$M_k(\mathbb{C})$的分次$*$ -恒等式。对于具有对合的自由分次代数中的每一个多线性单项式，我们都对应一个有向标号图。利用这些图，我们可以产生一组生成器的$(T,*)$理想的身份。这也导致了新的证明结果Kostant和Rowen的标准身份所满足的斜矩阵。最后我们确定了$M_k(\mathbb{C})$的$*$分次余维数的一个渐近公式.
Finite basis problem for identities with involution
考虑特征为零的域上具有对合的结合代数。证明了任何具有对合的代数都与具有分次对合的有限维$Z_4$ -分次代数的Grassmann包络满足相同的对合恒等式.作为结果，我们得到了正解的Specht问题的身份与对合：任何结合代数与对合域上的特征为零有一个有限的基础的身份与对合。这些结果类似于A. R. Kemer关于普通恒等式的定理。
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。
Graded identities with involution for the algebra of upper triangular matrices
本文研究上三角矩阵代数上带分次对合的群分次，证明其可归结为反射或辛对合，给出恒等式基与余维数增长，并确定指数，特别地证明 $UT_3(F)$ 的普通 $\ast$-指数为 $3$。
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。
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_…
Polynomial identities, central polynomials and cocharacters of $M_2(F)$ with $G$-graded involution
本文研究特征零域上矩阵代数 $M_2(F)$ 的 $G$-分次对合，确定其恒等式与中心多项式的生成元，并显式计算余特征与余维数。
Hook Theorem for Superalgebras with Superinvolution or Graded Involution
本文在特征零域上具有超对合或分次对合的PI-超代数中，证明了多重线性#-超恒等式的钩子定理，并由此导出Amitsur恒等式的类似物，以对称群表示刻画恒等式的组合特征。
On star-homogeneous-graded polynomial identities of upper triangular matrices over an arbitrary field
本文研究任意域上带精细群分次和齐次对合的上三角矩阵代数的分次多项式恒等式，给出基与渐近行为，并计算任意分次与对合下的指数和第二指数。
On the asymptotic behaviour of the graded-star-codimension sequence of upper triangular matrices
本文研究了无限域上赋予群分级和齐次对合的上三角矩阵代数，计算了其分级星余维数序列的渐近行为，发现该渐近增长仅取决于矩阵代数的大小，与分级和对合无关。
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。
Let $F$ be an algebraically closed field of characteristic zero, and $G$ be a finite abelian group. If $A=\oplus_{g\in G} A_g$ is a $G$-graded algebra, we study degree-inverting involutions on $A$, i.e., involutions $*$ on $A$ satisfying $(A_g)^*\subseteq A_{g^{-1}}$, for all $g\in G$. We describe such involutions for the full $n\times n$ matrix algebra over $F$ and for the algebra of $n\times n$ upper triangular matrices.
Finitely generated algebras with involution and their identities
考虑零特征域上具有对合的结合代数。证明了在这种情况下，对任意一个带对合的有限维生成结合代数，存在一个带对合的有限维代数，它满足与带对合的恒等式完全相同的恒等式.
Unitary SK_1 for a Graded Division Ring and its Quotient Division Ring
设E是中心上有限维的分次除环，其分次群是无挠的，q（E）是它的商除环.设τ是E上的保度酉对合。我们证明了SK_1（E，tau）与SK_1（q（E），tau）同构。
Involutions on tensor products of quaternion algebras
研究了具有对合的全可分解代数的可能分解，即具有对合的四元数代数的张量积。特别是，我们感兴趣的分解，其中一个或几个因素是分裂四元数代数$M_2(F)$，赋予一个正交对合。使用规范的理论，开发的蒂格诺尔，沃兹沃斯，我们构造的例子代数同构的张量积的四元数代数$k$分裂因子，赋予一个对合，这是完全可分解的，但不承认任何分解与$k$因子$M_2(F)$与对合。这推广了Sivatski的一个较早的结果，其中所考虑的代数是度为$8$，指数为$4$的代数，并赋予了某些正交对合.
Algebras with involution that become hyperbolic over the fonction field of a conic
本文研究了在四元数代数$Q$的二次曲线的函数域上具有第一类对合的中心单代数的双曲性。我们对这些代数进行了4次分类，并给出了这样一个例子：一个具有8次正交对合的除代数不包含具有其标准对合的$Q$，即使它包含$Q$，并且可完全分解为具有对合的四元数代数的一个张量积.
On split products of quaternion algebras with involution in characteristic two
对特征为2的域上带对合的分裂四元数代数的张量积能否表示为带对合的分裂四元数代数的张量积的问题，给出了肯定的答案.
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.
The Embedding Problem in Algebras with Involution
本文针对特征零代数闭域上的带对合单代数，研究在 $A$ 满足 $B$ 的带对合多项式恒等式时，是否存在保持对合的嵌入 $A\hookrightarrow B$。
Maximal prepositive cones on quaternion algebras with involution
本文研究了四元数代数中带反交换的预正锥，证明了对于广泛类别的四元数代数，每个预正锥都是最大的。
Classification of quantum tori with involution
具有分级对合的量子环面表现为A_1、C和BC型扩展仿射李代数的坐标代数。我们把它们归为具有对合的代数范畴。由此，我们得到了C型扩张仿射李代数的根系的精确信息。
On totally decomposable algebras with involution in characteristic two
给出了特征2域上具有对合的中心单代数可分解为具有对合的四元数代数的张量积的一个充要条件.本文还证明了最近由A. Dolphin，能分类正交型的全可分解中心单代数.
Pfister involutions
在对合的中心单代数的框架内，提出了Pfister形式的概念的类似物的存在性问题。特别地，研究了作为带对合的四元数代数的张量积分裂的带正交对合的代数。它被证明，直到度16，在任何扩展，代数分裂，对合是伴随的一个菲斯特形式。此外，还讨论了具有对合的代数的上同调不变量。
We introduce positive cones on algebras with involution. These allow us to prove analogues of Artin's solution to Hilbert's 17th problem, the Artin-Schreier theorem characterizing formally real fields, and to define signatures with respect to positive cones. We consider the space of positive cones of an algebra with involution and investigate its topological properties, showing in particular that it is a spectral space. As an application we solve the problem of the existence of positive involutions.
Classification of the direct limits of involution simple associative algebras and the corresponding dimension groups
对任意特征的代数闭域上的有限维对合单结合代数代数的（可数）直接极限进行了分类。这也会对相应的维度组进行分类。不变量集由两个超自然数和两个实参数组成。
We prove that there is a monadic adjunction between the category of bounded posets with involution and the category of orthomodular posets.
We provide a coherent picture of our efforts thus far in extending real algebra and its links to the theory of quadratic forms over ordered fields in the noncommutative direction, using hermitian forms and "ordered" algebras with involution.
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。
Let F be a field of characteristic different from $2$, and let $UT_2(F)$ be the algebra of $2\times 2$ upper triangular matrices over $F$. For every involution of the first kind on $UT_2(F)$, we describe the set of all $*$-central polynomials for this algebra.
Formally real involutions on central simple algebras
结合环$R$上的对合$#$是\textit{formally real}，如果形式为$r^# r$的非零元素之和，其中$r \in R$非零。假设$R$是一个中心单代数（即$R=M_n(D)$是某个整数$n$和中心除代数$D$），$#$是$R$上的一个形式为$r^# = a^{-1} r^\ast a$的对合，其中$\ast$是$R$上的某个转置对合，$a$是一个可逆矩阵，使得$a^\ast=\pm a$。在第1节中，我们描述了$#$的形式实在的$a$和$\ast|_D$。在后面的章节中，我们将这个结果应用于交叉积除代数上对合的形式实在的研究。我们可以刻画$D=(K/F,Φ)$上的对合扩展到分裂代数$D \otimes_F K \cong M_n(K)$上的形式实对合。每一个这样的对合是正式真正的，但我们表明，存在正式真正的对合$D$不是这种形式。特别地，存在形式上实的对合$#$，其厄米迹形式$x \mapsto \tr(x^#x)$不是半正定的。
Subnormal subgroups of almost locally simple artinian algebras with involutions
本文研究带对合的几乎局部单阿廷代数乘法群的次正规子群，证明若其迹集或范数集中心，则代数必为四元数除环或二阶矩阵环。
We associate to every central simple algebra with involution of orthogonal type in characteristic two a totally singular quadratic form which reflects certain anisotropy properties of the involution. It is shown that this quadratic form can be used to classify totally decomposable algebras with orthogonal involution. Also, using this form, a criterion is obtained for an orthogonal involution on a split algebra to be conjugated to the transpose involution.
We present new additive results for the core inverse in a Banach algebra with involution. We obtain necessary and sufficient conditions under which the sum of two core invertible elements in a Banach algebra with involution is core invertible. Then we apply our results to block complex matrices and obtain certain conditions under which a block complex is core invertible. These generalize many known results, e.g.,~\cite[Theorem 4.3]{XCZ}, ~\cite[Theorem 2.5]{XS}.
We give various formulas to compute the number of all involutions, i.e. elements of order 2, in an incidence algebra $I(X,\mathbb{K})$, where $X$ is a finite poset (star, Y and Rhombuses) and $\mathbb{K}$ is a finite field of characteristic different from 2. Using the techniques describing here we show an algorithm to calculate the number of involutions on any finite poset.
We study polynomial identities of algebras with involution of nonassociative algebras over a field of characteristic zero. We prove that the growth of the sequence of $*$-codimensions of a finite-dimensional algebra is exponentially bounded. We construct a series of finite-dimensional algebras with fractional $*$-PI-exponent. We also construct a family of infinite-dimensional algebras $C_α$ such that ${\rm exp}^*(C_α)$ does not exist.
In earlier work we developed the theory of signatures of hermitian forms over algebras with involution with respect to orderings on the base field of the algebra and obtained in particular that the total signature of a hermitian form is a continuous function from the space of orderings of that field to $\mathbb{Z}$. In this note we give another presentation of signatures and also introduce and study the stability index of algebras with involution.
We study gradings by abelian groups on associative algebras with involution over an arbitrary field. Of particular importance are the fine gradings (that is, those that do not admit a proper refinement), because any grading on a finite-dimensional algebra can be obtained from them via a group homomorphism (although not in a unique way). We classify up to equivalence the fine gradings on simple associative algebras with involution over the field of real numbers (or any real closed field) and, as a consequence, on the real forms of classical simple Lie algebras.
Central Simple Algebras with Involution: A Geometric Approach
设$k$是特征为零的代数闭基域。将中心单代数与不可约的、一般自由的$PGL_n$ -簇之间的范畴等价性推广到具有对合的中心单代数的范畴.具有对合的中心单代数的相关簇带有半直积$P_{n,τ}:=PGL_n\rtimes<τ>$的作用，其中$τ$是由$τ(h)=(h^{-1})^transpose$给出的$\PGLn$的自同构。对合的基本性质是根据$P_{n,τ}$对相关变种的作用来描述的，特别是根据这种作用的一般位置的稳定剂来描述的。
Deforming algebras with anti-involution via twisted associativity
本文证明用反对合扭曲乘法可得到 II 型 Hom-associative 代数，建立相应模理论与忠实函子，并提出图解 operad 推广路径。
Let $FI(X,K)$ be the finitary incidence algebra of a non-connected partially ordered set $X$ over a field $K$ of characteristic different from $2$. For the case where every multiplicative automorphism of $FI(X,K)$ is inner, we present necessary and sufficient conditions for two involutions on $FI(X,K)$ to be equivalent.
The Plesken Lie algebra for associative algebras with anti-involution: semisimple cellular algebras
本文将Plesken李代数从有限群复群代数推广到任意带反合结合代数，并证明半单胞腔代数的Plesken李代数可分解为正交李代数的直和，其大小由胞腔模维数决定。
$W$-algebras with involution: polynomial identities and asymptotic growth
本文针对特征零域上的带对合$W$-代数，建立了带对合的$W$-多项式恒等式理论，证明了$(W,*)$-指数与普通$*$-指数一致，并具体分析了四维上三角矩阵子代数的两种不等价结构，计算了其恒等式理想与余维数序列，最终证明它们生成不同的几乎多项式增长簇。
Some general criteria to produce explicit free algebras inside the division ring of fractions of skew polynomial rings are presented. These criteria are applied to some special cases of division rings with natural involutions, yielding, for instance, free subalgebras generated by symmetric elements both in the division ring of fractions of the group algebra of a torsion free nilpotent group and in the division ring of fractions of the first Weyl algebra.
In characteristic two, it is shown that a central simple algebra of degree equal to a power of two with anisotropic orthogonal involution is totally decomposable, if it becomes either anisotropic or metabolic over all extensions of the ground field. A similar result is obtained for the case where this algebra with involution is Brauer-equivalent to a quaternion algebra and it becomes adjoint to a bilinear Pfister form over all splitting fields of the algebra.
Mixed Witt rings of algebras with involution
本文通过将双线性形式与厄米特形式结合，为第一类带对合代数定义了混合Witt环，并推导了其基于对合迹形式的乘积公式与互反公式等基本性质。
Signature maps from positive cones on algebras with involution
本文直接从正锥描述带有对合的中心单代数上埃尔米特形式的符号，并修正了前文受影响的结果。
Continuity of total signature maps for Azumaya algebras with involution
本文将中心单代数的签名研究方法推广至带对合的Azumaya代数，实现了给定序下埃尔米特型签名的自然选取，证明所得全签名是基环实谱上的连续函数。
A Knebusch trace formula for Azumaya algebras with involution
本文提出了一种针对具有involution的Azumaya代数的踪公式，扩展了Knebusch的工作，并探讨了总踪序列与稳定性指数的关系。
We study totally decomposable symplectic and unitary involutions on central simple algebras of index 2 and on split central simple algebras respectively. We show that for every field extension, these involutions are either anisotropic or hyperbolic after extending scalars, and that the converse holds if the algebras are of 2-power degree. These results are new in characteristic 2, otherwise were shown in [3] and [6] respectively.
The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution
在1976年Procesi和Schacher开发了一个阿廷Schreier型理论的中心单代数与对合和profetured，在这样的代数一个完全积极的元素总是一个总和埃尔米特平方。本文构造了这个猜想的基本反例，并研究了这个猜想成立的情况。此外，一个Positivstellenarchy建立的非交换多项式，半正定的所有元组的矩阵的一个固定的大小。
A generic characterization of direct summands for orthogonal involutions
在二次型的代数理论中的“超越方法”是基于两个主要的结果，在60年代由Cassels和Pfister证明，并被称为表示定理和子形式定理。1996年，让-皮埃尔·泰尔在对合的中心单代数中证明了表示定理的推广。本文研究了正交对合的子形问题。给出了直和项的一般刻画;证明了除代数和指数至多为2的代数的子形定理的一个类似物。
An involution of a real commutative algebra $A$ is a real-linear homomorphism $f : A \rightarrow A$ such that $f^2 = \mathrm{Id}$. We show that there are six involutions of the algebra of bicomplex numbers, contrary to the actual number of four stated in the literature. We also characterize $n$-involutions satisfying the additional property $f^n = \mathrm{Id}$ for some integer $n \geq 2$. We show there are eight $n$-involutions and they occur only for $n = 2$ and $n= 4$. We use our result to give a new characterization of the invertible elements of the algebra of bicomplex numbers.
Let $\mathbb{F}$ be a field of characteristic $p$, and let $UT_n(\mathbb{F})$ be the algebra of $n \times n$ upper triangular matrices over $\mathbb{F}$ with an involution of the first kind. In this paper we describe: the set of all $*$-central polynomials for $UT_n(\mathbb{F})$ when $n\geq 3$ and $p\neq 2$ ; the set of all $*$-polynomial identities for $UT_3(\mathbb{F})$ when $\mathbb{F}$ is infinite and $p>2$.
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。
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