Generic algebras with involution of degree 8m

Hyperbolicity of orthogonal involutions

We show that a non-hyperbolic orthogonal involution on a central simple algebra over a field of characteristic different from 2 remains non-hyperbolic over some splitting field of the algebra.

Commuting involutions of Lie algebras, commuting varieties, and simple Jordan algebras

We study certain "σ-commuting varieties" associated with a pair of commuting involutions of a semisimple Lie algebra $\g$. The usual commuting variety of $\g$ and commuting varieties related to one involution are particular cases of our construction. We develop a general theory of σ-commuting varieties and point out some cases, when they have especially good properties. We show that, for a special choice of commuting involutions, the σ-commuting variety is isomorphic to the commuting variety of a simple Jordan algebra. As a by-product of our theory, we show that if $J$ is the Jordan algebra of…

A Combinatorial Approach to Involution and delta-Regularity I: Involutive Bases in Polynomial Algebras of Solvable Type

General revision. In particular the parts concerning involutive bases over rings have been significantly changed. In addition some proofs have been improved.

On the algebraic cobordism ring of involutions

We consider the cobordism ring of involutions of a field of characteristic not two, whose elements are formal differences of classes of smooth projective varieties equipped with an involution, and relations arise from equivariant K-theory characteristic numbers. We investigate in detail the structure of this ring. Concrete applications are provided concerning involutions of varieties, relating the geometry of the ambient variety to that of the fixed locus, in terms of Chern numbers. In particular, we prove an algebraic analog of Boardman's five halves theorem in topology, of which we provide s…

Critical varieties and motivic equivalence for algebras with involution

Motivic equivalence for algebraic groups was recently introduced in [9], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F_2, this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic eq…

具有正交对合的 Azumaya 代数容许非正常等距

Azumaya Algebras With Orthogonal Involution Admitting an Improper Isometry

本文研究具有正交对合的 Azumaya 代数，证明若存在非正常等距则 Brauer 类平凡，并指出该假设不足以推出矩阵代数结构。

Iwahori-Matsumoto involution and linear Koszul Duality

We use linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras studied in previous papers of the authors to give a geometric realization of the Iwahori-Matsumoto involution of affine Hecke algebras. More generally we prove that linear Koszul duality is compatible with convolution in a general context related to convolution algebras.

具有反演的有向图与通过库仑分支的移位扭曲杨代数

Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches II

本文通过库仑分支研究具有反演的有向图与移位扭曲杨代数之间的代数同态关系。

带对合的代数中不可分解幂等元的唯一性

Uniqueness of indecomposable idempotents in algebras with involution

通过推广 Eichler 的格分解结果，本文证明了带正对合的有限维 $\mathbb{Q}$-代数阶中不可分解埃尔米特幂等元分解的唯一性，并据此确立了任意域上极化阿贝尔簇及积分 Hodge 结构的唯一分解定理。

带有对合的箭图与通过库仑分支的位移扭曲杨氏量

Quivers with Involutions and Shifted Twisted Yangians via Coulomb Branches

本文研究了带有对合的箭图的3d $\mathcal{N}=4$规范理论对合固定部分的库仑分支，证明了从位移扭曲杨氏量到量子化库仑分支代数存在代数同态，从而为3D镜像对称提供了新实例。

双有理对合

A birational involution

本文基于格理论预测并利用Mukai模型与例外群作用描述了18次K3曲面Hilbert立方的反辛双有理对合，证明其不定轨迹与同调射影对偶性相关，并得出该对合是Mukai flop实例的结论。

椭圆纤维丛的Manin对合与离散可积系统

Manin involutions for elliptic pencils and discrete integrable systems

本文研究了平面双有理映射生成的离散可积系统的代数几何，给出了高次椭圆纤维丛的Manin对合几何描述，并刻画了基点特殊几何特征以确保Manin对合复合构成低次可积映射，同时将部分可积Kahan离散化识别为Manin对合的复合。

Geometry of four-folds with three non-commuting involutions

In this paper we adapt some techniques developed for K3 surfaces, to study the geometry of a family of projective varieties in $\Pl_K^2 \times \Pl_K^2 \times \Pl_K^2$ defined as the intersection of a form of degree $(2,2,2)$ and a form of degree $(1,1,1)$. Members of the family will be equipped with dominant rational self-maps and we will study the actions of those maps on divisors and compute the first dynamical degrees of the composition of any pair.

An $8$-Periodic Exact Sequence of Witt Groups of Azumaya Algebras with Involution

Given an Azumaya algebra with involution $(A,σ)$ over a commutative ring $R$ and some auxiliary data, we construct an $8$-periodic chain complex involving the Witt groups of $(A,σ)$ and other algebras with involution, and prove it is exact when $R$ is semilocal. When $R$ is a field, this recovers an $8$-periodic exact sequence of Witt groups of Grenier-Boley and Mahmoudi, which in turn generalizes exact sequences of Parimala--Sridharan--Suresh and Lewis. We apply this result in several ways: We establish the Grothendieck--Serre conjecture on principal homogeneous bundles and the local purity c…

On The Gersten-Witt Complex of an Azumaya Algebra with Involution

Let $(A,σ)$ be an Azumaya algebra with involution over a regular ring $R$. We prove that the Gersten-Witt complex of $(A,σ)$ defined by Gille is isomorphic to the Gersten-Witt complex of $(A,σ)$ defined by Bayer-Fluckiger, Parimala and the author. Advantages of both constructions are used to show that the Gersten-Witt complex is exact when $\dim R\leq 3$, $\mathrm{ind}\, A\leq 2$ and $σ$ is orthogonal or symplectic. This means that the Grothendieck-Serre conjecture holds for the group $R$-scheme of $σ$-unitary elements in $A$ under the same hypotheses; $R$ is not required to contain a field.