Tame group actions on central simple algebras

可数无限秩自由幂零李代数的驯顺自同构

Tame Automorphisms of Free Nilpotent Lie Algebras of Countably Infinite Rank

本文证明特征零域上可数无限秩自由幂零李代数L_{∞,c}的每个自同构都由自由李代数L_∞的自同构诱导。

Representation-tame algebras need not be homologically tame

We show that, also within the class of representation-tame finite dimensional algebras $Λ$, the big left finitistic dimension of $Λ$ may be strictly larger than the little. In fact, the discrepancies $Findim Λ- findim Λ$ need not even be bounded for special biserial algebras which constitute one of the (otherwise) most thoroughly understood classes of tame algebras. More precisely: For every positive integer $r$, we construct a special biserial algebra $Λ$ with the property that $findim Λ= r + 1$, while $Findim Λ= 2r + 1$. In particular, there are infinite dimensional representations of $Λ$ wh…

Baer and Mittag-Leffler modules over tame hereditary algebras

We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones.

Graded tame blocks of group algebras

In this paper we investigate gradings on tame blocks of group algebras whose defect group is dihedral. We classify gradings on an arbitrary dihedral block up to graded Morita equivalence. We do this by computing the group of outer automorphisms that fix the isomorphism classes of simple modules. We also show how to grade these blocks via transfer of gradings via dervied equivalences.

Integral bases of cluster algebras and representations of tame quivers

In \cite{CK2005} and \cite{SZ}, the authors constructed the bases of cluster algebras of finite types and of type $\widetilde{A}_{1,1}$, respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras of affine types.

关于驯顺型cluster-tilted代数上不可分解$τ$-刚性模的研究

On indecomposable $τ$-rigid modules over cluster-tilted algebras of tame type

本文证明了驯顺型cluster-tilted代数上不同的不可分解$τ$-刚性模具有不同的维数向量，并以此为基础建立了驯顺型cluster algebras的弱版本分母猜想，同时利用Iyama-Yoshino构造描述了相关的三角范畴subfactors。

Is tame open?

Is tame open? No answer so far. One may pose the Tame-Open Conjecture: Tame is open. But how to support it? No effective way to date. In this note, the rank of a wild algebra is introduced. The Wild-Rank Conjecture, which implies the Tame-Open Conjecture, is formulated. The Wild-Rank Conjecture is improved to the Basic-Wild-Rank Conjecture. A covering criterion on the rank of a basic wild algebra is given, which can be effectively applied to verify the Basic-Wild-Rank Conjecture for concrete algebras. It makes all conjectures much reliable.

Tamely ramified subfields of division algebras

For any number field K, it is unknown which finite groups appear as Galois groups of extensions L/K such that L is a maximal subfield of a division algebra with center K (a K-division algebra). For K=Q, the answer is described by the long standing Q-admissibility conjecture. We extend a theorem of Neukirch on embedding problems with local constraints in order to determine for every number field K, what finite solvable groups G appear as Galois groups of tame maximal subfields of K-division algebras, generalizing Liedahl's theorem for metacyclic G and Sonn's solution of the Q-admissibility conj…

Rigid and Schurian modules over cluster-tilted algebras of tame type

We give an example of a cluster-tilted algebra A with quiver Q, such that the associated cluster algebra has a denominator vector which is not the dimension vector of any indecomposable A-module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra A, we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid A-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomp…

Filtrations and Distortion in Infinite-Dimensional Algebras

A tame filtration of an algebra is defined by the growth of its terms, which has to be majorated by an exponential function. A particular case is the degree filtration used in the definition of the growth of finitely generated algebras. The notion of tame filtration is useful in the study of possible distortion of degrees of elements when one algebra is embedded as a subalgebra in another. A geometric analogue is the distortion of the (Riemannian) metric of a (Lie) subgroup when compared to the metric induced from the ambient (Lie) group. The distortion of a subalgebra in an algebra also refle…

自由 Poisson 代数的一个 Anick 型野生自同构

An Anick type wild automorphism of free Poisson algebras

本文在 3 生成自由 Poisson 代数中构造 Anick 型野生自同构 $δ$，证明其在多项式代数上诱导驯顺自同构且本身稳定驯顺。

表示驯服的Geiß-Leclerc-Schröer代数及关于根系的一个修正的GLS猜想

Representation-tame Geiß-Leclerc-Schröer algebras and a revised GLS conjecture on root systems

本文分类了表示驯服的GLS代数，研究其最小对称化子情形，通过扩展弦带描述AR箭图，并修正了仿射型GLS猜想，证明秩向量对应正根及非根向量。

从有限到无限长度模在温和余核代数上的表示

From finite to infinite length modules over tame hereditary algebras

本文基于有限维表示的基础，介绍无限维表示，重点描述纯射模块，并探讨无 torsion 自由可除模块的核心贡献。

Hasse Invariant for the Tame Brauer Group of a Higher Local Field

We generalize the Hasse invariant of local class field theory to the tame Brauer group of a higher dimensional local field, and use it to study the arithmetic of central simple algebras over such fields, which are given {\it a priori} as tensor products of standard cyclic algebras. We also compute the tame Brauer dimension (or {\it period-index bound}) and the cyclic length of a general henselian-valued field of finite rank and finite residue field.

Automorphisms of Weyl Algebra and a Conjecture of Kontsevich

We outline the proof of a conjecture of Kontsevich on the isomorphism between the group of polynomial symplectomorphisms in $2n$ variables and the group of automorphisms of the $n$-th Weyl algebra over complex numbers. Our proof uses lifting of polynomial symplectomorphisms to Weyl algebra automorphisms by means of approximation by tame symplectomorphisms and gauging of the lifted morphism. Approximation by tame symplectomorphisms is the symplectic version of the well-known theorem of D. Anick and is a result of our prior work.

On Automorphisms of the Tame Polynomial Automorphism Group in Positive Characteristic

In this paper we prove that over algebraically closed field $K$ of positive characteristic $\neq 2$ every automorphism of the group of origin-preserving automorphisms of the polynomial algebra $K[x_1,\ldots, x_n]$ ($n>3$) which fixes every diagonal matrix preserves, up to composition with a linear inner automorphism, every tame automorphism.

Coordinates and Automorphisms of Polynomial and Free Associative Algebras of Rank Three

We study z-automorphisms of the polynomial algebra K[x,y,z] and the free associative algebra K<x,y,z> over a field K, i.e., automorphisms which fix the variable z. We survey some recent results on such automorphisms and on the corresponding coordinates. For K<x,y,z> we include also results about the structure of the z-tame automorphisms and algorithms which recognize z-tame automorphisms and z-tame coordinates.

有限维 Jacobian 代数：有限性与驯顺性

Finite-dimensional Jacobian algebras: Finiteness and tameness

本文研究有限维 Jacobian 代数的表示型，证明$E$-有限性与$E$-驯顺性在突变下不变，并据此完成分类、证明 Demonet 猜想及 Reading 定理逆命题。

Tame division algebras of prime period over function fields of $p$-adic curves

Let F be a field of transcendence degree one over a p-adic field, and let l be a prime not equal to p. Results of Merkurjev and Saltman show that H^2(F,μ_l) is generated by Z/l-cyclic classes. We prove the "Z/l-length" in H^2(F,μ_l) equals the l-Brauer dimension, which Saltman showed to be two. It follows that all F-division algebras of period l are crossed products, either cyclic (by Saltman's cyclicity result) or tensor products of two cyclic division algebras. Our result was originally proved by Suresh assuming F contains μ_l.

Convex subshifts, separated Bratteli diagrams, and ideal structure of tame separated graph algebras

We introduce a new class of partial actions of free groups on totally disconnected compact Hausdorff spaces, which we call convex subshifts. These serve as an abstract framework for the partial actions associated with finite separated graphs in much the same way as classical subshifts generalize the edge shift of a finite graph. We define the notion of a finite type convex subshift and show that any such subshift is Kakutani equivalent to the partial action associated with a finite bipartite separated graph. We then study the ideal structure of both the full and the reduced tame graph C*-algeb…

Stratifications of derived categories from tilting modules over tame hereditary algebras

In this paper, we consider the endomorphism algebras of infinitely generated tilting modules of the form $R_{\mathcal U}\oplus R_{\mathcal U}/R$ over tame hereditary $k$-algebras $R$ with $k$ an arbitrary field, where $R_{\mathcal{U}}$ is the universal localization of $R$ at an arbitrary set $\mathcal{U}$ of simple regular $R$-modules, and show that the derived module category of $\End_R(R_{\mathcal U}\oplus R_{\mathcal U}/R)$ is a recollement of the derived module category $\D{R}$ of $R$ and the derived module category $\D{{\mathbb A}_{\mathcal{U}}}$ of the adèle ring ${\mathbb A}_{\mathcal{U…

温和余表示型的余分次Hopf代数

Coradically graded Hopf algebras of tame corepresentation type

本文通过链箭图方法，刻画了具有对偶Chevalley性质的有限维Hopf代数的余表示类型，并给出了温和余表示型的充要条件。

Exchange rings and real rank zero C*-algebras associated with finitely separated graphs

We introduce a generalisation of Condition (K) to finitely separated graphs and show that it is equivalent to essential freeness of the associated partial action as well as the exchange property of any of the associated tame algebras. As a consequence, we can show that any tame separated graph algebra with the exchange property is separative.

On the infinite tame-wild dichotomy conjecture and related problemns

We prove the tame-wild dichotomy conjecture, due to D. Simson, for infinite dimensional algebras and coalgebras. The key part of the approach is proving new representation theoretic characterizations local finiteness. Among other, we show that the Ext quiver of the category ${\rm f.d.-}A$ of finite dimensional representations of an arbitrary algebra $A$ is locally finite (i.e. $\dim(\Ext^1(S,T))<\infty$ for all simple finite dimensional $A$-modules $S,T$) if and only if for every dimension vector $\underline{d}$, the representations of $A$ of dimension vector $\underline{d}$ are all contained …

On wild Lie algebras

We give a criterion of tameness and wildness for a finite-dimensional Lie algebra over an algebraically closed field.

Tame 箭图与仿射基 I：典范基的霍尔代数方法

Tame quivers and affine bases I: a Hall algebra approach to the canonical bases

本文针对仿射型量子群，利用tame箭图表示导出的PBW基构造bar不变基，证明其与Lusztig典范基一致并给出双射，指标集按特定模与对称群不可约特征有序排列，方法推广自Deng-Du-Xiao的工作。

Algebras of quasi-quaternion type

We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe that symmetric tame algebras that are also 2-CY-tilted are of quasi-quaternion type. We present a combinatorial construction of such algebras by introducing the notion of triangulation quivers. The class of algebras that we get contains Erdmann's algebras of quaternion type on the one hand and the Jacobian algebras of the quivers with potentials associated by …

自由双交换李代数的自同构，I

Automorphisms of free metabelian Lie algebras, I

本文研究了自由双交换李代数的自同构性质，证明了特定条件下自同构的 tame 性及部分自同构的几乎 tame 性，推翻了领域内长期存在的结论。

驯服箭图与仿射基II：非对称情形

Tame quivers and affine bases II: nonsimply-laced cases

本文用Ringel-Hall代数方法将仿射典范基推广到一般对称化情形，证明所得bar不变基与Lusztig带符号典范基一致，并在$\tilde{B}_n,\tilde{C}_n$型中直接给出典范基。

Cyclic Length in the Tame Brauer Group of the Function Field of a p-Adic Curve

Let $F$ be the function field of a smooth curve over the $p$-adic number field $\Q_p$. We show that for each prime-to-$p$ number $n$ the $n$-torsion subgroup $\H^2(F,μ_n)={}_n\Br(F)$ is generated by $\Z/n$-cyclic classes; in fact the $\Z/n$-length is equal to two. It follows that the Brauer dimension of $F$ is two (first proved in \cite{Sa97}), and any $F$-division algebra of period $n$ and index $n^2$ is decomposable.