Bibinur Duisengalieva, Altyngul Naurazbekova, Ualbai Umirbaev
It is proved that the tame automorphism group of a differential polynomial algebra $k\{x,y\}$ over a field $k$ of characteristic $0$ in two variables $x,y$ with $m$ commuting derivations $δ_1, \ldots, δ_m$ is a free product with amalgamation. An example of a wild automorphism of the algebra $k\{x,y\}$ in the case of $m\geq 2$ derivations is constructed.
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We study automorphisms of the free associative algebra K<x,y,z> over a field K which fix the variable z. We describe the structure of the group of z-tame automorphisms and derive algorithms which recognize z-tame automorphisms and z-tame coordinates.
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。
We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: "Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of KQ is the path coalgebra of a quiver with relations".
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.
The class of refinement monoids (abelian monoids satisfying the Riesz refinement property) is subdivided into those which are tame, defined as being an inductive limit of finitely generated refinement monoids, and those which are wild, i.e., not tame. It is shown that tame refinement monoids enjoy many positive properties, including separative cancellation ($2x=2y=x+y \implies x=y$) and multiplicative cancellation with respect to the algebraic ordering ($mx\le my \implies x\le y$). In contrast, examples are constructed to exhibit refinement monoids which enjoy all the mentioned good properties…
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.
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。
It is proved that every z-automorphism (z-coordinate, respectively) of the free associative algebra F<x,y,z> over an arbitrary field F is stably tame.
An example of a non-triangulable locally nilpotent derivation and an example of a wild exponential automorphism of the free Novikov algebra $N\left\langle x,y,z\right\rangle$ in three variables $x,y,z$ over a field of characteristic zero are constructed.
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 …
We study automorphisms of the free associative algebra K<x,y,z> over a field K which fix z and such that the images of x, y are linear with respect to x, y. We prove that some of these automorphisms are wild in the class of all automorphisms fixing z, including the well known automorphism discovered by Anick, and show how to recognize the wild ones. This class of automorphisms induces tame automorphisms of the polynomial algebra K[x,y,z]. For n>2 the automorphisms of K<x_1,...,x_n,z> which fix z and are linear in the x's are tame.
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.
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.
In 2004 Shestakov and Umirbaev proved that the Nagata automorphism of the polynomial algebra in three variables is wild. We fix a Z-grading on this algebra and consider graded-wild automorphisms, i.e. such automorphisms that can not be decomposed onto elementary automorphisms respecting the grading. We describe all gradings allowing graded-wild automorphisms.
Erroneous proofs of the wildness of some automorphisms of free metabelian Lie algebras
本文指出 Bahturin–Nabiyev 与 Özcurt–Ekici 关于自由亚阿贝尔李代数野性自同构的证明存在不可修正错误，强调秩 n≥4 情形的驯顺/野性问题仍未解决。
We give a criterion of tameness and wildness for a finite-dimensional Lie algebra over an algebraically closed field.
An explicit tame factorization of the four-variable Anick automorphism
本文给出四变量多项式环中Anick自同构的显式 tame 分解，含16个固定z的初等自同构，定义在ℤ[1/2]上，未涉及特征2情形。
Let R be a PID. We construct and classify all coordinates of R[x,y] of the form p_2y+Q_2(p_1x+Q_1(y)) with p_1 and p_2 in qt(R) and Q_1 and Q_2 in qt(R)[y]. From this construction (with R=K[z]) we obtain non tame automorphisms s of K[x,y,z] (where K is a field of characteristic 0) such that the sub-group generated by s and the affine automorphisms contains all tame automorphisms.
Over a field of characteristic zero, we prove that the Freiheitssatz holds for brace algebras, the word problem for the brace algebras with a single defining relation is decidable, two generated subalgebras of free brace algebras are free, and that automorphisms of two generated free brace algebras are tame.
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。
We prove the Freiheitssatz for right-symmetric algebras and the decidability of the word problem for right-symmetric algebras with a single defining relation. We also prove that two generated subalgebras of free right-symmetric algebras are free and automorphisms of two generated free right-symmetric algebras are tame.
We prove that the group of tame automorphisms of a free Lie algebra (as well as of a free anticommutative algebra) rank 3 over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra (as well as of a free anticommutative algebra) of rank 3 over an arbitrary Euclidean ring analogous to the Anick automorphism for free associative algebras.
Let $Λ$ be a finite-dimensional algebra over an algebraically closed field, then $Λ$ is either tame or wild. Is there any homological description in terms of AR-translations on tameness? Or equivalently, is there any combinatorial description in terms of AR-quivers? The answer is yes. In the present paper we prove the following main theorem: ``$Λ$ is tame if and only if almost all modules are isomorphic to their Auslander-Reiten translations, if and only if they lie in homogeneous tubes". The method used in the paper is bimodule problem, which has been studied by several authors. We treat bimo…
Tangent Lie Algebras of Automorphism Groups of Free Algebras
本文为自由代数自同构群引入切李代数，证明其包含于常散度导子李代数，并利用该工具证明几乎所有已知野生自同构（除Nagata和Anick外）均为绝对野生，包括Bergman自同构及多数多项式幂零李代数簇中的自由代数。
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.
Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra K<x,y,z> over a field K of characteristic 0. In particular, the well-known Anick automorphism is wild. In this article we obtain a stronger result (the Strong Anick Conjecture that implies the Anick Conjecture). Namely, we prove that there exist wild coordinates of K<x,y,z>. In particular, the two nontrivial coordinates in the Anick automorphism are both wild. We establish a similar result for several large classes of automorphisms of K<x,y,z>. We also find a…
We construct an analogue of the Nagata automorphism of free nonassociative algebras and free commutative algebras of rank two over a Euclidean domain and prove that it is wild.
Let P be a free Poisson algebra in two variables over a field of characteristic zero. We prove that the automorphisms of P are tame and that the locally nilpotent derivations of P are triangulable.
Chein Automorphisms of Free Metabelian Anticommutative Algebras
本文研究了自由双交换交换代数中只移动一个变量的Chein自同构，证明了当秩为2时自同构是线性的，并指出度数为3的Chein自同构在n≥3时具有绝对野性。
We prove that the well-known Anick automorphism of the free associative algebra F<x,y,z> over an arbitrary field F of characteristic 0 is wild.
Motivated by Weyl algebra analogues of the Jacobian conjecture and the Tame Generators problem, we prove quantum versions of these problems for a family of analogues to the Weyl algebras. In particular, our results cover the Weyl-Hayashi algebras and tensor powers of a quantization of the first Weyl algebra which arises as a primitive factor algebra of $U_q^+(\mathfrak{so}_5)$.
It is proved that the Nagata automorphism (Nagata coordinates, respectively) of the polynomial algebra $F[x,y,z]$ over a field $F$ cannot be lifted to a $z$-automorphism ($z$-coordinate, respectively) of the free associative algebra $K<x,y,z>$. The proof is based on the following two new results which have their own interests: degree estimate of ${Q*_FF<x_1,...,x_n>}$ and tameness of the automorphism group ${\text{Aut}_Q(Q*_FF<x,y>)}$.
Automorphisms of Free Metabelian Lie Algebras, II
该研究证明了秩n≥4的自由亚阿贝尔李代数Mₙ的自同构几乎可驯服，且特征≠3时Aut(Mₙ)由线性、二次、三次自同构生成，n≥5时任意特征域均成立。
We study a structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semi direct product of abelian groups. Using this decomposition we describe a structure of the lower central series and the series of derived subgroups for the group of unitriangular automorphisms and prove that every element from the derived subgroup is a commutator. In addition we prove that the group of unitriangular automorphisms of a free associative algebra of rang more than 2 is not linear and describe some two-generated subgroups from th…
A differential analogue of the wild automorphism conjecture
建立了代数动力学中Reichstein-Rogalski-Zhang猜想的微分类比：若射影簇X上存在整体代数向量场v使得(X,v)无真不变子簇，则X为阿贝尔簇；同时研究了阿贝尔簇上具有该性质的向量场。
Analogous to a recent result of N. Kowalzig and U. Krähmer for twisted Calabi-Yau algebras, we show that the Hochschild cohomology ring of a Frobenius algebra with semisimple Nakayama automorphism is a Batalin-Vilkovisky algebra, thus generalizing a result of T.Tradler for finite dimensional symmetric algebras. We give a criterion to determine when a Frobenius algebra given by quiver with relations has semisimple Nakayama automorphism and apply it to some known classes of tame Frobenius algebras. We also provide ample examples including quantum complete intersections, finite dimensional Hopf a…
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