arXiv math/05112632005-11math.RAmath.GR

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

Karl-Hermann Neeb

备注：: Section II has been reformulated, in part. Prop. II.3. Due to a hint of Ph. Gille, Problem II is now taken care of

英文摘要：

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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Automorphisms, derivations and central extensions of Lie algebras arising from quantum tori

For a Lie algebra $\mathfrak g$ related to a quantum torus, we compute its automorphisms, derivations and universal central extension. This Lie algebra $\mathfrak g$ is isomorphic to a subalgebra of the Lie algebra of derivations over the quantum torus, and moreover contains a subalgebra isomorphic to the centerless higher rank Virasoro algebra.

2020-01-05math.RA

量子环面与量子空间的斜导子

Skew derivations of quantum tori and quantum spaces

本文研究量子环面与量子仿射空间的σ-导子结构，给出其分解形式，推广了已有结论，并将结果应用于迭代Ore扩张，提出构造量子环面的算法，还探讨了斜导子与正规化序列的联系。

2024-05-17math.QAmath.RA

Cuspidal modules for solenoidal Lie algebras over rational quantum tori

In this paper we classify all irreducible cuspidal modules over a solenoidal Lie algebra over a rational quantum torus, generalizing the results in [BF2], [Su] and [Xu2].

2019-04-13math.RAmath.RT

Rigidity of quantum tori and the Andruskiewitsch-Dumas conjecture

We prove the Andruskiewitsch-Dumas conjecture that the automorphism group of the positive part of the quantized universal enveloping algebra $U_q({\mathfrak{g}})$ of an arbitrary finite dimensional simple Lie algebra g is isomorphic to the semidirect product of the automorphism group of the Dynkin diagram of g and a torus of rank equal to the rank of g. The key step in our proof is a rigidity theorem for quantum tori. It has a broad range of applications. It allows one to control the (full) automorphism groups of large classes of associative algebras, for instance quantum cluster algebras.

2013-11-01math.QAmath.RA

The group of automorphisms of the Lie algebra of derivations of a field of rational functions

We prove that the group of automorphisms of the Lie algebra $\Der_K (Q_n)$ of derivations of the field of rational functions $Q_n=K(x_1,..., x_n)$ over a field of characteristic zero is canonically isomorphic to the group of automorphisms of the $K$-algebra $Q_n$.

2013-04-15math.AGmath.RA

Rationality problem for norm one tori

We classify stably/retract rational norm one tori in dimension $p-1$ where $p$ is a prime number and in dimension up to ten with some minor exceptions.

2019-05-23math.AGmath.NTmath.RA

更多相关论文收起更多论文

多规范一维托里有理性问题，II

The rationality problem for multinorm one tori, II

Ibaraki University（茨城大学）；College of Science, Rikkyo University（立教大学理学部）

本文研究有限étale代数关联的多规范一维托的稳定和退化有理性，针对因数次数的最大公约数d，证明d=1时托稳定有理，给出退化有理性的判据，并推广Endo-Miyata和Endo的结果。

2026-04-03math.AGmath.RA

Rigidity of quadratic Poisson tori

We prove a rigidity theorem for the Poisson automorphisms of the function fields of tori with quadratic Poisson structures over fields of characteristic 0. It gives an effective method for classifying the full Poisson automorphism groups of ${\mathbb{N}}$-graded connected cluster algebras equipped with Gekhtman-Shapiro-Vainshtein Poisson structures. Based on this, we classify the groups of algebraic Poisson automorphisms of the open Schubert cells of the full flag varieties of semisimple algebraic groups over fields of characteristic 0, equipped with the standard Poisson structures. Their coor…

2016-09-21math.QAmath.RAmath.SG

Rationality problem for norm one tori in small dimensions

We classify stably/retract rational norm one tori in dimension $n-1$ for $n=2^e$ $(e\geq 1)$ is a power of $2$ and $n=12, 14, 15$. Retract non-rationality of norm one tori for primitive $G\leq S_{2p}$ where $p$ is a prime number and for the five Mathieu groups $M_n\leq S_n$ $(n=11,12,22,23,24)$ is also given.

2019-05-26math.AGmath.NTmath.RA

关于满足带反自同构的广义有理恒等式的除环的一个注记

A note on division rings satisfying generalized rational identities with anti-automorphisms

本文研究带非平凡反自同构幂 $σ^m$ 的除环，证明其若满足相应广义有理恒等式，则必为中心有限除环。

2025-10-19math.RA

Rationality problem for algebraic tori

We give the complete stably rational classification of algebraic tori of dimensions $4$ and $5$ over a field $k$. In particular, the stably rational classification of norm one tori whose Chevalley modules are of rank $4$ and $5$ is given. We show that there exist exactly $487$ (resp. $7$, resp. $216$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $4$, and there exist exactly $3051$ (resp. $25$, resp. $3003$) stably rational (resp. not stably but retract rational, resp. not retract rational) algebraic tori of dimension $5$. We ma…

2017-03-06math.AGmath.NTmath.RA