Degenerate Sklyanin algebras and Generalized Twisted Homogeneous Coordinate rings

Quantum projective planes as certain graded twisted tensor products

Let $\mathbb{k}$ be an algebraically closed field. Building upon previous work, we classify, up to isomorphism of graded algebras, quadratic graded twisted tensor products of $\mathbb{k}[x,y]$ and $\mathbb{k}[z]$. When such an algebra is Artin-Schelter regular, we identify its point scheme and type. We also describe which three-dimensional Sklyanin algebras contain a subalgebra isomorphic to a quantum $\mathbb{P}^1$, and we show that every algebra in this family is a graded twisted tensor product of $\mathbb{k}_{-1}[x,y]$ and $\mathbb{k}[z]$.

Exotic Elliptic Algebras

This paper examines a general method for producing twists of a comodule algebra by tensoring it with a torsor then taking co-invariants. We examine the properties that pass from the original algebra to the twisted algebra and vice versa. We then examine the special case where the algebra is a 4-dimensional Sklyanin algebra viewed as a comodule algebra over the Hopf algebra of functions on the non-cyclic group of order 4 with the torsor being the 2x2 matrix algebra. The twisted algebra is an "exotic elliptic algebra". We show that the twisted algebra has many of the good properties that the Skl…

退化扭曲迹在型A量化克莱因奇点上的研究

Degenerate twisted traces on quantized Kleinian singularities of type A

本文研究型A量化克莱因奇点上的退化扭曲迹，揭示其维度与自同构及量子化参数的关系。

Noncommutative Blowups of Elliptic Algebras

We develop a ring-theoretic approach for blowing up many noncommutative projective surfaces. Let T be an elliptic algebra (meaning that, for some central element g of degree 1, T/gT is a twisted homogeneous coordinate ring of an elliptic curve E at an infinite order automorphism). Given an effective divisor d on E whose degree is not too big, we construct a blowup T(d) of T at d and show that it is also an elliptic algebra. Consequently it has many good properties: for example, it is strongly noetherian, Auslander-Gorenstein, and has a balanced dualizing complex. We also show that the ideal st…

关于 Clifford 代数中的中心化子与扭曲中心化子的注记

A Note on Centralizers and Twisted Centralizers in Clifford Algebras

本文研究了退化与非退化 Clifford 代数中的中心化子与扭曲中心化子，给出了固定次数子空间及其直和的显式形式，结果可应用于计算机科学、物理学和工程领域。

Hopf-Galois objects of Calabi-Yau Hopf algebras

By using the language of cogroupoids, we show that Hopf-Galois objects of a twisted Calabi-Yau Hopf algebra with bijective antipode are still twisted Calabi-Yau, and give their Nakayama automorphism explicitly. As applications, cleft Galois objects of twisted Calabi-Yau Hopf algebras and Hopf-Galois objects of the quantum automorphism groups of non-degenerate bilinear forms are proved to be twisted Calabi-Yau.

几何代数中的广义退化 Clifford 群与 Lipschitz 群

Generalized Degenerate Clifford and Lipschitz Groups in Geometric Algebras

本文在退化几何代数中定义并刻画广义 Clifford 群与 Lipschitz 群，借助中心化子、扭中心化子和范数函数揭示其结构，并讨论相关 Lie 代数及跨学科应用。

Degenerate affine Hecke algebras and centralizer construction for the symmetric groups

In our recent papers the centralizer construction was applied to the series of classical Lie algebras to produce the quantum algebras called (twisted) Yangians. Here we extend this construction to the series of the symmetric groups S(n). We study the `stable' properties of the centralizers of S(n-m) in the group algebra C[S(n)] as n increases with m fixed. We construct a limit centralizer algebra A and describe its algebraic structure. The algebra A turns out to be closely related with the degenerate affine Hecke algebras. We also show that the so-called tame representations of the infinite sy…

关于退化 Clifford 几何代数中的一些 Lie 群

On Some Lie Groups in Degenerate Clifford Geometric Algebras

本文在退化 Clifford 几何代数中引入并研究了五族 Lie 群，证明它们在伴随与扭曲伴随表示下保持特定子空间，并包含退化旋量群等子群，具有多学科应用价值。

Finite dimensional Hopf actions on algebraic quantizations

Let k be an algebraically closed field of characteristic zero. In joint work with J. Cuadra [arxiv.org/abs/1409.1644, arxiv.org/abs/1509.01165], we showed that a semisimple Hopf action on a Weyl algebra over a polynomial algebra k[z_1,...,z_s] factors through a group action, and this in fact holds for any finite dimensional Hopf action if s=0. We also generalized these results to finite dimensional Hopf actions on algebras of differential operators. In this work we establish similar results for Hopf actions on other algebraic quantizations of commutative domains. This includes universal envelo…

关于保持退化克莱因代数子空间的李群

On Lie Groups Preserving Subspaces of Degenerate Clifford Algebras

本文研究了保持退化克莱因代数子空间的李群及其代数，探讨其与海森堡群的关系及在等变神经网络中的应用。