Naihuan Jing, James J. Zhang
We prove Auslander-Gorenstein and $\GKdim$-Macaulay properties for certain invariant subrings of some quantum algebras, the Weyl algebras, and the universal enveloping algebras of finite dimensional Lie algebras.
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The study of invariants of group actions on commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. It has been of particular interest to understand what conditions on the group result in an invariant ring satisfying useful properties. In particular, Watanabe's Theorem states that the invariant subring of $k[x_1,\ldots,x_n]$ under the natural action of a finite subgroup of $SL_n(k)$ is always Gorenstein. In this paper, we study this question in the more general setting of group actions on noncommutative non-connected algebras $A$. We develop …
This is a survey article on the invariant rings of Hopf actions
We say that a subring $R_0$ of a ring $R$ is semi-invariant if $R_0$ is the ring of invariants in $R$ under some set of ring endomorphisms of some ring containing $R$. We show that $R_0$ is semi-invariant if and only if there is a ring $S\supseteq R$ and a set $X\subseteq S$ such that $R_0=\Cent_R(X):={r\in R \suchthat xr=rx \forall x\in X}$; in particular, centralizers of subsets of $R$ are semi-invariant subrings. We prove various properties of semi-invariant subrings and show how they can be used for various applications including: (1) The center of a semiprimary (resp. right perfect) ring …
We study important invariants and properties of the Veronese subalgebras of $q$-skew polynomial rings, including their discriminant, center and automorphism group, as well as cancellation property and the Tits alternative.
We study semisimple Hopf algebra actions on Artin-Schelter regular algebras and prove several upper bounds on the degrees of the minimal generators of the invariant subring, and on the degrees of syzygies of modules over the invariant subring. These results are analogues of results for group actions on commutative polynomial rings proved by Noether, Fogarty, Fleischmann, Derksen, Sidman, Chardin, and Symonds.
We study the invariant theory of a class of quantum Weyl algebras under group actions and prove that the fixed subrings are always Gorenstein. We also verify the Tits alternative for the automorphism groups of these quantum Weyl algebras.
We study Artin-Schelter Gorenstein fixed subrings of some Artin-Schelter regular algebras of dimension 2 and 3 under finite group actions, and prove a noncommutative version of the Kac-Watanabe and Gordeev theorem for these algebras.
We study invariants and quotient categories of fixed subrings of Artin-Schelter regular algebras under Hopf algebra actions.
Noncommutative invariants of finite and classical groups
本文研究了有限与经典群的非交换不变量，通过Schur functor将问题转化为向量不变量问题，并证明了特定条件下张量代数的不变子环的有限生成性。
We introduce and develop a structure theory of a new class of noncommutative rings - Galois orders, that generalize classical orders in noncommutative rings. Galois orders realized as certain subrings of invariants in skew semigroup rings. This class of rings contains many classical objects, in particular the Generalized Weyl algebras, the universal enveloping algebra of the general linear Lie algebra and certain rings of invariant differential operators on algebraic varieties.
In this paper, we introduce the theory of local cohomology and local duality to Notherian connected cochain DG algebras. We show that the notion of local cohomology functor can be used to detect the Gorensteinness of a homologically smooth DG algebra. For any Gorenstein homologically smooth locally finite DG algebra $\mathcal{A}$, we define a group homomorphism $\mathrm{Hdet}: \mathrm{Aut}_{dg}(\mathcal{A})\to k^{\times},$ called the homological determinant. As applications, we present a sufficient condition for the invariant DG subalgebra $\mathcal{A}^G$ to be Gorensten, where $\mathcal{A}$ i…
We prove that if $u:K \rightarrow M$ is a left minimal extension, then there exists an isomorphism between two subrings, $\textrm{End}_R^M(K)$ and $\textrm{End}_R^K(M)$ of $\textrm{End}_R(K)$ and $\textrm{End}_R(M)$ respectively, modulo their Jacobson radicals. This isomorphism is used to deduce properties of the endomorphism ring of $K$ from those of the endomorphism ring of $M$ in certain situations such us when $K$ is invariant under endomorphisms of $M,$ or when $K$ is invariant under automorphisms of $M$.
Reflection Groups and Rigidity of Quadratic Poisson Algebras
本文研究二次Poisson代数的不变量理论，证明了Shephard-Todd-Chevalley定理的Poisson版本，并揭示了多种代数族的Poisson刚性，即固定子环A^G与A同构当且仅当群G平凡。
Cohen-Macaulay representations of invariant subrings
本文分类了特征零任意域上有限 Cohen-Macaulay 型的二维完备局部环，推广了代数闭域情形的已有工作，证明其形式为不变子环并建立代数 McKay 对应，同时将其扩展至高维并证明存在非交换极小奇点解消。
Let $G$ be a finite subgroup of the linear group of a finite-dimensional complex vector $V$, $B={\operatorname S}(V)$ be the symmetric algebra, ${\mathcal D}=\mathcal D^G_B$ the ring of $G$-invariant differential operators, and ${\mathcal D}^-$ its subring of negative degree operators. We prove that $M\mapsto M^{ann}= {\operatorname Ann}_{\mathcal D^-}(M)$ defines an isomorphism between the category of ${\mathcal D}$-submodules of $B$ and a category of modules formed as lowest weight spaces. This is applied to a construction of simple ${\mathcal D}$-submodules of $B$ when $G$ is a generalized …
Tilting theory for hypersurface singularities of dimension one
本文研究一维奇异超曲面的标准silting对象的自同态代数，证明其为Iwanaga-Gorenstein或Gorenstein，并给出显式表示及Auslander-Reiten箭图。
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