Finite Group Of Automorphisms Research Articles

Hopf module duality applied to X-outer Galois theory

We work over a fixed ground field k in arbitrary characteristic. Let R be a prime algebra, and let Q denote the symmetric Martindale quotient ring of R. We know from [10, Section 6.4] that to study group actions on R, it is fundamental to consider the extended actions on Q, and those actions which are X-outer, or outer as automorphisms of Q, have nice properties, which could be generalized for X-outer (in some appropriate sense) Hopf algebra actions. To recall Kharchenko’s Galois-type correspondence theorem, let G be a finite group of X-outer automorphisms of R. It was proved in [2, Theorem 3.10.2], [11, Theorem B] that

Embedding groups of graph automorphisms in surfaces

Necessary and sufficient conditions (n.a.s.c) are found for a subgroup of the automorphism group of a finite graph to be realizable as the restriction to an invariant spine of some group of homeomorphisms of a compact surface. Also, n.a.s.c. are found for the restricted case when the surface is required to be orientable. The conditions are formulated in terms of the action of stabilizers of vertices on their stars. In both cases, a parametrization of the possible representations is given. Several examples are treated, as well as an application to deciding whether a given finite group of outer automorphisms of a free group is realizable via a surface homeomorphism.

On Griess Algebras

In this paper we prove that for any commutative (but in general non-associative) algebra $A$ with an invariant symmetric non-degenerate bilinear form there is a graded vertex algebra $V = V_0 \oplus V_2 \oplus V_3\oplus ...$, such that $\dim V_0 = 1$ and $V_2$ contains $A$. We can choose $V$ so that if $A$ has a unit $e$, then $2e$ is the Virasoro element of $V$, and if $G$ is a finite group of automorphisms of $A$, then $G$ acts on $V$ as well. In addition, the algebra $V$ can be chosen with a non-degenerate invariant bilinear form, in which case it is simple.

Vertex operator algebras, generalized doubles and dual pairs

Let V be a simple vertex operator algebra and G a finite automorphism group. Then there is a natural right G-action on the set of all inequivalent irreducible V-modules. Let \({\cal S}\) be a finite set of inequivalent irreducible V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra \(A_{\alpha}(G,{\cal S})\) for a suitable 2-cocycle \(\alpha\) naturally determined by the G-action on \({\cal S}\) such that \(A_{\alpha}(G,{\cal S})\) and the vertex operator algebra \(V^G\) form a dual pair on the sum of V-modules in \({\cal S}\) in the sense of Howe. In particular, every irreducible V-module is completely reducible \(V^G\)-module.

The relationship between skew group algebras and orbifold theory

Let V be a simple vertex operator algebra and let G be a finite automorphism group of V . In [C. Dong, G. Yamskulna, Math. Z., to appear], it was shown that any irreducible V -module is a completely reducible V G -module where V G is the G -invariant vertex operator subalgebra of V . In this paper, we give an alternative proof of this fact using the theory of skew group algebras. We also extend this result to any irreducible g -twisted V -module when g is in the center of G and V is a g -rational vertex operator algebra.

Algebra Structure on the Hochschild Cohomology of the Ring of Invariants of a Weyl Algebra under a Finite Group

Let An be the nth Weyl algebra, and let G⊂Sp2n(C)⊂Aut(An) be a finite group of linear automorphisms of An. In this paper, we compute the multiplicative structure on the Hochschild cohomology HH•(AGn) of the algebra of invariants of G. We prove that, as a graded algebra, HH•(AGn) is isomorphic to the graded algebra associated to the center of the group algebra CG with respect to a filtration defined in terms of the defining representation of G.

On Gorenstein surfaces dominated by P2

AbstractIn this paper we prove that a normal Gorenstein surface dominated by P2 is isomorphic to a quotient P2/G, where G is a finite group of automorphisms of P2 (except possibly for one surface ). We can completely classify all such quotients. Some natural conjectures when the surface is not Gorenstein are also stated.

The Generalized Center Galois Extensions of Rings

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and Ii = {c - gi(c) | c ∈ C} for each gi ∈ G. Then, B is called a center Galois extension with Galois group G if BIi = B for each gi ≠ 1 in G, and a weak center Galois extension with group G if BIi = Bei for some nonzero idempotent ei in C for each gi ≠ 1 in G. When ei is a minimal element in the Boolean algebra generated by {ei | gi ∈ G} Bei is a center Galois extension with Galois group Hi for some subgroup Hi of G. Moreover, the central Galois algebra B(1 – ei) is characterized when B is a Galois algebra with Galois group G.

The McKay correspondence as an equivalence of derived categories

Let G G be a finite group of automorphisms of a nonsingular three-dimensional complex variety M M , whose canonical bundle ω M \omega _M is locally trivial as a G G -sheaf. We prove that the Hilbert scheme Y = G Y = G - Hilb ⁡ M \operatorname {Hilb}M parametrising G G -clusters in M M is a crepant resolution of X = M / G X=M/G and that there is a derived equivalence (Fourier–Mukai transform) between coherent sheaves on Y Y and coherent 𝐺-sheaves on M M . This identifies the K theory of Y Y with the equivariant K theory of M M , and thus generalises the classical McKay correspondence. Some higher-dimensional extensions are possible.

TORSION-FREE RINGS WITH FINITE AUTOMORPHISM GROUPS

The paper considers finite rank torsion-free rings, i.e, subrings of finite-dimensional rational algebras. The main result of the paper is the characterization of finite rank torsion-free rings with finite automorphism groups. Nontrivial examples of rings R with AutR finite have been constructed.

Homological Properties of Quantum Polynomials

We study the endomorphim semigroup of a general quantum polynomial ring, its finite groups of automorphisms, and homological properties, as a module over the skew group ring of a finite group of automorphisms. Moreover, properties of the division ring of fractions are considered.

On Azumaya algebras with a finite automorphism group

Let B be a ring with 1, C the center of B, and G a finite automorphism group of B. It is shown that if B is an Azumaya algebra such that B = ⊕∑g∈GJg where Jg = {b ∈ B|bx = g(x)b for all x ∈ B}, then there exist orthogonal central idempotents {fi ∈ C|i = 1, 2, …, m for some integer m} and subgroups Hi of G such that where Bfi is a central Galois algebra with Galois group for each i = 1, 2, …, m and D is contained in C.

On weak center Galois extensions of rings

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and BG the set of elements in B fixed under each element in G. Then, the notion of a center Galois extension of BG with Galois group G (i.e., C is a Galois algebra over CG with Galois group G|C≅G) is generalized to a weak center Galois extension with group G, where B is called a weak center Galois extension with group G if BIi = Bei for some idempotent in C and Ii = {c − gi(c) | c ∈ C} for each gi ≠ 1 in G. It is shown that B is a weak center Galois extension with group G if and only if for each gi ≠ 1 in G there exists an idempotent ei in C and {bkei ∈ Bei; ckei ∈ Cei, k = 1, 2, …, m} such that and gi restricted to C(1 − ei) is an identity, and a structure of a weak center Galois extension with group G is also given.

ON THE FIXED POINTS OF A FINITE GROUP ACTING ON A RELATIVELY FREE LIE ALGEBRA

We show that if F is a free Lie algebra of rank at least 2 and if G is a non-trivial finite group of automorphisms of F then the fixed point subalgebra F^G is not finitely generated. Some similar results are proved for relatively free Lie algebras.

On characterizations of a center Galois extension

Let B be a ring with 1, C the center of B, G a finite automorphism group of B, and BG the set of elements in B fixed under each element in G. Then, it is shown that B is a center Galois extension of BG (that is, C is a Galois algebra over CG with Galois group G|C≅G) if and only if the ideal of B generated by {c − g(c) | c ∈ C} is B for each g ≠ 1 in G. This generalizes the well known characterization of a commutative Galois extension C that C is a Galois extension of CG with Galois group G if and only if the ideal generated by {c − g(c) | c ∈ C} is C for each g ≠ 1 in G. Some more characterizations of a center Galois extension B are also given.

On central commutator Galois extensions of rings

LetBbe a ring with1,Ga finite automorphism group ofBof ordernfor some integern,BGthe set of elements inBfixed under each element inG, andΔ=VB(BG)the commutator subring ofBGinB. Then the type of central commutator Galois extensions is studied. This type includes the types of Azumaya Galois extensions and GaloisH-separable extensions. Several characterizations of a central commutator Galois extension are given. Moreover, it is shown that whenGis inner,Bis a central commutator Galois extension ofBGif and only ifBis anH-separable projective group ringBGGf. This generalizes the structure theorem for central Galois algebras with an inner Galois group proved by DeMeyer.

The maximum order of finite groups of automorphisms of K 3 surfaces

Let G be a finite group of automorphisms of a K 3 surface X . We shall show that | G | ≤ 3840 and if | G | = 3840, then the pair ( X, G ) is unique up to isomorphisms.

X-Inner Automorphisms of Semi-Commutative Quantum Algebras

Many important quantum algebras such as quantum symplectic space, quantum Euclidean space, quantum matrices, q-analogs of the Heisenberg algebra, and the quantum Weyl algebra are semi-commutative. In addition, enveloping algebras U(L+) of even Lie color algebras are also semi-commutative. In this paper, we generalize work of Montgomery and examine the X-inner automorphisms of such algebras. The theorems and examples in our paper show that for algebras R of this type, the non-identity X-inner automorphisms of R tend to have infinite order. Thus if G is a finite group of automorphisms of R, then the action of G will be X-outer and this immediately gives useful information about crossed products R∗tG.

A Galois-Type Correspondence Theory for Actions of Finite-Dimensional Pointed Hopf Algebras on Prime Algebras

The objective of this paper is to present a Galois-type correspondence theory for X-outer actions of finite-dimensional Hopf algebras over a field k on prime algebras R. Noncommutative Galois theory was initiated by Noether in her work on inner automorphisms of simple algebras and was w x further studied by Jacobson J1 , who established a Galois correspondence w theory for a finite group of automorphisms acting on a division ring. In J2, x Chap 6 Jacobson used the product of the ring and the group of automorphisms to develop a Galois theory for a ring of linear transformations of a vector space. A major advance is due to Kharchenko, who developed a Galois correspondence theory for actions of automorphisms and derivaŽ . tions on prime semiprime rings. His definitions of inner and outer actions, which were named X-outer and X-inner actions, involve the Martindale ring of quotients of a prime ring R. Montgomery and Passman presented the X-outer case separately and used trace forms of minimal length to show in an elegant way Kharchenko’s Galois correspondence

Quantum Galois theory for finite groups

Dong and Mason [DM1] initiated a systematic research for a vertex operator algebra with a finite automorphism group, which is referred to as the “operator content of orbifold models” by physicists [DVVV]. The purpose of this paper is to extend one of their main results. We will assume that the reader is familiar with the vertex operator algebras (VOA), see [B],[FLM]. Throughout this paper, V denotes a simple vertex operator algebra, G is a finite automorphism group of V , C denotes the complex number field, and Z denotes rational integers. Let H be a subgroup of G and Irr(G) denote the set of all irreducible CG-characters. In their paper [DM1], they studied the sub VOA V H = {v ∈ V : h(v) = v for all h ∈ H} of H-invariants and the subspace V χ on which G acts according to χ ∈ Irr(G). Especially, they conjectured the following Galois correspondence between sub VOAs of V and subgroups of G and proved it for an Abelian or dihedral group G [DM1, Theorem 1] and later for nilpotent groups [DM2], which is an origin of their title of [DM1].