Finite Group Of Automorphisms Research Articles

On invariants of modular free Lie algebras

Suppose that L(X) is a free Lie algebra of finite rank over a field of positive characteristic. Let G be a nontrivial finite group of homogeneous automorphisms of L(X). It is known that the subalgebra of invariants H = LG is infinitely generated. Our goal is to describe how big its free generating set is. Let \( Y = \bigcup\limits_{n = 1}^\infty {{Y_n}} \) be a homogeneous free generating set of H, where elements of Yn are of degree n with respect to X. We describe the growth of the generating function of Y and prove that |Yn| grow exponentially.

Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ⩾ 2

We obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ⩾ 2 into any complex space Z. After lifting to the normalization of the subvariety F(Ω) ⊂ Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou’s theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f: Ω → D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described.

K3 surfaces with a symplectic automorphism of order 11

We classify possible finite groups of symplectic automorphisms of K3 surfaces of order divisible by 11. The characteristic of the ground field must be equal to 11. The complete list of such groups consists of five groups: the cyclic group C_{11} of order 11, C_{11} ⋊ C_5 , \mathrm{PSL2}(\mathbb F_{11}) and the Mathieu groups M_{11} , M_{22} . We also show that a surface X admitting an automorphism g of order 11 admits a g -invariant elliptic fibration with the Jacobian fibration isomorphic to one of explicitly given elliptic K3 surfaces.

Artin characters, Hurwitz trees and the lifting problem

We study finite groups of automorphisms of the p-adic open disk. In particular, we generalize results of Green, Matignon and Henrio from cyclic groups of order p to arbitrary finite groups. As an application, we produce a counterexample to a question of Chinburg, Guralnick and Harbater, concerning the local lifting problem for generalized quaternion groups.

Automorphisms of Coxeter Groups and Lusztig’s Conjectures for Hecke Algebras with Unequal Parameters

AbstractLet (W,S) be a Coxeter system, let G be a finite solvable group of automorphisms of (W, S) and let ϕ be a weight function which is invariant under G. Let ϕG denote the weight function on WG obtained by restriction from ϕ. The aim of this paper is to compare the a-function, the set of Duflo involutions and the Kazhdan-Lusztig cells associated with (W, ϕ) and to (WG,ϕG), provided that Lusztig’s Conjectures hold.

Finite groups of symplectic automorphisms of K3 surfaces in positive characteristic

We show that Mukai's classification of finite groups which may act symplectically on a complex K3 surface extends to positive characteristic p under assumptions that (i) the order of the group is coprime to p and (ii) either the surface or its quotient is not birationally isomorphic to a supersingular K3 surface with Artin invariant 1. In the case without assumption (ii) we classify all possible new groups which may appear. We prove that assumption (i) on the order of the group is always satisfied if p > 11. For p = 2,3,5,11, we give examples of K3 surfaces with finite symplectic automorphism groups of order divisible by p which are not contained in Mukai's list.

Finitely Presented MV-algebras with Finite Automorphism Group

We address the question, which MV-algebras have finite automorphism group. We prove that finitely presented MV-algebras whose maximal spectral space has topological dimension not exceeding 1 do have finite automorphism group. We give examples to show that finite presentability is an essential hypothesis. Our proof produces as an interesting by-product a complete graph–theoretic isomorphism invariant for the class of MV-algebras involved.

SAGBI bases for rings of invariant Laurent polynomials

Let k be a field, let L n = k[x ±1 1 , x ±1 n ] be the Laurent polynomial ring in n variables and let G be a finite group of k-algebra automorphisms of L n . We give a necessary and sufficient condition for the ring of invariants L G n to have a SAGBI basis. We show that if this condition is satisfied, then L G n has a SAGBI basis relative to any choice of coordinates in L n and any term order.

Writing representations over proper division subrings

Let E be a division ring, and G a finite group of automorphisms of E whose elements are distinct modulo inner automorphisms of E . Let F = E G be the division subring of elements of E fixed by G. Given a representation ρ : A → E d × d of an F -algebra A , we give necessary and sufficient conditions for ρ to be writable over F . (Here E d × d denotes the algebra of d × d matrices over E , and a matrix A writes ρ over F if A −1 ρ ( A ) A ⊆ F d × d .) We give an algorithm for constructing an A, or proving that no A exists. The case of particular interest to us is when E is a field, and ρ is absolutely irreducible. The algorithm relies on an explicit formula for A, and a generalization of Hilbert's Theorem 90 that arises in galois cohomology. The algorithm has applications to the construction of absolutely irreducible group representations (especially for solvable groups), and to the recognition of class C 5 in Aschbacher's matrix group classification scheme [M. Aschbacher, On the maximal subgroups of the finite classical groups, Invent. Math. 76 (1984) 469–514, MR0746539; Shangzhi Li, On the subgroup structure of classical groups, in: Group Theory in China, in: Math. Appl. (China Ser.), vol. 365, Kluwer Acad. Publ., Dordrecht, 1996, pp. 70–90, MR1447199. [1,13]].

Symmetries, quotients and Kähler-Einstein metrics

We consider Fano manifolds M that admit a collection of finite automorphism groups G1, …, Gk, such that the quotients M/Gi are smooth Fano manifolds possessing a Kähler-Einstein metric. Under some numerical and smoothness assumptions on the ramification divisors, we prove that M admits a Kähler-Einstein metric too.

TWO-DIMENSIONAL VERSAL G-COVERS AND CREMONA EMBEDDINGS OF FINITE GROUPS

Let G be a finite group. A Galois cover ω : X → Y with Galois group G is called a versal G-cover if any G-cover is induced from ω : X → Y . In this paper, we prove that, if the essential dimension of G is equal to 2, then there exists a versal G-cover ω : X → Y such that (i) X is a smooth rational surface, (ii) G is a finite automorphism group of X, (iii) the action of G on X is minimal, and (iv) Y = X/G. We also give some examples of finite groups which have non-conjugate embeddings into the Cremona group Cr2(C).

On a Generalized Wilson Functional Equation

Abstract We solve the functional equation where 𝐺 is a locally compact Abelian group, {𝑘0 = 𝑖𝑑𝐺, 𝑘1, . . . , 𝑘𝑁–1} is a finite group of continuous automorphisms of 𝐺, 𝑓 is a bounded continuous complex-valued function on 𝐺 and 𝑔 is an almost periodic function on 𝐺.

Modularity in Orbifold Theory for Vertex Operator Superalgebras

This paper is about the orbifold theory for vertex operator superalgebras. Given a vertex operator superalgebra V and a finite automorphism group G of V, we show that the trace functions associated to the twisted sectors are holomorphic in the upper half plane for any commuting pairs in G under the C_2-cofinite condition. We also establish that these functions afford a representation of the full modular group if V is C_2-cofinite and g-rational for any g in G.

On intertwining operators and finite automorphism groups of vertex operator algebras

Let V be a simple vertex operator algebra and G a finite automorphism group. We give a construction of intertwining operators for irreducible V G -modules which occur as submodules of irreducible V-modules by using intertwining operators for V. We also determine some fusion rules for a vertex operator algebra as an application.

On Compact Complex Manifolds with Finite Automorphism Group

It is known that compact complex manifolds of general type and Kobayashi hyperbolic manifolds have finite automorphism groups. We give criteria for finiteness of the automorphism group of a compact complex manifold which allow us to produce large classes

Determining whether a finite group of outer automorphisms of a free group is geometric

In this article we give necessary and sufficient conditions for a given finite group of outer automorphisms to be induced by the action of a group of orientation-preserving homeomorphisms on the fundamental group of a punctured surface. When the group is abelian, necessary and sufficient conditions can also be given in the absence of orientability assumptions. These properties are formulated in terms of the finite automorphism groups which project into the given outer automorphism group: each non-trivial automorphism in any such group can fix at most a cyclic subgroup of the fundamental group.

Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable*1

Let G be a finite group of automorphisms of a non-solvable Lie algebra L of finite dimension over a field of characteristic zero. This paper is concerned with the relationship between the structure of G and that of L . By imposing the condition that the fixed-point subalgebra of each non-identity element of G is solvable, we are able to determine the structure of the group G and that of the quotient algebra of L by its solvable radical, Solv( L ), except when G is a group of odd order in which every Sylow subgroup is cyclic and the center is non-trivial. In this case there exists an element x ∈ L , x ∉Solv( L ), which is fixed by every element of G of prime order. We note that the imposed condition is satisfied by every finite group of automorphisms of a three-dimensional simple Lie algebra, regular groups of automorphisms of a non-solvable Lie algebra which are abelian of type ( p , p ), p prime, and by the center of a finite group of automorphisms of a semisimple Lie algebra without proper semisimple invariant subalgebra. We exhibit several examples of groups satisfying that condition. Also, we give some results relating the structures of G and L in more general cases.

Groups of automorphisms of Lie algebras such that the fixed-point subalgebra of each non-identity element is solvable

Let G be a finite group of automorphisms of a non-solvable Lie algebra L of finite dimension over a field of characteristic zero. This paper is concerned with the relationship between the structure of G and that of L. By imposing the condition that the fixed-point subalgebra of each non-identity element of G is solvable, we are able to determine the structure of the group G and that of the quotient algebra of L by its solvable radical, Solv( L), except when G is a group of odd order in which every Sylow subgroup is cyclic and the center is non-trivial. In this case there exists an element x∈ L, x∉Solv( L), which is fixed by every element of G of prime order. We note that the imposed condition is satisfied by every finite group of automorphisms of a three-dimensional simple Lie algebra, regular groups of automorphisms of a non-solvable Lie algebra which are abelian of type ( p, p), p prime, and by the center of a finite group of automorphisms of a semisimple Lie algebra without proper semisimple invariant subalgebra. We exhibit several examples of groups satisfying that condition. Also, we give some results relating the structures of G and L in more general cases.

Uniform product of Ag, n( V) for an orbifold model V and G-twisted Zhu algebra

Let V be a vertex operator algebra and G a finite automorphism group of V. For each g∈ G and nonnegative rational number n∈ Z/|g| , an associative algebra A g, n ( V) plays an important role in the theory of vertex operator algebras, but the given product in A g, n ( V) depends on the eigenspaces of g. We show that if V has no negative weights then there is a uniform definition of products on V and we introduce a G-twisted Zhu algebra A G, n ( V) which covers all A g, n ( V). Let V be a simple vertex operator algebra with no negative weights and let S be a finite set of inequivalent irreducible twisted V-modules which is closed under the action of G. There is a finite dimensional semisimple associative algebra A α(G, S) for a suitable 2-cocycle naturally determined by the G-action on S . We show that a duality theorem of Schur–Weyl type holds for the actions of A α(G, S) and V G on the direct sum of twisted V-modules in S as an application of the theory of A G, n ( V). It follows as a natural consequence of the result that for any g∈ G every irreducible g-twisted V-module is a completely reducible V G -module.

Twisted modules over vertex algebras on algebraic curves

We extend the geometric approach to vertex algebras developed by the first author to twisted modules, allowing us to treat orbifold models in conformal field theory. Let V be a vertex algebra, H a finite group of automorphisms of V, and C an algebraic curve such that H⊂Aut( C). We show that a suitable collection of twisted V-modules gives rise to a section of a certain sheaf on the quotient X= C/ H. We introduce the notion of conformal blocks for twisted modules, and analyze them in the case of the Heisenberg and affine Kac–Moody vertex algebras. We also give a chiral algebra interpretation of twisted modules.