Finite Order Automorphisms Research Articles

Symplectic rigidity of O’Grady’s tenfolds

We prove that any symplectic automorphism of finite order of an irreducible holomorphic symplectic manifold of O’Grady’s 10 10 -dimensional deformation type is trivial.

Twisted regular representations for vertex operator algebras

This paper is to study what we call twisted regular representations for vertex operator algebras. Let V be a vertex operator algebra, let σ1,σ2 be commuting finite-order automorphisms of V and let σ=(σ1σ2)−1. Among the main results, for any σ-twisted V-module W and any nonzero complex number z, we construct a weak σ1⊗σ2-twisted V⊗V-module Dσ1,σ2(z)(W) inside W⁎. Let W1,W2 be σ1-twisted, σ2-twisted V-modules, respectively. We show that P(z)-intertwining maps from W1⊗W2 to W⁎ are the same as homomorphisms of weak σ1⊗σ2-twisted V⊗V-modules from W1⊗W2 into Dσ1,σ2(z)(W). We also show that a P(z)-intertwining map from W1⊗W2 to W⁎ is equivalent to an intertwining operator of type (W′W1W2), which is a twisted version of a result of Huang and Lepowsky. Finally, we show that for each τ-twisted V-module M with τ any finite-order automorphism of V, the coefficients of the q-graded trace function lie in Dτ,τ−1(−1)(V) and generate a τ⊗τ−1-twisted V⊗V-submodule isomorphic to M⊗M′.

Rational Reidemeister trace of an outer automorphism of finite order

Let H be a group of type FP over the rationals and let ϕ:H→H be an automorphism whose class in Out(H) has finite order and such that the Lefschetz number of some iterate ϕi is nonzero. We show that under a suitable Bass conjecture assumption, this iterate ϕi is conjugate to a finite order automorphism, and the intersection of the fixed point set of ϕ with the center of H is of bounded exponent. In the torsion-free case it follows that the center of the HNN extension 〈H,t|∀h∈H:tht−1=ϕ(h)〉 is infinite cyclic. These results are related to Geoghegan and Nicas' first order Euler characteristic of a group G with a finite Eilenberg-MacLane complex; this is a morphism χ1(G) from the center of G to H1(G;k) where k is a ring of coefficients. We give a complete determination of χ1(G) with rational coefficients.

Automorphisms of finite order, periodic contractions, and Poisson-commutative subalgebras of {mathcal {S}}({mathfrak g})

Let {mathfrak g} be a semisimple Lie algebra, vartheta in textsf{Aut}({mathfrak g}) a finite order automorphism, and {mathfrak g}_0 the subalgebra of fixed points of vartheta . Recently, we noticed that using vartheta one can construct a pencil of compatible Poisson brackets on {mathcal {S}}({mathfrak g}), and thereby a ‘large’ Poisson-commutative subalgebra {mathcal {Z}}({mathfrak g},vartheta ) of {mathcal {S}}({mathfrak g})^{{mathfrak g}_0}. In this article, we study invariant-theoretic properties of ({mathfrak g},vartheta ) that ensure good properties of {mathcal {Z}}({mathfrak g},vartheta ). Associated with vartheta one has a natural Lie algebra contraction {mathfrak g}_{(0)} of {mathfrak g} and the notion of a good generating system (=g.g.s.) in {mathcal {S}}({mathfrak g})^{mathfrak g}. We prove that in many cases the equality mathrm{ind,}{mathfrak g}_{(0)}=mathrm{ind,}{mathfrak g} holds and {mathcal {S}}({mathfrak g})^{mathfrak g} has a g.g.s. According to V. G. Kac’s classification of finite order automorphisms (1969), vartheta can be represented by a Kac diagram, mathcal {K}(vartheta ), and our results often use this presentation. The most surprising observation is that {mathfrak g}_{(0)} depends only on the set of nodes in mathcal {K}(vartheta ) with nonzero labels, and that if vartheta is inner and a certain label is nonzero, then {mathfrak g}_{(0)} is isomorphic to a parabolic contraction of {mathfrak g}.

Dynamics of actions of automorphisms of discrete groups $G$ on $\mathrm{Sub}_G$ and applications to lattices in Lie groups

For a locally compact Hausdorff group G and the compact space \mathrm{Sub}_G of closed subgroups of G endowed with the Chabauty topology, we study the dynamics of actions of automorphisms of G on \mathrm{Sub}_G in terms of distality and expansivity. We prove that an infinite discrete group G , which is either polycyclic or a lattice in a connected Lie group, does not admit any automorphism which acts expansively on \mathrm{Sub}^c_G , the space of cyclic subgroups of G , while only the finite order automorphisms of G act distally on \mathrm{Sub}^c_G . For an automorphism T of a connected Lie group G which keeps a lattice \Gamma invariant, we compare the behaviour of the actions of T on \mathrm{Sub}_G and \mathrm{Sub}_\Gamma in terms of distality. Under certain necessary conditions on the Lie group G , we show that T acts distally on \mathrm{Sub}_G if and only if it acts distally on \mathrm{Sub}_\Gamma . We also obtain certain results about the structure of lattices in a connected Lie group.

Dimension formulae and generalised deep holes of the Leech lattice vertex operator algebra

We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{\operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite-order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $\operatorname{Aut}(V)$. Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra $V_\Lambda$ with non-trivial fixed-point Lie subalgebra and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space. This provides the first uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $\Lambda$ and the 23 Niemeier lattices with non-vanishing root system found by Conway, Parker and Sloane.

Serre Functors and Graded Categories

We study Serre structures on categories enriched in pivotal monoidal categories, and apply this to study Serre structures on two types of graded k-linear categories: categories with group actions and categories with graded hom spaces. We check that Serre structures are preserved by taking orbit categories and skew group categories, and describe the relationship with graded Frobenius algebras. Using a formal version of Auslander-Reiten translations, we show that the derived category of a d-representation finite algebra is fractionally Calabi-Yau if and only if its preprojective algebra has a graded Nakayama automorphism of finite order. This connects various results in the literature and gives new examples of fractional Calabi-Yau algebras.

Permutohedral complexes and rational curves with cyclic action

We define a moduli space of rational curves with finite-order automorphism and weighted orbits, and we prove that the combinatorics of its boundary strata are encoded by a particular polytopal complex that also captures the algebraic structure of a complex reflection group acting on the moduli space. This generalizes the situation for Losev-Manin's moduli space of curves (whose boundary strata are encoded by the permutohedron and related to the symmetric group) as well as the situation for Batyrev-Blume's moduli space of curves with involution, and it extends that work beyond the toric context.

The integral monodromy of isolated quasihomogeneous singularities

The integral monodromy on the Milnor lattice of an isolated quasihomogeneous singularity is subject of an almost untouched conjecture of Orlik from 1972. We prove this conjecture for all iterated Thom-Sebastiani sums of chain type singularities and cycle type singularities. The main part of the paper is purely algebraic. It provides tools for dealing with sums and tensor products of ${\mathbb Z}$-lattices with automorphisms of finite order and with cyclic generators. The calculations are involved. They use fine properties of unit roots, cyclotomic polynomials, their resultants and discriminants.

Classification of Cartan embeddings which are austere submanifolds

We classify automorphisms of finite order on compact connected simple Lie groups by which the induced Cartan embedding is an austere embedding. Furthermore, we consider the problem whether an austere Cartan embedding is weakly reflective or not.

The integral monodromy of the cycle type singularities

The middle homology of the Milnor fiber of a quasihomogeneous polynomial with an isolated singularity is a ${\mathbb Z}$-lattice and comes equipped with an automorphism of finite order, the integral monodromy. Orlik (1972) made a precise conjecture, which would determine this monodromy in terms of the weights of the polynomial. Here we prove this conjecture for the cycle type singularities. A paper of Cooper (1982) with the same aim contained two mistakes. Still it is very useful. We build on it and correct the mistakes. We give additional algebraic and combinatorial results.

Automorphisms of Finite Order of Soluble Groups of Finite Rank

We study the effect on sections of a soluble-by-finite group G of finite rank of an almost fixed-point-free automorphism φ of G of finite order. We also elucidate the structure of G if φ has order 4 and if G is also (torsion-free)-by-finite. The latter extends recent work of Xu, Zhou and Liu.

PERIODIC AUTOMORPHISMS, COMPATIBLE POISSON BRACKETS, AND GAUDIN SUBALGEBRAS

Let \U0001d52e be a finite-dimensional Lie algebra. The symmetric algebra (\U0001d52e) is equipped with the standard Lie–Poisson bracket. In this paper, we elaborate on a surprising observation that one naturally associates the second compatible Poisson bracket on (\U0001d52e) to any finite order automorphism ϑ of \U0001d52e. We study related Poisson-commutative subalgebras (\U0001d52e; ϑ) of \U0001d4ae(\U0001d52e) and associated Lie algebra contractions of \U0001d52e. To obtain substantial results, we have to assume that \U0001d52e = \U0001d524 is semisimple. Then we can use Vinberg’s theory of ϑ-groups and the machinery of Invariant Theory.If \U0001d524 = \U0001d525⊕⋯⊕\U0001d525 (sum of k copies), where \U0001d525 is simple, and ϑ is the cyclic permutation, then we prove that the corresponding Poisson-commutative subalgebra (\U0001d52e; ϑ) is polynomial and maximal. Furthermore, we quantise this (\U0001d524; ϑ) using a Gaudin subalgebra in the enveloping algebra \U0001d4b0(\U0001d524).

Classification of level zero irreducible integrable modules for twisted full toroidal Lie algebras

In this paper, we first construct the twisted full toroidal Lie algebra by an extension of a centreless Lie torus LT which is a multiloop algebra twisted by several automorphisms of finite order and equipped with a particular grading. We then provide a complete classification of all the irreducible integrable modules with finite-dimensional weight spaces for this twisted full toroidal Lie algebra having a non-trivial LT-action and where the centre of the underlying Lie algebra acts trivially.

Representations of twisted toroidal Lie algebras from twisted modules over vertex algebras

Given a simple finite-dimensional Lie algebra and an automorphism of finite order, one defines the notion of a twisted toroidal Lie algebra. In this paper, we construct representations of twisted toroidal Lie algebras from twisted modules over affine and lattice vertex algebras.

On the Solvability of ℤ3-Graded Novikov Algebras

Symmetries of algebraic systems are called automorphisms. An algebra admits an automorphism of finite order n if and only if it admits a Zn-grading. Let N=N0⊕N1⊕N2 be a Z3-graded Novikov algebra. The main goal of the paper is to prove that over a field of characteristic not equal to 3, the algebra N is solvable if N0 is solvable. We also show that a Z2-graded Novikov algebra N=N0⊕N1 over a field of characteristic not equal to 2 is solvable if N0 is solvable. This implies that for every n of the form n=2k3l, any Zn-graded Novikov algebra N over a field of characteristic not equal to 2,3 is solvable if N0 is solvable.

Fgc extended affine Lie algebras as fixed point subalgebras

It is known that the centreless cores of fgc extended affine Lie algebras can be viewed as fixed point subalgebras. In this paper, we study fixed point subalgebras of fgc extended affine Lie algebras. Our main result is that the fixed point subalgebras of untwisted extended affine Lie algebras under a finite abelian group (which is determined by some commuting finite order automorphisms) are fgc extended affine Lie algebras.

Plethysm and cohomology representations of external and symmetric products

We prove refined generating series formulae for characters of (virtual) cohomology representations of external products of suitable coefficients, e.g., (complexes of) constructible or coherent sheaves, or (complexes of) mixed Hodge modules on spaces such as (possibly singular) complex quasi-projective varieties. These formulae generalize our previous results for symmetric and alternating powers of such coefficients, and apply also to other Schur functors. The proofs of these results are reduced via an equivariant Künneth formula to a more general generating series identity for abstract characters of tensor powers V⊗n of an element V in a suitable symmetric monoidal category A. This abstract approach applies directly also in the equivariant context for spaces with additional symmetries (e.g., finite group actions, finite order automorphisms, resp., endomorphisms), as well as for introducing an abstract plethysm calculus for symmetric sequences of objects in A.

Fixed rings of quantum generalized Weyl algebras

Generalized Weyl algebras (GWAs) appear in diverse areas of mathematics including mathematical physics, noncommutative algebra, and representation theory. We study the invariants of quantum GWAs under finite order automorphisms. We extend a theorem of Jordan and Wells and apply it to determine the fixed ring of quantum GWAs under diagonal automorphisms. We further study properties of the fixed rings including global dimension, the Calabi–Yau property, rigidity, and simplicity.

Inner Finite Order Automorphisms of Type A

Abstract available with the full text.