Finite Order Automorphisms Research Articles

A note on Clifford-Klein forms

We consider the problem of finding Clifford-Klein forms in a class of homogeneous spaces determined by inclusions of real Lie algebras of a special type which we call strongly regular. This class of inclusions is described in terms of their Satake diagrams. For example, the complexifications of such inclusions contain the class of subalgebras generated by automorphisms of finite order. We show that the condition of strong regularity implies the restriction on the real rank of subalgebras. This in part explains why the known examples of Clifford-Klein forms are rare. We make detailed calculations of some known examples from the point of view of the Satake diagrams.

Automorphisms of finite order of nilpotent groups III

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose \(\phi \) is an automorphism of G of finite order m. Let \(\psi \) and \(\gamma \) denote the associated maps of G given by $$\begin{aligned} \psi :~\mathrm{g} \mapsto \mathrm{g.g} \phi .\mathrm{g} \phi ^{2}. {\ldots }~\mathrm{g}\phi ^{\mathrm{m}-1}~\quad \mathrm{and} \quad \gamma : \mathrm{g} \mapsto \mathrm{g}^{-1}.g\phi \quad \mathrm{for} \quad \mathrm{g} \in \mathrm{G}. \end{aligned}$$ In a previous paper we proved that G\(\psi \).ker\(\psi \) and G\(\gamma \).ker\(\gamma \) are both very large in that they contain subgroups of finite index in G. Here we prove that \({ }\) and \({ }\) are both very small, namely finite. Thus \(\psi \) and \(\gamma \) determine two very weak splittings of G.

Finite order automorphisms and a parametrization of nilpotent orbits in p -adic lie algebras

Let $k$ be a field with a nontrivial discrete valuation which is complete and has perfect residue field. Let $G$ be the group of $k$-rational points of a reductive, linear algebraic group ${\bf G}$ equipped with a finite cyclic group $L$ of order $m$ acting on ${\bf G}$ by algebraic automorphisms defined over $k$. We assume that the Lie algebra of $G$ decomposes into a direct sum of eigenspaces under the action of $L$ (with generator $\theta$); we let ${\frak g}^i$ denote the eigenspace corresponding to eigenvalue $\xi^i$, where $\xi$ is a primitive $m^{th}$ root of unity, that is, $X \in {\frak g}^i$ if and only if $\theta(X)=\xi^iX$. If ${\bf H}$ is a $k$-subgroup of ${\bf G}^{L}$, the group of $L$-fixed points, which contains the neutral component of ${\bf G}^{L}$, then $H={\bf H}(k)$ acts on each eigenspace of ${\frak g}$. Let $r \in \Bbb{R}$. Under mild restrictions on the residual characteristic of $k$, the set of nilpotent $H$-orbits in the $\xi$-eigenspace ${\frak g}^1$ is parametrized by equivalence classes of noticed Moy-Prasad cosets of depth $r$ which lie in ${\frak g}^1$.

Non-crystallographic nets: characterization and first steps towards a classification

Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroup F(N) of the automorphism group of NC nets (N, T). As a consequence, NC nets are unstable nets (they display vertex collisions in any barycentric representation) and, conversely, stable nets are crystallographic nets. The labelled quotient graphs of NC nets are characterized by the existence of an equivoltage partition (a partition of the vertex set that preserves label vectors over edges between cells). A classification of NC nets is proposed on the basis of (i) their relationship to the crystallographic net with a homeomorphic barycentric representation and (ii) the structure of the subgroup F(N).

Automorphisms of finite order of Nilpotent groups

Let $$\phi $$ be an automorphism of prime order $$p$$ of the nilpotent group $$G$$ . Frequently one needs to consider the map $$\psi : G\rightarrow G$$ given by $$\begin{aligned} g\psi =g\cdot g\phi \cdot g\phi ^2\cdot \ldots \cdot g\phi ^{p-1}. \end{aligned}$$ This $$\psi $$ is not usually a homomorphism of $$G$$ and $$ker\psi $$ and $$G\psi $$ need not be subgroups. Under relatively mild restrictions we prove that $$G\psi $$ is finite if and only if $$ker\psi $$ contains a subgroup of $$G$$ of finite index and $$ker\psi $$ is finite if and only if $$G\psi $$ contains a subgroup of finite index. These results do not extend to polycyclic groups or Cernikov groups.

Hirota equations associated with simply laced affine Lie algebras: The cuspidal classE6of e6(1)

In this paper we derive Hirota equations associated with the simply laced affine Lie algebras g(1), where g is one of the simply laced complex Lie algebras an,dn,e6,e7 or e8, defined by finite order automorphisms of g which we call Lepowsky automorphisms. In particular, we investigate the Hirota equations for Lepowsky automorphisms of e6 defined by the cuspidal class E6 of the Weyl group W(E6) of e6. We also investigate the relationship between the Lepowsky automorphisms of the simply laced complex Lie algebras g and the conjugate canonical automorphisms defined by Kac. This analysis is applied to identify the canonical automorphisms for the cuspidal class E6 of e6.

Multiloop algebras, iterated loop algebras and extended affine Lie algebras of nullity 2

Let \mathbb M_n be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to n -tuples of commuting finite order automorphisms. It is a classical result that \mathbb M_1 is the class of all derived algebras modulo their centres of affine Kac-Moody Lie algebras. This combined with the Peterson-Kac conjugacy theorem for affine algebras results in a classification of the algebras in \mathbb M_1 . In this paper, we classify the algebras in \mathbb M_2 , and further determine the relationship between \mathbb M_2 and two other classes of Lie algebras: the class of all loop algebras of affine Lie algebras and the class of all extended affine Lie algebras of nullity 2.

Semibounded Unitary Representations of Double Extensions of Hilbert–Loop Groups

A unitary representation of a, possibly infinite dimensional, Lie group G is called semibounded if the corresponding operators id�(x) from the derived representation are uniformly bounded from above on some non-empty open subset of the Lie algebra g of G. We classify all irreducible semibounded representations of the groups b L'(K) which are double extensions of the twisted loop group L'(K), where K is a simple Hilbert– Lie group (in the sense that the scalar product on its Lie algebra is invariant) and ' is a finite order automorphism of K which leads to one of the 7 irreducible locally affine root systems with their canonicalZ-grading. To achieve this goal, we extend the

Extended affinization of invariant affine reflection algebras

The class of invariant affine reflection algebras is the most general known extension of the class of affine Kac--Moody Lie algebras, introduced in 2008. We develop a method known as ``affinization'' for the class of invariant affine reflection algebras, and show that starting from an algebra belonging to this class together with a certain finite order automorphism, and applying the so called ``affinization method'', we obtain again an invariant affine reflection algebra. This can be considered as an important step towards the realization of invariant affine reflection algebras.

Cyclotomic associators and finite type invariants for tangles in the solid torus

The universal Vassiliev-Kontsevich invariant is a functor from the category of tangles to a certain graded category of chord diagrams, compatible with the Vassiliev filtration and whose associated graded is an isomorphism. The Vassiliev filtration has a natural extension to tangles in any thickened surface $M\times I$ but the corresponding category of diagrams lacks some finiteness properties which are essential to the above construction. We suggest to overcome this obstruction by studying families of Vassiliev invariants which, roughly, are associated to finite coverings of $M$. In the case $M=\mathbb{C}^*$, it leads for each positive integer $N$ to a filtration on the space of tangles in $\mathbb{C}^* \times I$ (or "B-tangles"). We first prove an extension of the Shum--Reshetikhin--Turaev theorem in the framework of braided module category leading to B-tangles invariants. We introduce a category of "$N$-chord diagrams", and use a cyclotomic generalization of Drinfeld associators, introduced by Enriquez, to put a braided module category structure on it. We show that the corresponding functor from the category of B-tangles is a universal invariant with respect to the $N$ filtration. We show that Vassiliev invariants in the usual sense are well approximated by $N$ finite type invariants. We show that specializations of the universal invariant can be constructed from modules over a metrizable Lie algebra equipped with a finite order automorphism preserving the metric. In the case the latter is a "Cartan" automorphism, we use a previous work of the author to compute these invariants explicitly using quantum groups. Restricted to links, this construction provides polynomial invariants.

VERTEX OPERATOR ALGEBRAS AND WEAK JACOBI FORMS

Let V be a strongly regular vertex operator algebra. For a state h ∈ V1 satisfying appropriate integrality conditions, we prove that the space spanned by the trace functions Tr M qL(0)-c/24ζh(0) (M a V-module) is a vector-valued weak Jacobi form of weight 0 and a certain index 〈h, h〉/2. We discuss refinements and applications of this result when V is holomorphic, in particular we prove that if g = eh(0) is a finite-order automorphism then Tr V qL(0)-c/24g is a modular function of weight 0 on a congruence subgroup of SL 2(ℤ).

Finite order automorphisms and real forms of affine Kac-Moody algebras in the smooth and algebraic category

Let $\mathfrak{g}$ be a real or complex (finite dimensional) simple Lie algebra and $\sigma\in\mathrm{Aut}\mathfrak{g}$. The authors study automorphisms of the twisted loop algebra $L(\mathfrak{g},\sigma)$ of smooth $\sigma$-periodic maps from $\mathbb{R}$ to $\mathfrak{g}$ as well as of the ``smooth'' affine Kac-Moody algebra $\hat L(\mathfrak{g},\sigma)$, which is a $2$-dimensional extension of $L(\mathfrak{g},\sigma)$. It turns out that these automorphisms which either preserve or reverse the orientation of loops, and are correspondingly called to be of first and second kind, can be described essentially by curves of automorphisms of $\mathfrak{g}$. If the order of the automorphisms is finite, then the corresponding curves in $\mathrm{Aut}\mathfrak{g}$ allow us to define certain invariants and these turn out to parametrize the conjugacy classes of the automorphisms. If their order is $2$ the authors carry this out in detail and deduce a complete classification of involutions and real forms (which correspond to conjugate linear involutions) of smooth affine Kac-Moody algebras. The resulting classification can be seen as an extension of Cartan's classification of symmetric spaces, i.e. of involutions on $\mathfrak{g}$. If $\mathfrak{g}$ is compact, then conjugate linear extensions of involutions from $\hat L(\mathfrak{g},\sigma)$ to conjugate linear involutions on $\hat L(\mathfrak{g}_{\mathbb{C}},\sigma_{\mathbb{C}})$ yield a bijection between their conjugacy classes and this gives existence and uniqueness of Cartan decompositions of real forms of complex smooth affine Kac-Moody algebras. The authors show that their methods work equally well also in the algebraic case where the loops are assumed to have finite Fourier expansions.

Fixed points of finite groups acting on generalised Thompson groups

We study centralisers of finite order automorphisms of the generalised Thompson groups Fn,∞ and conjugacy classes of finite subgroups in finite extensions of Fn,∞. In particular, we show that centralisers of finite automorphisms in Fn,∞ are either of type FP∞ or not finitely generated.

Finite order automorphisms on contragredient Lie superalgebras

We add combinatorial data to the Dynkin diagrams to classify all finite order automorphisms on contragredient Lie superalgebras, and study their invariant subalgebras.

Periodic automorphisms of the free Lie algebra of rank 3

We prove that each automorphism of finite order of the free Lie algebra of rank 3 over an algebraically closed field is conjugate to a linear automorphism if the field characteristic fails to divide the automorphism order.

Good Index Behaviour of θ-Representations, I

Let Q be an algebraic group with ${\mathfrak q}={\mathrm{Lie\,}} Q$ and V a Q-module. The index of V is the minimal codimension of the Q-orbits in the dual space V *. There is a general inequality, due to Vinberg, relating the index of V and the index of a Q v -module $V/{\mathfrak q}{\cdot}v$ for v ∈ V. A pair (Q, V) is said to have GIB if Vinberg’s inequality turns into an equality for all v ∈ V. In this article, we are interested in the GIB property of θ-representations, where θ is a finite order automorphism of a simple Lie algebra ${\mathfrak g}$ . An automorphism of order m defines a ${{\mathbb Z}}/m{{\mathbb Z}}$ -grading ${\mathfrak g} =\bigoplus{\mathfrak g}_i$ . If G 0 is the identity component of G θ , then it acts on ${\mathfrak g}_1$ and this action is called a θ-representation. We classify inner automorphisms of ${\mathfrak gl}_n$ and all finite order autmorphisms of the exceptional Lie algebras such that $(G_0,{\mathfrak g}_1)$ has GIB and ${\mathfrak g}_1$ contains a non-zero semisimple element.

Almost fixed-point-free automorphisms of soluble groups

Suppose G is either a soluble (torsion-free)-by-finite group of finite rank or a soluble linear group over a finite extension field of the rational numbers. We consider the implications for G if G has an automorphism of finite order m with only finitely many fixed points. For example, if m is prime then G is a finite extension of a nilpotent group and if m = 4 then G is a finite extension of a centre-by-metabelian group. This extends the special cases where G is polycyclic, proved recently by Endimioni (2010); see [3].

Mini-Workshop: Symmetric Varieties and Involutions of Algebraic Groups

The topics of this conference all in some way evolved from the classical theory of real and complex Lie groups. Indeed, one of the important mathematical goals during the 1950's was to find analogs of the semisimple Lie groups of exceptional type over arbitrary fields. Chevalley completed the first crucial step by producing his famous basis theorem for simple complex Lie algebras, and later Steinberg succeeded in describing these analogs group-theoretically. An important development due to Tits was the theory of groups with a BN -pair and invented buildings; these buildings belong to arbitrary Chevalley groups as naturally as the projective spaces belong to the special linear groups. Since then the theories of algebraic groups and of buildings developed into various directions. However, due to their common origin both theories often lead naturally to similar questions which were attacked by completely different means. In the context of this conference, the PhD thesis by Bernhard Mühlherr and the work by Aloysius Helminck and coauthors on involutions of algebraic groups illustrate this in a quite remarkable way. Both projects contributed strongly to the understanding of the geometry of involutions of algebraic groups, but surprisingly each one has gone unnoticed by the researchers of the other until recently. One of the main objectives of this conference was to bring these two theories closer to each other. The first two lectures on Monday morning familiarised all participants with the concept of buildings; Pierre-Emmanuel Caprace explained the foundations of buildings from the simplicial point of view, while Bernhard Mühlherr introduced the chamber system approach to buildings and explained the power of filtrations when studying sub-geometries of (twin) buildings that arise from the action of certain subgroups of the isometry group of the (twin) building. As these two lectures were of an introductory nature and since their content is already well documented (we refer to the recently published book by Abramenko and Brown for the theory of buildings and to the contributions of Alice Devillers and of Hendrik Van Maldeghem to the Oberwolfach report 20/2008 for an account on filtrations and their powerful applications), we do not include acstracts of these lectures. On Monday afternoon and on Friday morning Aloysius Helminck presented the theory of involutions of algebraic groups, while in Monday's final lecture Max Horn showed how to combine Helminck's theory with M"uhlherr's PhD thesis in order to obtain general and powerful results on the geometry of involutions of groups with a root group datum, a class of groups that contains the semisimple algebraic groups, the split Kac–Moody groups, and the split finite groups of Lie type. Most of Tuesday and part of Wednesday were focussed on the Tits centre conjecture. In a series of two lectures Gerhard Röhrle and Michael Bate presented an algebraic-group approach towards proving the conjecture, while on Tuesday afternoon Katrin Tent presented a combinatorial approach and on Wednesday morning Linus Kramer reported on metric considerations in the context of the Tits centre conjecture. It is our impression that these four lectures have triggered additional activity towards proving the centre conjecture, and that one or more of these approaches will be successful in the near future. The fourth talk on Tuesday afternoon was given by Yiannis Sakellaridis on spherical varieties and automorphic forms, while the second talk on Wednesday by Lizhen Ji presented compactifications of locally symmetric spaces. Thursday's talks by Sergey Shpectorov and by Paul Levy concentrated on involutions of affine buildings, respectively automorphisms of finite order of semisimple Lie algebras. The remaining three talks were more topologically in nature. Guy Rousseau presented his theory of microaffine buildings, hovels, and Kac–Moody groups over ultrametric fields on Thursday. Thursday's fourth talk was by Bertrand Rémy on Satake compactifications of buildings via Berkovich theory. The conference was concluded by Pierre-Emmanuel Caprace's report on aspects of the structure of locally compact groups. We are particularly pleased by the lively interaction between the participants during the long afternoon breaks (each morning's lectures finished at 11.30 a.m. while the afternoon sessions only started at 4.20 p.m.) and during the evenings.

Fixed point subalgebras of root graded Lie algebras

We study the subalgebra of fixed points of a root graded Lie algebra under a certain class of finite order automorphisms. As the centerless core of extended affine Lie algebras or equivalently irreducible centerless Lie tori are examples of root graded Lie algebras, our work is an extension of some recent result about the subalgebra of fixed points of a Lie torus under a certain finite order automorphism.

Lie Tori and Their Fixed Point Subalgebras

We study the fixed point subalgebra of a centerless irreducible Lie torus under a certain finite order automorphism. We investigate which axioms of a Lie torus hold for the fixed points and which do not. We relate our study to some recent results about the fixed points of extended affine Lie algebras under a class of finite order automorphisms.