Automorphisms Of Compact Groups Research Articles

Homogeneous hypercomplex structures II - Coset spaces of compact Lie groups

In this second in a series paper we complete our classification of hypercomplex manifolds, on which a compact group of automorphisms acts transitively. The description of the spaces as well as the proofs of our results use only the structure theory of reductive groups.

The Plancherel formula for countable groups

We discuss a Plancherel formula for countable groups, which provides a canonical decomposition of the regular representation of such a group Γ into a direct integral of factor representations. Our main result gives a precise description of this decomposition in terms of the Plancherel formula of the FC-center Γfc of Γ (that is, the normal subgroup of Γ consisting of elements with a finite conjugacy class); this description involves the action of an appropriate totally disconnected compact group of automorphisms of Γfc. As an application, we determine the Plancherel formula for linear groups. In an appendix, we use the Plancherel formula to provide a unified proof for Thoma’s and Kaniuth’s theorems which respectively characterize countable groups which are of type I and those whose regular representation is of type II.

Generic analytic polyhedron with non-compact automorphism group

In this paper we prove the following rigidity theorem: a generic analytic polyhedron with non-compact automorphism group is biholomorphic to the product of a complex manifold with compact automorphism group and a polydisk. Moreover, this complex manifold and the dimension of this polydisk can be explicitly described in terms of the limit set of the automorphism group.

Nilpotent Gelfand pairs and Schwartz extensions of spherical transforms via quotient pairs

It has been shown [1,2,9,10] that for several nilpotent Gelfand pairs (N,K) (i.e., with N a nilpotent Lie group, K a compact group of automorphisms of N and the algebra L1(N)K commutative) the spherical transform establishes a 1-to-1 correspondence between the space S(N)K of K-invariant Schwartz functions on N and the space S(Σ) of functions on the Gelfand spectrum Σ of L1(N)K which extend to Schwartz functions on Rd, once Σ is suitably embedded in Rd. We call this property (S).We present here a general bootstrapping method which allows to establish property (S) to new nilpotent pairs (N,K), once the same property is known for a class of quotient pairs of (N,K) and a K-invariant form of Hadamard formula holds on N.We finally show how our method can be recursively applied to prove property (S) for a significant class of nilpotent Gelfand pairs.

Spherical spectral synthesis

We make an attempt to extend L. Schwartz’s classical result on spectral synthesis to several dimensions. Due to counterexamples of D. I. Gurevich this is impossible for translation invariant varieties. Our idea is to replace translations by proper euclidean motions in higher dimensions. For this purpose we introduce the basic concepts of spectral analysis and synthesis in the non-commutative setting based on Gelfand pairs, where “translation will be replaced by invariance with respect to a compact group of automorphisms. The role of exponential functions will be played by spherical functions. As an application we obtain the extension of L. Schwartz’s fundamental result.

Kähler–Einstein metrics along the smooth continuity method

We show that if a Fano manifold M is K-stable with respect to special degenerations equivariant under a compact group of automorphisms, then M admits a Kahler–Einstein metric. This is a strengthening of the solution of the Yau–Tian–Donaldson conjecture for Fano manifolds by Chen–Donaldson–Sun (Int Math Res Not (8):2119–2125, 2014), and can be used to obtain new examples of Kahler–Einstein manifolds. We also give analogous results for twisted Kahler–Einstein metrics and Kahler–Ricci solitons.

A BOUNDED KOHN NIRENBERG DOMAIN

Building on the famous domain of Kohn and Nirenberg we give an example of a domain which shares the important features of the Kohn Nirenberg domain, but which can also be shown to be -bounded. As an application, we remark that this example has compact automorphism group.

ENDOMORPHISMS AND AUTOMORPHISMS OF LOCALLY COVARIANT QUANTUM FIELD THEORIES

In the framework of locally covariant quantum field theory, a theory is described as a functor from a category of spacetimes to a category of *-algebras. It is proposed that the global gauge group of such a theory can be identified as the group of automorphisms of the defining functor. Consequently, multiplets of fields may be identified at the functorial level. It is shown that locally covariant theories that obey standard assumptions in Minkowski space, including energy compactness, have no proper endomorphisms (i.e. all endomorphisms are automorphisms) and have a compact automorphism group. Further, it is shown how the endomorphisms and automorphisms of a locally covariant theory may, in principle, be classified in any single spacetime. As an example, the endomorphisms and automorphisms of a system of finitely many free scalar fields are completely classified.

The nub of an automorphism of a totally disconnected, locally compact group

AbstractTo any automorphism,$\alpha $, of a totally disconnected, locally compact group,$G$, there is associated a compact,$\alpha $-stable subgroup of$G$, here called thenubof$\alpha $, on which the action of$\alpha $is ergodic. Ergodic actions of automorphisms of compact groups have been studied extensively in topological dynamics and results obtained transfer, via the nub, to the study of automorphisms of general locally compact groups. A new proof that the contraction group of$\alpha $is dense in the nub is given, but it is seen that the two-sided contraction group need not be dense. It is also shown that each pair$(G, \alpha )$, with$G$compact and$\alpha $ergodic, is an inverse limit of pairs that have ‘finite depth’ and that analogues of the Schreier refinement and Jordan–Hölder theorems hold for pairs with finite depth.

Contraction groups, ergodicity, and distal properties of automorphisms of compact groups

Given an automorphism $\tau$ of a compact group $G$, we study the factorization of $C(\tau,K)$, the contraction group of $\tau$ modulo a closed $\tau$-invariant subgroup $K$, into the product $C(\tau)K$, of the contraction group $C(\tau)$ of $\tau$, and $K$. We prove that the factorization $C(\tau,K)=C(\tau)K$ holds for every closed $\tau$-invariant subgroup $K$ if and only if $G$ contains arbitrarily small closed normal $\tau$-invariant subgroups $N$ with finite-dimensional quotients $G/N$. For metrizable groups, we obtain that $C(\tau)K$ is a dense subgroup of $C(\tau,K)$, for every closed $\tau$-invariant subgroup $K$. These results are used to link the contraction group to the properties of the dynamical system $(G,\tau)$. It follows that $\tau$ is distal if and only if $C(\tau)$ is trivial, while ergodicity of $\tau$ implies that $C(\tau)$ is nontrivial. When $G$ is metrizable, the closure of $C(\tau)$ is the largest closed $\tau$-invariant subgroup on which $\tau$ acts ergodically and, at the same time, it is the smallest among closed normal $\tau$-invariant subgroups $N$ such that $\tau$ acts distally on $G/N$. If $\tau$ is ergodic, then its restriction to any closed connected normal $\tau$-invariant subgroup $N$ with finite-dimensional quotient $G/N$ is also ergodic. Moreover, when $G$ is connected, the largest closed $\tau$-invariant subgroup on which $\tau$ acts ergodically is necessarily connected.

On generalized Gajda's functional equation of D'Alembert type

Let G be a locally compact group and K be a compact group of automorphisms of G. We consider the functional equation

The automorphism group of a domain with an exponentially flat boundary point

We study the automorphisms group action on a bounded domain in C n having a boundary point that is exponentially flat. Such a domain typically has a compact automorphism group. Our results enable us to generate many new examples.

A dichotomy in orbit growth for commuting automorphisms

We consider asymptotic orbit-counting problems for certain expansive actions by commuting automorphisms of compact groups. A dichotomy is found between systems with asymptotically more periodic orbits than the topological entropy predicts, and those for which there is no excess of periodic orbits.

Gelfand pairs on the Heisenberg group and Schwartz functions

Let H n be the ( 2 n + 1 ) -dimensional Heisenberg group and K a compact group of automorphisms of H n such that ( K ⋉ H n , K ) is a Gelfand pair. We prove that the Gelfand transform is a topological isomorphism between the space of K-invariant Schwartz functions on H n and the space of Schwartz function on a closed subset of R s homeomorphic to the Gelfand spectrum of the Banach algebra of K-invariant integrable functions on H n .

On certain structural properties of Banach algebras associated with automorphisms

In the Hilbert space setting (i.e., in the framework of C∗-algebra theory), analogous objects are closely related with a crossed product (see, e.g., [4]), and many works are devoted to the description of their structures. In particular, Landstad [3] found necessary and sufficient conditions (in terms of duality theory) under which a C∗-algebra is isomorphic to the crossed product (of a certain algebra and a locally compact automorphism group). In the case of a discrete group, in [1, Chap. 2], there are necessary and sufficient conditions under which a C∗-algebra is isomorphic to a crossed product in terms of the action of the group (the so-called topologically free action (see Sec. 2.8 of the present paper) and also in terms of fulfillment of a certain inequality (Property (∗), (2.1) of the present paper) that guarantees the existence of an expectation of the algebra B(A, Tg) on the algebra A (see (2.2) and (2.3)). The goal of the present paper is the study of interrelations between the properties (topologically free action, Property (∗), and dual action of the group) mentioned above in the general Banach space setting. Along with this, we study property (∗∗) (Sec. 2.2 ) of reconstructing an element of the algebra B(A, Tg) by its “Fourier coefficients,” which naturally arises here.

On the normalizing groupoids and the commensurability groupoids for inclusions of factors associated to ergodic equivalence relations–subrelations

It is shown that for the inclusion of factors ( B ⊆ A ) : = ( W ∗ ( S , ω ) ⊆ W ∗ ( R , ω ) ) corresponding to an inclusion of ergodic discrete measured equivalence relations S ⊆ R , S is normal in R in the sense of Feldman–Sutherland–Zimmer [J. Feldman, C.E. Sutherland, R.J. Zimmer, Subrelations of ergodic equivalence relations, Ergodic Theory Dynam. Systems 9 (1989) 239–269] if and only if A is generated by the normalizing groupoid of B. Moreover, we show that there exists the largest intermediate equivalence subrelation N R ( S ) which contains S as a normal subrelation. We further give a definition of “commensurability groupoid” as a generalization of normality. We show that the commensurability groupoid of B in A generates A if and only if the inclusion B ⊆ A is discrete in the sense of Izumi–Longo–Popa [M. Izumi, R. Longo, S. Popa, A Galois correspondence for compact groups of automorphisms of von Neumann algebras with a generalization to Kac algebras, J. Funct. Anal. 155 (1998) 25–63]. We also show that there exists the largest equivalence subrelation Comm R ( S ) such that the inclusion B ⊆ W ∗ ( Comm R ( S ) , ω ) is discrete. It turns out that the intermediate equivalence subrelations N R ( S ) and Comm R ( S ) ⊆ R thus defined can be viewed as groupoid-theoretic counterparts of a normalizer subgroup and a commensurability subgroup in group theory.

Canonical extension of endomorphisms of type III factors

We extend the notion of the canonical extension of automorphisms of type III factors to the case of endomorphisms with finite statistical dimensions. Following the automorphism case, we introduce two notions for endomorphisms of type III factors: modular endomorphisms and Connes-Takesaki modules. Several applications to compact groups of automorphisms and subfactors of type III factors are given from the viewpoint of ergodic theory.

Modular Categories and Orbifold Models

In this paper, we try to answer the following question: given a modular tensor category $\A$ with an action of a compact group $G$, is it possible to describe in a suitable sense the ``quotient'' category $\A/G$? We give a full answer in the case when $\A=\vec$ is the category of vector spaces; in this case, $\vec/G$ turns out to be the category of representation of Drinfeld's double $D(G)$. This should be considered as category theory analog of topological identity ${pt}//G=BG$. This implies a conjecture of Dijkgraaf, Vafa, E. Verlinde and H. Verlinde regarding so-called orbifold conformal field theories: if $\V$ is a vertex operator algebra which has a unique irreducible module, $\V$ itself, and $G$ is a compact group of automorphisms of $\V$, and some not too restricitive technical conditions are satisfied, then $G$ is finite, and the category of representations of the algebra of invariants, $\V^G$, is equivalent as a tensor category to the category of representations of Drinfeld's double $D(G)$. We also get some partial results in the non-holomorphic case, i.e. when $\V$ has more than one simple module.

Construction de réseaux en théorie de Kac-Moody

We show that a Kac-Moody group (see [ 10]) over a sufficiently large finite field is a (non-uniform) lattice in the locally compact automorphism group of the product of the buildings of its twinning. The same argument proves that a spherical parabolic subgroup is a (non-uniform) lattice in the automorphism group of the building of opposite sign.

A Galois Correspondence for Compact Groups of Automorphisms of von Neumann Algebras with a Generalization to Kac Algebras

LetMbe a factor with separable predual andGa compact group of automorphisms ofMwhose action is minimal, i.e.,MG′∩M=C, whereMGdenotes theG-fixed point subalgebra. Then every intermediate von Neumann algebraMG⊂N⊂Mhas the formN=MHfor some closed subgroupHofG. An extension of this result to the case of actions of compact Kac algebras on factors is also presented. No assumptions are made on the existence of a normal conditional expectation ontoN.