Commutativity of automorphisms of subfactors modulo inner automorphisms

We introduce a new algebraic invariant of a subfactor . We show that this is an abelian group and that if the subfactor is strongly amenable, then the group coincides with the relative Connes invariant introduced by Y. Kawahigashi. We also show that this group is contained in the center of in many interesting examples such as quantum subfactors with level , but not always contained in the center. We also discuss its relation to the most general setting of the orbifold construction for subfactors.

     

On inner automorphisms and certain central automorphisms of groups

Let G be a group, let M and N be two normal subgroups of G. We denote by Aut _N ^M (G), the set of all automorphisms of G which centralize G/M and N. In this paper we investigate the structure of a group G in which one of the Inn(G) = Aut _N ^M (G), Aut _N ^M (G) ≤ Inn(G) or Inn(G) ≤ Aut _N ^M (G) holds. We also discuss the problem: “what conditions on G is sufficient to ensure that G has a non-inner automorphism which centralizes G/M and N”.     

Local automorphisms

We show that any linear map on a finite dimensional CSL algebra which at each point is equal to the value of some automorphism of is either an automorphism or can be factored as an automorphism and the transpose of a self-adjoint summand of . New examples of local mappings are constructed.

     

Trivial automorphisms

We prove that the statement ‘For all Borel ideals I and J on ω, every isomorphism between Boolean algebras P(ω)/I and P(ω)/J has a continuous representation’ is relatively consistent with ZFC. In this model every isomorphism between P(ω)/I and any other quotient P(ω)/J over a Borel ideal is trivial for a number of Borel ideals I on ω. We can also assure that the dominating number, σ, is equal to ?₁ and that \({2^{{\aleph _1}}} > {2^{{\aleph _0}}}\) . Therefore, the Calkin algebra has outer automorphisms while all automorphisms of P(ω)/Fin are trivial. Proofs rely on delicate analysis of names for reals in a countable support iteration of Suslin proper forcings.     

Finitary automorphisms of groups

We introduce the so-called multiscale limit for spectral curves associated with real finite-gap sine-Gordon solutions. This technique allows us to solve the old problem of calculating the density of the topological charge for real finite-gap sine-Gordon solutions directly from the θ-functional formulae.     

Polynomial automorphisms of lattices

Presented by Walter Taylor.     

Bounded automorphisms of surfaces

The equivalence among bounded homeomorphisms of a surface, reducible homeomorphisms of the surface, bounded automorphisms of the surface group, induced or exchange automorphisms of the surface group, and the stabilizer of the translation length function of the tree action is established.     

Nonlinear equivariant automorphisms

We show that the natural representation of SL₃ × SL₅ × SL₁₃ allows nonlinear equivariant automorphisms; more exactly, the group of polynomial automorphisms on ?³ ? ?⁵ ? ?¹³ commuting with the simple SL₃ × SL₅ × SL₁₃-action is isomorphic to ? ? ?. This is the first example of a simple module with nonlinear equivariant automorphisms.     

Near automorphisms of cycles

Let f be a permutation of V(G). Define δ_f(x,y)=|d_G(x,y)-d_G(f(x),f(y))| and δ_f(G)=∑δ_f(x,y) over all the unordered pairs {x,y} of distinct vertices of G. Let π(G) denote the smallest positive value of δ_f(G) among all the permutations f of V(G). The permutation f with δ_f(G)=π(G) is called a near automorphism of G. In this paper, we study the near automorphisms of cycles C_n and we prove that π(C_n)=4⌊n/2⌋-4, moreover, we obtain the set of near automorphisms of C_n.     

On automorphisms of biproducts

We study certain subgroups of the full group of Hopf algebra automorphisms of a biproduct. In the process, interesting subgroups of certain permutation groups come into play.     

Power automorphisms of finitep-groups

For a finite groupG letA(G) denote the group of power automorphisms, i.e. automorphisms normalizing every subgroup ofG. IfG is ap-group of class at mostp, the structure ofA (G) is shown to be rather restricted, generalizing a result of Cooper ([2]). The existence of nontrivial power automorphisms, however, seems to impose restrictions on thep-groupG itself. It is proved that the nilpotence class of a metabelianp-group of exponentp ² possessing a nontrival power automorphism is bounded by a function ofp. The “nicer” the automorphism—the lower the bound for the class. Therefore a “type” for power automorphisms is introduced. Several examples ofp-groups having large power automorphism groups are given.     

On automorphisms of A-groups

Let G be an A-group (i.e. a group in which xx ^α  = x ^α x for all and let denote the subgroup of Aut(G) consisting of all automorphisms that leave invariant the centralizer of each element of G. The quotient is an elementary abelian 2-group and natural analogies exist to suggest that it might always be trivial. It is shown that, in fact, for any odd prime p and any positive integer r, there exist infinitely many finite pA-groups G for which has rank r. Received: 23 March 2008, Revised: 20 May 2008     

Chromatic automorphisms of graphs

The vertex-coloring group and the full automorphism group of an n-colorable graph are independent if and only if n is an integer ≥ 3.     

p-Groups of automorphisms of abelianp-groups

We consider the action of a p-group G on an Abelian p-group A, with the latter treated as a faithful right ℤG-module. Our aim is to establish a connection between exponents of the kernels under the induced action of G on elementary p-groups A/pA and Ω₁(A) = {x ε A|px=0}; the kernels are denoted by C_G(A/pA) and C_G(Ω₁(A)), respectively. It is proved that if the exponent of one of the kernels C_G(A/pA) or C_G(Ω₁(A)) is finite then the other also has a finite exponent bounded in terms of the first; moreover, these kernels are nilpotent. In one case we impose the additional restriction . And the wreath product of a quasicyclic group and an arbitrary p-group G shows that this condition cannot be dropped. The results obtained are used to confirm, for one particular case, the conjecture on the boundedness of a derived length of a finite group with an automorphism of order 2 all of whose fixed points are central. (The solubility of such groups, and also the reduction to the case of 2-groups, were established in [1].) Translated fromAlgebra i Logika, Vol. 39, No. 3, pp. 359–371, May–June, 2000.     

Zero-dimensional automorphisms with specification

Necessary and sufficient conditions for zero-dimensional abelian automorphisms to satisfy specification are given.