Tag - Automorphisms of structures

Antonio Viruel: Permutation representations of finite groups via evolution algebras

18th December, 2023

In the wake of the influential work by Elduque-Labra, it is known that every finite-dimensional evolution K-algebra X such that X2 = X, namely X is idempotent, has a finite group of automorphisms. Building on this foundation, works of Costoya et al. show that given any finite group G, there exists an idempotent finite-dimensional evolution algebra X such that Aut(X) ≅ G. Moreover, when the base field is sufficiently large in comparison to the group G, such an X can be selected to be simple. As a result, Sriwongsa-Zou propose that idempotent finite-dimensional evolution algebras can be classified based on the isomorphism type of their group of automorphisms and dimension. Within this context, we establish that the natural representation of highly transitive groups cannot be realized as the complete group of automorphisms of an idempotent finite-dimensional evolution algebra. For instance, for any sufficiently large integer n, there exists no evolution algebra X such that X2 = X, dim X = n, and Aut(X) is isomorphic to the alternating group An. However, we demonstrate that for any (not necessarily faithful) permutation representation ρ : G → Sn and any field K, there exists a finite-dimensional evolution K-algebra X such that X2 = X, Aut(X) ≅ G and the induced representation given by the Aut(X)-action on the natural idempotents of X is ρ.

Antonio Peralta: How can we apply Jordan structures to reinterpret Wigner-Uhlhorn theorem?

1st April, 2022

So far to date, much has been written about E. Wigner's and U. Uhlhron's theorems and their importance for physics and mathematics. For the sake of conciseness, let us go straight to some of the starring results. There are six mathematical models employed in quantum mechanics, among them we have: 1. The C-algebra B(H) of bounded operators; 2. The Jordan algebra B(H)sa of bounded self-adjoint operators; 3. The orthomodular lattice L of closed subspaces of H, equivalently, the lattice of all projections in B(H), where H is a complex Hilbert space.

The natural automorphisms of these mathematical models (i.e., the bijections on these sets preserving the corresponding relevant structure: associative product and involution, Jordan product, and orthogonality and order between subspaces or projections) represent the symmetry groups of quantum mechanics and are endowed with natural topologies induced by the probabilistic structure of quantum mechanics. It is known that these symmetry groups are all isomorphic when dim(H) ≥ 3. The last restriction exclude rank two, where there are no more than two orthogonal projections. This equivalence can be seen as the celebrated Wigner unitary-antiunitary theorem.

By replacing the set of projections P(H) by the wider set PI(H) = U(B(H)), of all partial isometries on H, L. Molnár proved the following result: Let be a complex Hilbert space with dim(H) ≥ 3. Suppose that Φ: U(B(H)) → U(B(H)) is a bijective transformation which preserves the natural partial ordering and the orthogonality between partial isometries in both directions. If Φ is continuous (in the operator norm) at a single element of U(B(H)) different from 0, then Φ extends to a real linear triple isomorphism.

During this talk we shall present new results, obtained in collaboration with Y. Friedman, showing that an extension of the previous results is possible in the case of a bijection between the lattices of tripotents of two Cartan factors and atomic JBW-triples non-containing rank-one Cartan factors. These new result provide new models to understand the quantum models. We shall also see how the results provide new alternatives to complement recent studies by J. Hamhalter proving that the set of partial isometries with its partial order and orthogonality relation is a complete Jordan invariant for von Neumann algebras.

Michal Ferov: Automorphism groups of Cayley graphs of Coxeter groups: when are they discrete?

4th March, 2022

The group of automorphisms of a connected locally finite graph is naturally a totally disconnected locally compact topological group, when equipped with the permutation topology. It therefore makes sense to ask for which graphs is the topology not discrete. We show that in case of Cayley graphs of Coxeter groups, one can fully characterize the discrete ones in terms of the symmetries of the corresponding Coxeter system.

Oksana Bezushchak: Locally matrix algebras and algebras of Mackey

21st October, 2021

n this talk we will discuss:

1. Tensor decompositions of locally matrix algebras and their parametrization by Steinitz numbers.

2. Automorphisms and derivations of locally matrix algebras.

3. Automorphisms and derivations of Mackey algebras and Mackey groups. In particular, we describe automorphisms of all infinite simple finitary torsion groups (in the classification of J.Hall) and derivations of all infinite-dimensional simple finitary Lie algebras (in the classification of A.Baranov and H.Strade).

Chloe Papin: A Whitehead Algorithm for Generalized Baumslag-Solitar Groups

15th October, 2021

Baumslag-Solitar groups BS(p,q) =< a,t | tapt-1 = aq > were first introduced as examples of non-Hopfian groups. They may be described using graphs of cyclic groups. In analogy with the study of Out(Fn) one can study their automorphisms through their action on an "outer space". After introducing generalized Baumslag-Solitar groups and their actions on trees, I will present an analogue of a Whitehead algorithm which takes an element of a free group and decides whether there exists a free factor which contains that element.

Ualbai Umirbaev: Automorphism groups of free algebras

4th December, 2020

There are many interesting results on the structure of the automorphism group Aut(Fn) and the outer automorphism group Out(Fn) of the free group Fn of rank n. Unfortunately, the theory of automorphism groups of free algebras over a field is not very rich and many problems are still open. I will describe some results and recall some open questions on the structures of the automorphism groups of:

   1.  the polynomial algebra K[x1,x2,…,xn] of rank n over a field K;
   2.  the free associative algebra K<x1,x2,...,xn> of rank n over K; and
   3.  the free Lie algebra Lie<x1,x2,...,xn> of rank n over K.

Laura Ciobanu: On computing fixed subgroups of endomorphisms in free groups

1st October, 2020

Given an endomorphism h of a free group F, the fixed subgroup of h consists of those elements x ∈ F for which h(x)=x. In this talk I will give some background on fixed subgroups in free groups, and then present an algorithm which computes the fixed subgroup and the stable image for any endomorphism of the free group of rank 2. This answers, for rank 2, a question posed by Stallings in 1984 and a more recent question of Ventura. I will explain why general endomorphisms are more difficult than automorphisms, and in what ways our algorithm needs the restriction on the rank. This is joint work with Alan Logan.