Automorphism Group Research Articles

Semifinite von Neumann algebras in gauge theory and gravity

von Neumann algebras have been playing an increasingly important role in the context of gauge theories and gravity. The crossed product presents a natural method for implementing constraints through the commutation theorem, rendering it a useful tool for constructing gauge-invariant algebras. The crossed product of a Type III algebra with its modular automorphism group is semifinite, which means that the crossed product regulates divergences in local quantum field theories. In this article, we find a sufficient condition for the semifiniteness of the crossed product of a Type III algebra with locally compact group containing the modular automorphism group. Our condition surprisingly implies the centrality of the modular flow in the symmetry group, and we provide evidence for the necessity of this condition. Under these conditions, we construct an associated trace that computes physical expectation values. We comment on the importance of this result and its implications for subregion physics in gauge theory and gravity. Published by the American Physical Society 2025

Automorphism Groups Associated to a Family of Hopf Algebras

Automorphism Groups Associated to a Family of Hopf Algebras

A note on separability in outer automorphism groups

Abstract We give a criterion for separability of subgroups of certain outer automorphism groups. This answers questions of Hagen and Sisto, by strengthening and generalizing a result of theirs on mapping class groups.

Chromatic quasisymmetric class functions for combinatorial Hopf monoids

We study the chromatic quasisymmetric class function of a linearized combinatorial Hopf monoid. Given a linearized combinatorial Hopf monoid H, and an H-structure h on a set N, there are proper colorings of h, generalizing graph colorings and poset partitions. We show that the automorphism group of h acts on the set of proper colorings. The chromatic quasisymmetric class function enumerates the fixed points of this action, weighting each coloring with a monomial. For the Hopf monoid of graphs this invariant generalizes Stanley’s chromatic symmetric function and specializes to the orbital chromatic polynomial of Cameron and Kayibi. We deduce various inequalities for the associated orbital polynomial invariants. We apply these results to several examples related to enumerating graph colorings, poset partitions, generic functions on matroids or generalized permutohedra, and others.

A remark on the number of automorphisms of finite abelian groups

Let \mathrm{Ab}_{0} be the class of finite abelian groups and consider the function f\colon \mathrm{Ab}_{0} \to (0,\infty) given by f(G)=\lvert{\operatorname{Aut}(G)\rvert/\lvert{(G)}\rvert} , where \operatorname{Aut}(G) is the automorphism group of a finite abelian group G . In this short note, we prove that the image of f is a dense set in [0,\infty) .

Symmetric and Kähler–Einstein Fano polygons

We investigate singular symmetric and Kähler–Einstein Fano polytopes. More precisely, we show that every symmetric Fano polytope is Kähler–Einstein generalizing the work by Batyrev and Selivanova, and study the automorphism groups of symmetric and Kähler–Einstein Fano polygons in detail. In particular, every finite subgroup of G L 2 ( Z ) GL_2(\mathbb {Z}) is an automorphism group of a Kähler–Einstein Fano polygon.

On automorphism groups of smooth hypersurfaces

We show that smooth hypersurfaces in complex projective spaces with automorphism groups of maximum size are isomorphic to Fermat hypersurfaces, with a few exceptions. For the exceptions, we give explicitly the defining equations and automorphism groups.

When is a polarised abelian variety determined by its 𝑝-divisible group?

We study the Siegel modular variety A g ⊗ F ¯ p \mathscr {A}_g\otimes \overline {\mathbb {F}}_p of genus g g and its supersingular locus S g \mathscr {S}_g . As our main result we determine precisely when S g \mathscr {S}_g is irreducible, and we list all x x in A g ⊗ F ¯ p \mathscr {A}_g\otimes \overline {\mathbb {F}}_p for which the corresponding central leaf C ( x ) \mathscr {C}(x) consists of one point, that is, for which x x corresponds to a polarised abelian variety which is uniquely determined by its associated polarised p p -divisible group. The first problem translates to a class number one problem for quaternion Hermitian lattices. The second problem also translates to a class number one problem, whose solution involves mass formulae, automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus g = 4 g=4 .

Physics with non-unital algebras? An invitation to the Okubo algebra

Abstract This paper discusses the potential application of the Okubonions, i.e. the Okubo algebra, within quantum chromodynamics (QCD). The Okubo algebra lacks a unit element and sits in the adjoint representation of its automorphism group SU(3), being fundamentally different from the more better-known octonions O. A physical interpretation is proposed in which the Okubonions represent gluons, the gauge bosons of QCD SU(3) color symmetry, while the octonions represent quarks. However, it is shown that the respective SU(3) symmetries for Okubonions and octonions are distinct and inequivalent subgroups of Spin(8) that share no common SU(2) subgroup. The unusual properties of Okubonions are suggested as a possible source for exotic QCD phenomena like asymptotic freedom and color confinement, though actual mechanisms remain to be investigated.

Discrete quantum subgroups of free unitary quantum groups

AbstractWe classify all compact quantum groups whose C*‐algebra sits inside that of the free unitary quantum groups . In other words, we classify all discrete quantum subgroups of , thereby proving a quantum variant of Kurosh's theorem to some extent. This yields interesting families which can be described using free wreath products and free complexifications. They can also be seen as quantum automorphism groups of specific quantum graphs which generalize finite rooted regular trees, providing explicit examples of quantum trees.

Uniform difference between the length spectra of Out(F2) and the genus two handlebody group

In this paper, we compare the translation length spectrum of the genus [Formula: see text] handlebody group in the Teichmüller space with that of the outer automorphism group of the free group of rank [Formula: see text] in the outer space. To this end, we analyze the homomorphism from the handlebody group to the outer automorphism group, induced by the inclusion of the boundary of the handlebody to the handlebody itself. With this homomorphism in hand, we reveal a crucial relationship: the translation length of a fully irreducible outer automorphism provides a lower bound for the translation lengths of pseudo-Anosov maps in its preimage. Notably, in the case of genus two, the difference between the translation length of each fully irreducible outer automorphism and the minimum translation length among the pseudo-Anosov maps in its preimage is bounded above by [Formula: see text]. This result partially addresses a question posed by Hensel and has practical implications, including providing a lower bound for the geodesic counting problem in the genus two handlebody group.

Domination sets and automorphism group of a graph associated to a finite vector space

Domination sets and automorphism group of a graph associated to a finite vector space

A Complex Structure for Two-Typed Tangent Spaces

This study concerns Riemannian manifolds with two types of tangent vectors. Let it be given that there are two subspaces of a tangent space with the property that each tangent vector has a unique decomposition as the sum of a vector in one subspace and a vector in the other subspace. Then, these tangent spaces can be complexified in such a way that the theory of the modular operator applies and that the complexified subspaces are invariant for the modular automorphism group. Notions coming from Kubo–Mori theory are introduced. In particular, the admittance function and the inner product of the Kubo–Mori theory can be generalized to the present context. The parallel transport operators are complexified as well. Suitable basis vectors are introduced. The real and imaginary contributions to the connection coefficients are identified. A version of the fluctuation–dissipation theorem links the admittance function to the path dependence of the eigenvalues and eigenvectors of the Hamiltonian generator of the modular automorphism group.

Hypersurfaces with large automorphism groups

We find sharp upper bounds on the order of the automorphism group of a hypersurface in complex projective space in every dimension and degree. In each case, we prove that the hypersurface realizing the upper bound is unique up to isomorphism and provide explicit generators for the automorphism group.

The lowest discriminant ideal of a Cayley–Hamilton Hopf algebra

Discriminant ideals of noncommutative algebras A A , which are module finite over a central subalgebra C C , are key invariants that carry important information about A A , such as the sum of the squares of the dimensions of its irreducible modules with a given central character. There has been substantial research on the computation of discriminants, but very little is known about the computation of discriminant ideals. In this paper we carry out a detailed investigation of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in the sense of De Concini, Reshetikhin, Rosso and Procesi, whose identity fiber algebras are basic. The lowest discriminant ideals are the most complicated ones, because they capture the most degenerate behaviour of the fibers in the exact opposite spectrum of the picture from the Azumaya locus. We provide a description of the zero sets of the lowest discriminant ideals of Cayley–Hamilton Hopf algebras in terms of maximally stable modules of Hopf algebras, irreducible modules that are stable under tensoring with the maximal possible number of irreducible modules with trivial central character. In important situations, this is shown to be governed by the actions of the winding automorphism groups. The results are illustrated with applications to the group algebras of central extensions of abelian groups, big quantum Borel subalgebras at roots of unity and quantum coordinate rings at roots of unity.

A determinant for automorphisms of groups

Let H and K be groups. In this paper we introduce a concept of determinant for endomorphisms of H × K and some concepts of incompatibility for group pairs as a measure of how far H and K are from being isomorphic. With the aid of the tools developed, we give a characterisation of invertible automorphisms of H × K by means of their determinants and an explicit description of Aut ( H × K ) as a group of 2-by-2 matrices, in case H or K belong to some relevant classes of groups. Many theoretical and practical applications of the determinants will be presented, together with examples and an analysis on some computational advantages of the determinants. In particular, we give a matrix characterisation of the whole automorphism group of some relevant classes of groups.

Quantum automorphism groups of trees

Quantum automorphism groups of trees

Efficient Learning of Long-Range and Equivariant Quantum Systems

In this work, we consider a fundamental task in quantum many-body physics – finding and learning ground states of quantum Hamiltonians and their properties. Recent works have studied the task of predicting the ground state expectation value of sums of geometrically local observables by learning from data. For short-range gapped Hamiltonians, a sample complexity that is logarithmic in the number of qubits and quasipolynomial in the error was obtained. Here we extend these results beyond the local requirements on both Hamiltonians and observables, motivated by the relevance of long-range interactions in molecular and atomic systems. For interactions decaying as a power law with exponent greater than twice the dimension of the system, we recover the same efficient logarithmic scaling with respect to the number of qubits, but the dependence on the error worsens to exponential. Further, we show that learning algorithms equivariant under the automorphism group of the interaction hypergraph achieve a sample complexity reduction, leading in particular to a constant number of samples for learning sums of local observables in systems with periodic boundary conditions. We demonstrate the efficient scaling in practice by learning from DMRG simulations of 1D long-range and disordered systems with up to 128 qubits. Finally, we provide an analysis of the concentration of expectation values of global observables stemming from the central limit theorem, resulting in increased prediction accuracy.

Edge Laplacians and Edge Poisson Transforms for Graphs

Abstract For a finite graph, we establish natural isomorphisms between eigenspaces of a Laplace operator acting on functions on the edges and eigenspaces of a transfer operator acting on functions on one-sided infinite non-backtracking paths. Interpreting the transfer operator as a classical dynamical system and the Laplace operator as its quantization, this result can be viewed as a quantum-classical correspondence. In contrast to previously established quantum-classical correspondences for the vertex Laplacian which exclude certain exceptional spectral parameters, our correspondence is valid for all parameters. This allows us to relate certain spectral quantities to topological properties of the graph such as the cyclomatic number and the 2-colorability. The quantum-classical correspondence for the edge Laplacian is induced by an edge Poisson transform on the universal covering of the graph which is a tree of bounded degree. In the special case of regular trees, we relate both the vertex and the edge Poisson transform to the representation theory of the automorphism group of the tree and study associated operator valued Hecke algebras.

Automorphisms of large-type free-of-infinity Artin groups

We compute explicitly the automorphism and outer automorphism group of all large-type free-of-infinity Artin groups. Our strategy involves reconstructing the associated Deligne complexes in a purely algebraic manner, i.e. in a way that is independent from the choice of standard generators for the groups.