Hopf Algebras on Schurian Quivers

Let H be a Hopf algebra with a finite-dimensional, nontrivial space of skew primitive elements, over an algebraically closed field of characteristic zero. We prove that H contains either the polynomial algebra as a Hopf subalgebra, or a certain Schurian simple-pointed Hopf subalgebra. As a consequence, a complete list of the locally finite, simple-pointed Hopf algebras is obtained. Also, the graded automorphism group of a Hopf algebra on a Schurian Hopf quiver is determined, and the relation between this group and the automorphism groups of the corresponding Hopf quiver, is clarified.     

Local automorphisms of operator algebras on Banach spaces

In this paper we extend a result of Semrl stating that every 2-local automorphism of the full operator algebra on a separable infinite dimensional Hilbert space is an automorphism. In fact, besides separable Hilbert spaces, we obtain the same conclusion for the much larger class of Banach spaces with Schauder bases. The proof rests on an analogous statement concerning the 2-local automorphisms of matrix algebras for which we present a short proof. The need to get such a proof was formulated in Semrl's paper.

     

Coleman Automorphisms of Permutational Wreath Products

Let N be a finite nontrivial nilpotent group and H a finite centerless permutation group on a finite set Ω (i.e., H acts faithfully on Ω). Let G = N?H = N^|Ω| ? H be the corresponding permutational wreath product of N by H. It is shown that every Coleman automorphism of G is an inner automorphism. This generalizes a well-known result due to Petit Lobão and Sehgal stating that the normalizer property holds for complete monomial groups with nilpotent base groups.     

Functional analytic aspects of non-commutative harmonic analysis

Given a homogeneous space X = G/H with an invariant measure it is shown, using Grothendieck's inequality, that a G-invariant Hilbert subspace of the space of distributions of order zero on X is actually contained in L_loc²(X). Moreover, if θ is an automorphism on G appropriately related to H, it is shown that, under condition that H-orbits are smooth, an H-bi-invariant distribution of positive type on G satisfies the identity Ť^θ = T if the corresponding Hilbert space is contained in L_loc²(X). This shows that, under the smooth orbit condition, G-invariant Hilbert subspaces of L_loc² (X) have a unique decomposition into irreducible Hilbert spaces as in the case of generalized Gelfand pairs.     

Sporadic groups and automorphisms of linear spaces

This article is a contribution to the study of the automorphism groups of finite linear spaces. In particular we look at almost simple groups and prove the following theorem: Let G be an almost simple group and let 𝒮 be a finite linear space on which G acts as a line‐transitive automorphism group. Then the socle of G is not a sporadic group. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 353–362, 2000     

Automorphisms of Constant Weight Codes and of Divisible Designs

We show that the automorphism group of a divisible design is isomorphic to a subgroup H of index 1 or 2 in the automorphism group of the associated constant weight code. Only in very special cases H is not the full automorphism group.     

Coleman automorphisms of holomorphs of finite abelian groups

Let K be a finite abelian group and let H be the holomorph of K. It is shown that every Coleman automorphism of H is an inner automorphism. As an immediate consequence of this result, it is obtained that the normalizer property holds for H.     

Lip Automorphism Germ and Lip Automorphism

Let X be a metric space, ε^n(X) be the standard trivial Lip n-bundle over X, and Φ be a Lip automorphism germ of ε^n(X). This paper proves that there is a Lip automorphism Φ‘ of ε^n(X) such that the germ of Φ‘ is Φ.     

Local automorphisms of standard operator algebras

Let X be an infinite-dimensional separable real or complex Banach space and A a closed standard operator algebra on X. Then every local automorphism of A is an automorphism. The assumptions of infinite-dimensionality, separability, and closeness are all indispensable.     

Weakly power automorphisms of groups

An automorphism α of a group G is called a weakly power automorphism if it maps every non-periodic subgroup of G onto itself. The aim of this paper is to investigate the behavior of weakly power automorphisms. In particular, among other results, it is proved that all weakly power automorphisms of a soluble non-periodic group G of derived length at most 3 are power automorphisms, i.e. they fix all subgroups of G. This result is best possible, as there exists a soluble non-periodic group of derived length 4 admitting a weakly power automorphism, which is not a power automorphism.     

Decomposition of Jordan Automorphisms of Strictly Upper Triangular Matrix Algebra Over Commutative Rings

Over a 2-torsionfree commutative ring R with identity, the algebra of all strictly upper triangular n + 1 by n + 1 matrices is denoted by n ₁. In this article, we prove that any Jordan automorphism of n ₁ can be uniquely decomposed as a product of a graph automorphism, a diagonal automorphism, a central automorphism and an inner automorphism for n ≥ 3. In the cases n = 1, 2, we also give a decomposition for any Jordan automorphism of n ₁.     

Eigenvalues and automorphisms of a graph

Let G be the automorphism group of a graph Γ and let λ be an eigenvalue of the adjacency matrix of Γ. In this article, (i) we derive an upper bound for rank(G), (ii) if G is vertex transitive, we derive an upper bound for the extension degree of ?(λ) over ?, (iii) we study automorphism groups of graphs without multiple eigenvalues, (iv) we study spectra of quotient graphs associated with orbit partitions.     

Linear Maps Preserving Generalized Invertibility

Let H be an infinite-dimensional complex separable Hilbert space and the algebra of all bounded linear operators on H. Let be a bijective continuous unital linear map preserving generalized invertibility in both directions. Then the ideal of all compact operators is invariant under ϕ and the induced linear map on the Calkin algebra is either an automorphism or an antiautomorphism.     

On noninner automorphisms of 2-generator finite p-groups

A longstanding conjecture asserts that every finite nonabelian p-group admits a noninner automorphism of order p. In this paper we give some necessary conditions for a possible counterexample G to this conjecture, in the case when G is a 2-generator finite p-group. Then we show that every 2-generator finite p-group with abelian Frattini subgroup has a noninner automorphism of order p.     

Standardness and Standard Automorphisms of Chevalley Groups, I: the Case of Rank at Least Two

The author studies the linkage between the standardness and the standard automorphisms of Chevalley groups over rings.It is proved that if H is any standard subgroup of G(R),then each of its automorphisms can be extended to an automorphism of G(R,I),restricted to an automorphism of E(R,I),and an automorphism of E(R,I) can be extended to one of G(R,I).The case of Chevalley groups of rank at least two is treated in this paper.Further results about the case of Chevalley groups of rank one,the case of non-commutative ground ring and some others exceptions will appear elsewhere.     

Hilbert compacts,compact ellipsoids,and compact extrema

We consider a system of so-called Hilbert compacts K(H) in a Hilbert space H; those Hilbert compacts admit a two-sided estimate by compact ellipsoids in H. For functionals in H, we introduce the notion of a compact extremum achieved at a certain base with respect to the imbedding in K(H). An example of the K-extremum of a variational functional in the Sobolev space W ₂¹ is considered.     

Divisible Designs Admitting GL(3, q) as an Automorphism Group

Starting from a 3-dimensional projective space we construct divisible designs admitting GL(3,q) as an automorphism group.     

Embeddings into Orthomodular Lattices with Given Centers,State Spaces and Automorphism Groups

We prove that, given a nontrivial Boolean algebra B, a compact convex set S and a group G, there is an orthomodular lattice L with the center isomorphic to B, the automorphism group isomorphic to G, and the state space affinely homeomorphic to S. Moreover, given an orthomodular lattice J admitting at least one state, L can be chosen such that J is its subalgebra.