Twisted conjugacy classes of the unit element

We study twisted conjugacy classes of the unit element in different groups. Fel’shtyn and Troitsky showed that the twisted conjugacy class of the unit element of an abelian group is a subgroup for every automorphism. The structure is investigated of a group whose twisted conjugacy class of the unit element is a subgroup for every automorphism (inner automorphism).     

Test elements for monomorphisms of free lie algebras and lie superalgebras †

A test criterion for an endomorphism φ of the free Lie (super) algebra L of finite rank to be an automorphism is obtained: φ is an automorphism of L if and only if for an element u ∈ L with the maximal rank the element φ(u) belongs to the orbit of u with respect to the automorphism group of L. In particular, test elements for monomorphisms of L are exactly the elements of the maximal rank     

Lambda-rings of automorphisms

Under composition, the set of automorphisms of a single algebra forms a group. The topic of this paper is the full set of all automorphisms of all algebra structures in a given abstract class on a fixed finite set. A lambda-ring, known as the automorphism type ring, is associated with this full set of automorphisms, and its structure is described. For certain classes (such as the variety of lattices), the full set of automorphisms is represented up to conjugation in the symmetric group by the automorphisms of a single, so-called representative algebra. In this case, the automorphism type ring is a subring of the ring of class functions on the automorphism group of the representative algebra.     

Proper holomorphic self-maps of symmetric powers of balls

We show that each proper holomorphic self map of a symmetric power of the unit ball is an automorphism naturally induced by an automorphism of the unit ball, provided the ball is of dimension at least two.     

Generalized Whittaker vectors for holomorphic and quaternionic representations

The conjugacy class of parabolic subgroups with Heisenberg unipotent radical in a simple Lie groups over ³ not of type C_nC_{n} contains an element defined over  for each quaternionic real form. In this paper we study the Whittaker models for quaternionic discrete series of these real forms and prove results analogous and by analogous methods to the case of simple Lie groups over  that are the automorphism groups of tube type Hermitian symmetric domain and (so-called Bessel models) for holomorphic representations. In particular we calculate the decomposition of the space of Whittaker vectors under the action of the stabilizer of the corresponding character in a Levi factor of the Heisenberg parabolic subgroup.     

Homoclinic processes and invariant measures for hyperbolic toral automorphisms

For every hyperbolic toral automorphism T, the present author has defined in his previous paper some unbounded T-invariant second-order difference operators related to the so-called homoclinic group of T. These operators were considered in the space L₂ with respect to the Haar measure. It is shown in the present paper that such operators give rise to transition semigroups in the space of continuous functions on the torus and generate dynamically invariant Markov processes. This leads almost immediately to a family of invariant measures for the automorphism T.Along with a short discussion, some open questions about properties of these measures are posed. Bibliography: 9 titles.     

On the upper-semicontinuity of automorphism groups of model domains in almost complex manifolds

We will prove that the automorphism groups of the strongly pseudoconvex model domains in almost complex manifolds are isomorphically embedded into the automorphism group of the unit ball.     

Word length in surface groups with characteristic generating sets

A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite union of orbits of the automorphism group, and show that the translation length of any element with a nonzero crossing number is positive, and bounded below by a constant depending only (and explicitly) on a bound on the crossing numbers of generating elements. This answers a question of Benson Farb.

     

Fixed Points of Zircon Automorphisms

A zircon is a poset in which every principal order ideal is finite and equipped with a so-called special matching. We prove that the subposet induced by the fixed points of any automorphism of a zircon is itself a zircon. This provides a natural context in which to view recent results on Bruhat orders on twisted involutions in Coxeter groups.     

Generalized projection operators in Geometric Algebra

Given an automorphism and an anti-automorphism of a semigroup of a Geometric Algebra, then for each element of the semigroup a (generalized) projection operator exists that is defined on the entire Geometric Algebra. A single fundamental theorem holds for all (generalized) projection operators. This theorem makes previous projection operator formulas [2] equivalent to each other. The class of generalized projection operators includes the familiar subspace projection operation by implementing the automorphism ‘grade involution’ and the anti-automorphism ‘inverse’ on the semigroup of invertible versors. This class of projection operators is studied in some detail as the natural generalization of the subspace projection operators. Other generalized projection operators include projections ontoany invertible element, or a weighted projection ontoany element. This last projection operator even implies a possible projection operator for the zero element.     

Explicit formula of holomorphic automorphism group on complex homogeneous bounded domains

We known that the maximal connected holomorphic automorphism group Aut (D)(0) is a semi-direct product of the triangle group T(D) and the maximal connected isotropic subgroup Iso(D)(0) of a fixed point in the complex homogeneous bounded domain D and any complex homogeneous bounded domain is holomorphic isomorphic to a normal Siegel domain D(VN,F). In this paper, we give the explicit formula of any holomorphic automorphism in T(D(VN, F)) and Iso(D(VN,F))(0), where G(0) is the unit connected component of the Lie group G.     

Hall polynomials and the fixed space of an automorphism of a finite abelian p-group

Recent work on Hall polynomials is used to study the fixed space of a random automorphism of a finite abelian p-group. An expression is found for the chance that an automorphism of an abelian p-group of type l\lambda fixes only the identity. A formula is obtained for the chance that a given subgroup H of type n\nu is the fixed space of an automorphism of an abelian p-group of type (k^l). There results generalize work of Rudvalis and Shinoda on the fixed space of an element of GL (n, q).     

Null-quadrics of codimension 4 in ℂ7

In this paper we study the automorphisms of nondegenerate quadrics of type (4, 3). In particular, we classify the so-called null-quadrics (we prove that there are exactly two null-quadrics up to equivalence), and find their automorphism groups. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 657–665, November, 1997. Translated by S. S. Anisov     

Real-local fields and the canonical operation of the automorphism group on the space of orderings

The object of our investigation is the canonical operation of the automorphism group of a formally real field F on X_F , the space of orderings of F. For a naturally distinguished class of formally real fields, the so-called real-local fields, the Baer-Krull-bijection induces on X_F the structure of a module over the endomorphism ring of the group of archimedean classes of F. We show that Aut F acts on X_F by affinities with respect to that module structure. Subsequently, this “arithmetization” of the operation is exemplarily applied to the question of transitivity (“When can any two orderings of F be transformed into each other by some automorphism of F?"), and to the investigation of the subgroup of Aut F generated by all order automorphism groups of F.     

What is the probability an automorphism fixes a group element?

Extending the notion of probability to the automorphisms of a group, we find the probability of an arbitrarily chosen automorphism of a group fixing an arbitrary element of the group.     

On the Automorphism Group of the Centralizer of an Idempotent in the Full Transformation Monoid

Let $e$ be an idempotent in the monoid $T(X)$ of all functions from a set $X$ into itself. Let $C(e)$ be the centralizer of $e$ in $T(X)$. It has recently been shown that the unit and automorphism groups of $C(e)$ are canonically isomorphic. Our goal is to furnish an alternative proof of this fact and make the observation that automorphism group of $C(e)$ is isomorphic to the direct product ${\Pi}_{i\in I}( \Sym(A_i)\wr \Sym(B_i))$ of wreath products of symmetric groups, where the sets $I$, $A_i$, $B_i$ are defined in terms of $e$.     

Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras

We extend the Larson–Sweedler theorem [Amer. J. Math. 91 (1969) 75] to weak Hopf algebras by proving that a finite dimensional weak bialgebra is a weak Hopf algebra iff it possesses a non-degenerate left integral. We show that the category of modules over a weak Hopf algebra is autonomous monoidal with semisimple unit and invertible modules. We also reveal the connection of invertible modules to left and right grouplike elements in the dual weak Hopf algebra. Defining distinguished left and right grouplike elements, we derive the Radford formula [Amer. J. Math. 98 (1976) 333] for the fourth power of the antipode in a weak Hopf algebra and prove that the order of the antipode is finite up to an inner automorphism by a grouplike element in the trivial subalgebra A^T of the underlying weak Hopf algebra A.     

Automorphic orbit problem for polynomial algebras

It is proved that every endomorphism preserving the automorphic orbit of a non-trivial element of the rank two polynomial algebra over the complex number field is an automorphism.     

Descent from the form ring and buchsbaum rings

An element it of a finitely generated free Lie algebra L is a test element if any endomorphism of L fixing u is an automorphism. We prove that test elements of L are precisely those elements not contained in any proper retract of L. In addition we characterize retracts of free Lie algebras.     

Automorphism groups for measurable spaces

In [1], the notion of a rigid separable measurable space was defined, and such spaces were shown to exist. We expand upon this idea and ask what are the possible automorphism groups for such spaces. We show that there is only one such non-trivial group which is Abelian (a countable product of two element groups), and that this group is realised as the automorphism group of some separable space. A particular class of such spaces is characterised in terms of rigid components.Finally, an example of a measurable space which is rigid in the strict sense is constructed. This answers a question of K.P.S. Bhaskara Rao and B.V. Rao.