Exchange Relation Planar Algebras

An inclusion of II ₁ factors NM of finite index has as an invariant, a double sequence of finite-dimensional algebras known as the standard invariant. Planar algebras were introduced by V. Jones as a geometric tool for computing standard invariants of existing subfactors as well as generating standard invariants for new subfactors. In this paper we define a class of planar algebras, termed exchange relation planar algebras, that provides a general framework for understanding several classes of known subfactor inclusions: the Fuss–Catalan algebras (i.e. those coming from the presence of intermediate subfactors) and all depth 2 subfactors. In addition, we present a new class of planar algebras (and thus a new class of subfactors) coming from automorphism subgroups of finite groups.     

Automorphisms and Derivations of Exceptional Simple Lie Algebras of Family <Emphasis Type="Italic">R</Emphasis>

In the paper we describe automorphisms and derivations of simple Lie algebras of family R. The Lie algebra of the automorphism group is found. Bibliography: 11 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 305, 2003, pp. 165–186.     

Rigidity of Branch Groups Acting on Rooted Trees

Automorphisms of groups acting faithfully on rooted trees are studied. We find conditions under which every automorphism of such a group is induced by a conjugation from the full automorphism group of the rooted tree. These results are applied to known examples such as Grigorchuk groups, Gupta–Sidki group, etc.     

Fixed points of automorphisms of Lie rings and locally finite groups

Let P be a locally finite group of prime exponent p. We prove that if P admits a finite soluble automorphism group G of order n coprime to p, such that the fixed point group C _P(G)is soluble of derived length d, then P is nilpotent of class bounded by a function of p, n, and d. A similar statement is shown to hold for Lie (p - 1)-Engel algebras; it is analogous to the Bergman-Isaacs theorem proved for associative rings, provided the condition of being soluble for an automorphism group is added. Our proof is based on a generalization of Kreknin's theorem concerning the solubility of Lie rings with a regular automorphism of finite order. This generalization, giving an affirmative answer to a question of Winter and extending one of his results to the case of infinitedimensional Lie algebras, is interesting in its own right. Moreover, we use a generalization of Higgins' theorem on the nilpotency of soluble Lie Engel algebras. Translated fromAlgebra i Logika, Vol. 34, No. 6, pp. 706-723, November-December, 1995.Supported by RFFR grant No. 94-01-00048-a and by ISF grant NQ7000.     

Similarity automorphism invariance of some P-properties of linear transformations on Euclidean Jordan algebras

Recently, Gowda and Sznajder [Gowda, M.S., Sznajder, R.: Automorphism invariance of P- and GUS-properties of linear transformations on Euclidean Jordan algebras. Math. Oper. Res. 31, 109–123 (2006)] have introduced and studied automorphism invariance of some P-properties for linear transformations. This paper deals with this automorphism invariance of some other complementarity properties, such as \(\hbox {E}_0,\,\hbox {P}_0\) , S, Z-properties. Particularly, we answer Gowda and Sznajder in positive that order P-property is algebra automorphism invariant in simple Jordan algebras. By replacing transposition with the invertibility in the concept of automorphism invariance, we propose a notion of similarity automorphism invariance. Most complementarity properties of linear transformations are also shown to be similarity invariant under algebra automorphisms and cone automorphisms.     

Z n -Orthograded Monocomposition Algebras

We study NQM algebras A having an orthogonal automorphism of finite order n 3 (called Z _n -orthograded NQM algebras). The Z ₃-orthograded NQM algebras of dimension 7 are treated in more detail. In particular, we find all algebras A which are not bi-isotropic in this class, and for every algebra A, determine an automorphism group Aut,A and an orthogonal automorphism group Ortaut,A. In constructing and classifying (up to isomorphism) NQM algebras, use is made of orthogonal decompositions of the algebras.     

Cohomomorphisms of separable algebras

Cohomomorphism algebras are investigated. The main result is the construction of a basis for the case of finite direct sums of full matrix algebras. The basis is used to study the simplest ring properties of cohomomorphism algebras.Translated fromAlgebra i Logika, Vol. 33, No. 4, pp. 415–421, July-August, 1994.     

Locally finite triangulated categories

A k-linear triangulated category is called locally finite provided for any indecomposable object Y in . It has Auslander–Reiten triangles. In this paper, we show that if a (connected) triangulated category has Auslander–Reiten triangles and contains loops, then its Auslander–Reiten quiver is of the form :

Full-size image (<1K)

By using this, we prove that the Auslander–Reiten quiver of any locally finite triangulated category is of the form , where Δ is a Dynkin diagram and G is an automorphism group of . For most automorphism groups G, the triangulated categories with as their Auslander–Reiten quivers are constructed. In particular, a triangulated category with as its Auslander–Reiten quiver is constructed.     

Okubo algebras in characteristic 3 and their automorphisms

Associated to any eight-dimensional non-unital composition algebra with associative norm, there are outer automorphisms of order 3 of the corresponding spin group, such tiat the fixed subgroup is the automorphism group of the composition algebra. Over fields of characteristic ≠ 3 these are simple algebraic groups of types G ₂ or A ₂, related respectively to the para-octonion and the Okubo algebras

A connection between the Okubo algebras over fields of characteristic 3 with some simple noncommutative Jordan algebras will be used to compute explicitly the automorphism groups and Lie algebras of derivations of these algebras. In contrast to the other characteristics, ths groups will no longer be of type A ₂ and will either be trivial or contain a large unipotent radical.     

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.

     

Bedingungen für die Zweispiegeligkeit der Automorphismengruppen von Cayleyalgebren

In this paper we derive necessary and sufficient conditions for the bireflectionality of the automorphism group of a Cayley algebra over a field of characteristic not 2. These are of particular interest for split Cayley algebras since their automorphism groups are the Chevalley groups of type G ₂. As an application we show the bireflectionality of the automorphism groups of Cayley algebras over real closed fields.     

Cluster automorphism groups of cluster algebras with coefficients

We study the cluster automorphism group of a skew-symmetric cluster algebra with geometric coefficients. We introduce the notion of gluing free cluster algebra, and show that under a weak condition the cluster automorphism group of a gluing free cluster algebra is a subgroup of the cluster automorphism group of its principal part cluster algebra (i.e., the corresponding cluster algebra without coefficients). We show that several classes of cluster algebras with coefficients are gluing free, for example, cluster algebras with principal coefficients, cluster algebras with universal geometric coefficients, and cluster algebras from surfaces (except a 4-gon) with coefficients from boundaries. Moreover, except four kinds of surfaces, the cluster automorphism group of a cluster algebra from a surface with coefficients from boundaries is isomorphic to the cluster automorphism group of its principal part cluster algebra; for a cluster algebra with principal coefficients, its cluster automorphism group is isomorphic to the automorphism group of its initial quiver.     

A new proof of the equivalence of injectivity and hyperfiniteness for factors on a separable Hilbert space

In 1975 A. Connes proved the fundamental result that injective factors on a separable Hilbert space are hyperfinite. In this paper a new proof of this result is presented in which the most technical parts of Connes proof are avoided. Particularly the proof does not rely on automorphism group theory. The starting point in this approach is Wassermann's simple proof of injective ? semidiscrete together with Choi and Effros' characterization of semidiscrete von Neumann algebras as those von Neumann algebras N for which the identity map on N has an approximate completely positive factorization through n × n-matrices.     

Dendriform equations

We investigate solutions for a particular class of linear equations in dendriform algebras. Motivations as well as several applications are provided. The latter follow naturally from the intimate link between dendriform algebras and Rota–Baxter operators, e.g. the Riemann integral map or Jackson's q-integral.     

Tropical matrix groups

We study the structure of groups of finitary tropical matrices under multiplication. We show that the maximal groups of \(n \times n\) tropical matrices are precisely the groups of the form \(G \times \mathbb {R}\) where G is a group admitting a 2-closed permutation representation on n points. Each such maximal group is also naturally isomorphic to the full linear automorphism group of a related tropical polytope. Our results have numerous corollaries, including the fact that every automorphism of a projective (as a module) tropical polytope of full rank extends to an automorphism of the containing space.     

Structure of weak automorphism groups of mono-unary algebras

A weak automorphism of an algebra on A with the set T of term operations is a permutation of of A such that . In this note we describe the groups of weak automorphisms of mono-unary algebras. Some examples of such algebras without weak automorphisms are given. Received January 31, 1996; accepted in final form March 16, 1999.     

Mendelsohn designs admitting the affine group

All Mendelsohn designs containing a Frobenius group with cyclic complement of orderv – 1 as a subgroup of the automorphism are found. Furthermore, the automorphism group of each of the designs is constructed. These designs generalize Mendelsohn's construction of Mendelsohn designs containing a certain doubly transitive automorphism group.The research on this paper was partially supported by North Texas State Faculty Research Grant #35524.     

Finite gradings of special linear Lie algebras

In this paper, we study gradings of simple classical Lie algebras with arbitrary Abelian groups and the interconnection of such gradings and automorphism groups of Lie algebras. We give a complete classification of gradings of special linear Lie algebras that are specified by inner automorphisms in the case of an algebraically closed field of zero characteristic.