Assuming the continuum hypothesis, we show that the Calkin algebra has 2 ℵ 1 outer automorphisms. Let H be a separable infinite dimensional Hilbert space, let L(H) be the algebra of bounded operators on H, let K(H) be the algebra of compact operators on H, and let Q = L(H)/K(H) be the Calkin algebra. A long-standing problem asks whether every automorphism of Q is inner, that is, of the form x →...

The purpose of this note is to define a graph whose vertex set finite group G $G$ , edge contained in that the commuting and contains enhanced power . We call deep Two elements are joined if only their inverse images every central extension commute. give conditions for be equal either graph, show automorphisms act as graph.

The problem of classifying the automorphisms of a free group of infinite countable rank is investigated. Quite a reasonable generating set for the group AutF∞ is described. Some new subgroups of this group and structural results for them are presented. The main result says that the group of all automorphisms is generated (modulo the IA-automorphisms) by strings and lower triangular automorphisms.

In 2], 3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian Chern-Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of at connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the correspondi...

In 2], 3] we suggested a new quantum algebra, the moduli algebra, which is conjectured to be a quantum algebra of observables of the Hamiltonian Chern-Simons theory. This algebra provides the quantization of the algebra of functions on the moduli space of at connections on a 2-dimensional surface. In this paper we classify unitary representations of this new algebra and identify the correspondi...

This research was motivated by universal algebraic geometry. One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? For answer of this question (see [8],[10]) we must consider the variety Θ, to which our algebras belongs, the category Θ of all finitely generated free algebras of Θ and research how the group AutΘ of all the automorphi...

We present an explicit construction for the central extension of the group Map(X, G) where X is a compact manifold and G is a Lie group. If X is a complex curve we obtain a simple construction of the extension by the Picard variety Pic(X). The construction is easily adapted to the extension of Aut(E), the gauge group of automorphisms of a nontrivial vector bundle E.

In this paper we study a key exchange protocol similar to DiffieHellman key exchange protocol using abelian subgroups of the automorphism group of a non-abelian nilpotent group. We also generalize group no.92 of HallSenior table [15], for arbitrary prime p and show that for those groups, the group of central automorphisms commute. We use these for the key exchange we are studying. MSC: 94A62, 2...

If W is an infinite rank 3 Coxeter group, whose Coxeter diagram has no infinite bonds, then the automorphism group of W is generated by the inner automorphisms and any automorphisms induced from automorphisms of the Coxeter diagram. Indeed Aut(W ) is the semi-direct product of Inn(W ) and the group of graph automorphisms.