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.

Let K be a field and P partially ordered set (poset). FI(P,K) I(P,K) the finitary incidence algebra space of over K, respectively, let D(P,K)=FI(P,K)⊕I(P,K) idealization FI(P,K)-bimodule I(P,K). In first part this paper, we show that D(P,K) has an anti-automorphism (involution) if only (involution). We also present characterization anti-automorphisms involutions on D(P,K). second part, obtain c...

Let A1 = K〈X,Y | [Y,X] = 1〉 be the (first) Weyl algebra over a field K of characteristic zero. It is known that the set of eigenvalues of the inner derivation ad(Y X) of A1 is Z. Let A1 → A1, X 7→ x, Y 7→ y, be a K-algebra homomorphism, i.e. [y, x] = 1. It is proved that the set of eigenvalues of the inner derivation ad(yx) of theWeyl algebra A1 is Z and the eigenvector algebra of ad(yx) is K〈x...

In this paper we study θ-cyclic codes over the ring R = F2 + vF2 = {0, 1, v, v+1} where v = v. This is the only ring of order four that is not a field and has a non-trivial ring automorphism. We describe generator polynomials of θ-cyclic codes defined over this ring. We also describe the generator polynomials of the duals of free θ-cyclic codes with respect to Euclidean and Hermitian inner prod...

theory of schemes led to progress on classical and concrete problems about curves and surfaces. 3. A theorem of Farah The following theorem appears in a paper by by Ilijas Farah (“All automorphisms of the Calkin algebra are inner”, Annals of Mathematics 173 (2011) 619–661) If Todorcevic’s axiom (TA) holds then all automorphisms of the Calkin algebra are inner. Since the focus of this lecture se...

An automorphism σ of a simple finite dimensional complex Lie algebra g is called torsion, if σ has finite order in the group Aut(g) of all automorphisms of g. The torsion automorphisms of g were classified by Victor Kac in [12], as an application of his results on infinite dimensional Lie algebras. Those torsion automorphisms contained in the identity component G = Aut(g)◦ are called inner; the...

Let S be either a sphere with 5 punctures or a torus with 3 punctures. We prove that the automorphism group of the complex of curves of S is isomorphic to the extended mapping class group M S. As applications we prove that surfaces of genus 1 are determined by their complexes of curves, and any isomorphism between two subgroups of M S of nite index is the restriction of an inner automorphism of...

Let $F_m$ be the free metabelian Lie algebra of rank $m$ over a field $K$ characteristic 0. An automorphism $\varphi$ is called central if commutes with every inner $F_m$. Such automorphisms form centralizer $\text{\rm C}(\text{\rm Inn}(F_m))$ group Inn}(F_m)$ in Aut}(F_m)$. We provide an elementary proof to show that Inn}(F_m))=\text{\rm Inn}(F_m)$.