LOCAL AUTOMORPHISMS OF SEMISIMPLE ALGEBRAS AND GROUP ALGEBRAS

Let F be a field of characteristic not 2, and let A be a finite-dimensional semisimple F -algebra. All local automorphisms of A are characterized when all the degrees of A are larger than 1. If F is further assumed to be an algebraically closed field of characteristic zero, K a finite group, F K the group algebra of K over F , then all local automorphisms of F K are also characterized.

Boolean Automorphisms of a Hypercube Coincide with the Linear Isometries

Boolean homomorphisms of a hypercube, which correspond to the morphisms in the category of finite Boolean algebras, coincide with the linear isometries of the category of finite binary metric vector spaces.

MULTIPLICATIVE GROUP AUTOMORPHISMS OF INVERTIBLE UPPER TRIANGULAR MATRICES OVER FIELDS

Suppose F is a field of characteristic not 2 and F* its multiplicative group. Let T*n(F) be the multiplicative group of invertible upper triangular n x n matrices over F and STn(F) its subgroup {(aij) E T*n(F)aii = 1, i}. This paper proves that f: T*n(F) → T*n(F) is a group automorphism if and only if there exist a matrix Q in T*n(F) and a field automorphism rs of F such that either where A = ((aij)), A-T is the transpose inverse of A, J = Ei n+1-i, and : i= 1T*n(F) → F* is a homomorphism which satisfies {(xIn)(x)x F*} = F* and {x F*(xIn)(x) = 1} = {1}. Simultaneously, they also determine the automorphisms of STn(F).

THE BANACH-LIE GROUP OF LIE TRIPLE AUTOMORPHISMS OF AN H^＊-ALGEBRA^＊

We study the Banach-Lie group Ltaut（A） of Lie triple automorphisms of a complex associative H^＊-algebra A. Some consequences about its Lie algebra, the algebra of Lie triple derivations of A, Ltder（A）, are obtained. For a topologically simple A, in the infinite-dimensional case we have Ltaut（A）0 = Aut（A） implying Ltder（A） = Der（A）. In the finite-dimensional case Ltaut（A）0 is a direct product of Aut（A） and a certain subgroup of Lie derivations δ from A to its center, annihilating commutators.

Invariants for Automorphisms of the Underlying Algebras Relative to Lie Algebras of Cartan Type

Let X denote a finite or infinite dimensional Lie algebra of Cartan type W, S, H or K over a field of characteristic p 〉 3. In this paper it is proved that certain filtrations of the underlying algebras are invariant under the admissible groups relative to Lie algebras of Cartan type X.

Notes on Automorphisms of Prime Rings

Let R be a prime ring, L a noncentral Lie ideal and a nontrivialautomorphism of R such that us（u）ut = 0 for all u 2 L, where s; t are fixednon-negative integers. If either charR 〉 s ＋ t or charR = 0, then R satisfies s4, thestandard identity in four variables. We also examine the identity （σ（[x; y]）-[x; y]）n =0 for all x; y ∈ I, where I is a nonzero ideal of R and n is a fixed positive integer. Ifeither charR 〉 n or charR = 0, then R is commutative.

On the Structure of Infinitesimal Automorphisms of the Poisson-Lie Group <i>SU</i>(2,R)

We study the Poisson-Lie structures on the group SU(2,R). We calculate all Poisson-Lie structures on SU(2,R) through the correspondence with Lie bialgebra structures on its Lie algebra su(2,R). We show that all these structures are linearizable in the neighborhood of the unity of the group SU(2,R). Finally, we show that the Lie algebra consisting of all infinitesimal automorphisms is strictly contained in the Lie algebra consisting of Hamiltonian vector fields.

(Right)Partial Generalized Automorphisms of Semiprime Rings

In this paper, we define the concept of (right) partial generalized automorphisms and discuss the extension problem, and also give a characterization of (right) partial generalized automorphisms of semiprime rings. Finally, we study the centralizing problem of right partial generalized automorphisms.

Centralizing Automorphisms of Lie Ideals in Prime Rings of Characteristic 2

Let R be a prime ring with an automorphism σ≠1, an identity map. Let L be a noncentral Lie ideal of R such that \xσ, x] ∈Z for all x ∈ L, where Z is the center of R. Then L is contained in the center of R, unless char(R) = 2 and dimcRC = 4.

On symplectic automorphisms of elliptic surfaces acting on CH_(0)

Let S be a complex smooth projective surface of Kodaira dimension one. We show that the group Auts(S) of symplectic automorphisms acts trivially on the Albanese kernel CH_(0)(S)albof the 0-th Chow group CH_(0)(S), unless possibly if the geometric genus and the irregularity satisfy pg(S) = q(S) ∈ {1, 2}. In the exceptional cases, the image of the homomorphism Auts(S) → Aut(CH_(0)(S)alb) has the order at most 3. Our arguments actually take care of the group Autf(S) of fibration-preserving automorphisms of elliptic surfaces f : S → B. We prove that if σ ∈ Autf(S) induces the trivial action on Hi,0(S) for i > 0, then it induces the trivial action on CH_(0)(S)alb. As a by-product we obtain that if S is an elliptic K3 surface, then Autf(S)∩Auts(S)acts trivially on CH_(0)(S)alb.

Local Jordan Derivations and Local Jordan Automorphisms of Upper Triangular Matrix Algebras

Let R be a commutative ring with identity, Tn （R） the R-algebra of all upper triangular n by n matrices over R. In this paper, it is proved that every local Jordan derivation of Tn （R） is an inner derivation and that every local Jordan automorphism of Tn（R） is a Jordan automorphism. As applications, we show that local derivations and local automorphisms of Tn （R） are inner.

Orthogonal graphs of characteristic 2 and their automorphisms

The (singular) orthogonal graph O(2ν + δ,q) over a field with q elements and of characteristic 2 (where ν 1, and δ = 0,1 or 2) is introduced. When ν = 1, O(2 · 1,q), O(2 · 1 + 1,q) and O(2 · 1 + 2,q) are complete graphs with 2, q + 1 and q2 + 1 vertices, respectively. When ν 2, O(2ν + δ,q) is strongly regular and its parameters are computed. O(2ν + 1,q) is isomorphic to the symplectic graph Sp(2ν,q). The chromatic number of O(2ν + δ,q) except when δ = 0 and ν is odd is computed and the group of graph automorphisms of O(2ν + δ,q) is determined.

Semilinear automorphisms of classical groups and quivers

For a classical group G over a field F together with a finite-order automorphism θ that acts compatibly on F, we describe the fixed point subgroup of θ on G and the eigenspaces of θ on the Lie algebra g in terms of cyclic quivers with involution. More precise classification is given when g is a loop Lie algebra, i.e.,when F = C((t)).

A Set of Almost Automorphic Functions and Applications

For a set S of real numbers, we introduce the concept of S-almost automorphic functions valued in a Banach space. It generalizes in particular the space of Z-almost automorphic functions. Considering the space of S-almost automorphic functions, we give sufficient conditions of the existence and uniqueness of almost automorphic solutions of a differential equation with a piecewise constant argument of generalized type. This is done using the Banach fixed point theorem.

Algebra Automorphisms of Quantized Enveloping Algebras Uq(■)

The algebra automorphisms of the quantized enveloping algebra Uq(g) are discussed, where q is generic. To some extent, all quantum deformations of automorphisms of the simple Lie algebra g have been determined.

An algebraic invariant for Jordan automorphisms on B(H):The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H)the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that φ is a surjective map from B(H) onto itself. If for everyλ∈ {-1, 1, 2, 3, 1/2, 1/3} and A, B ∈ B(H), A - λB ∈ I(H) (→)φ(A) - λφ(B) ∈ I(H), then φis a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that φ(A) = TAT-1 for all A ∈ B(H), or φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A - iB ∈ I(H) (→)φ(A) - iφ(B) ∈ I(H), here i is the imaginary unit, then φ is either an automorphism or an anti-automorphism.

Automorphisms for Some Symmetric Multiparameter Quantized Weyl Algebras and Their Localizations（Dedicated to Professor Yingbo Zhang on her 70th birthday）

We study a family of ＂symmetric＂ multiparameter quantized Weyl alge- bras A-q,A n（K） and some related algebras. We compute the Nakayama automorphism of A-q,A n（K）, give a necessary and sufficient condition for A-q,A n（K） to be Calabi-Yau, and prove that A-q,A n（K） is cancellative. We study the automorphisms and isomorphism problem for A-q,A n（K） and .A-q,A n（K[t]）. Similar results are established for the Maltsiniotis multiparam- eter quantized Weyl algebraA-q,A n（K） and its polynomial extension. We prove a quantum analogue of the Dixmier conjecture for a simple localization （A-q,A n（K））z and determine its automorphism group.

Automorphisms of certain Borel subgroups over a connnected ring

Let G be the Chevalley-Demazure group scheme determined by a finite-dimensional complex simple Lie algebra L and its adjoint representation, R a commutative ring with identity and R~* the multiplicative group consisting of all units in R. Denote by G(R) the Chevalley group over R with respect to G. Let △ be a root system of L, △^+ the set of all positive roots with respect to some simple root system of △. Let E(R) be the el-

Finite-order inner automorphisms of the first kind on affine Lie algebras

SOME authors classified order two and order three automorphisms of affine Lie algebras. In this letter,

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.