GENERALIZED DERIVATIONS ON PARABOLIC SUBALGEBRAS OF GENERAL LINEAR LIE ALGEBRAS

Let P be a parabolic subalgebra of a general linear Lie algebra gl(n, F) over a field F, where n ≥ 3, F contains at least n different elements, and char(F) = 2. In this article, we prove that generalized derivations, quasiderivations, and product zero derivations of P coincide, and any generalized derivation of P is a sum of an inner derivation, a central quasiderivation, and a scalar multiplication map of P. We also show that any commuting automorphism of P is a central automorphism, and any commuting derivation of P is a central derivation.     

Approximate （m,n）-Cauchy-Jensen Additive Mappings in C^＊-algebras

Concerning the stability problem of functional equations, we introduce a general (m, n)-Cauchy-Jensen functional equation and establish new theorems about the Hyers-Ulam stability of general (m, n)-Cauchy-Jensen additive mappings in C*-algebras, which generalize the results obtained for Cauchy-Jensen type additive mappings.     

全矩阵代数的抛物子代数上具有可导性的非线性映射

Let F be a field of characteristic 0, Mn（F） the full matrix algebra over F, t the subalgebra of Mn（F） consisting of all upper triangular matrices. Any subalgebra of Mn（F） containing t is called a parabolic subalgebra of Mn（F）. Let P be a parabolic subalgebra of Mn（F）. A map φ on P is said to satisfy derivability if φ（x·y） = φ（x）·y＋x·φ（y） for all x,y ∈ P, where φ is not necessarily linear. Note that a map satisfying derivability on P is not necessarily a derivation on P. In this paper, we prove that a map φ on P satisfies derivability if and only if φ is a sum of an inner derivation and an additive quasi-derivation on P. In particular, any derivation of parabolic subalgebras of Mn（F） is an inner derivation.     

A chain rule formula for higher derivations and inverses of polynomial maps

The multidimensional chain rule formula for analytic functions and its generalization to higher derivatives perfectly work in the algebraic setting in characteristic zero. In positive characteristic one runs into problems due to denominators in these formulas. In this article we show a direct analog of these formulas using higher derivations which are defined in any characteristic. We also use these formulas to show how higher derivations to different coordinate systems are related to each other. Finally, we apply this to polynomial automorphisms in arbitrary characteristic and obtain a formula for the inverse of such a polynomial automorphism.     

Representations of Bihom-Lie superalgebras

The purpose of this article is to study cohomology and representations of Bihom-Lie superalgebras. In particular, trivial representations and adjoint representations of Bihom-Lie superalgebras are studied in detail. Derivations and central extensions of Bihom-Lie superalgebras are also discussed as an application.     

A differential identity for Green functions

If P is a differential operator with constant coefficients, an identity is derived to calculate the action of e ^P on the product of two functions. In many-body theory, P describes the interaction Hamiltonian and the identity yields a hierarchy of Green functions. The identity is first derived for scalar fields and the standard hierarchy is recovered. Then the case of fermions is considered and the identity is used to calculate the generating function for the Green functions of an electron system in a time-dependent external potential.     

A linear counterexample to the Fourteenth Problem of Hilbert in dimension eleven

A family of -actions on affine space is constructed, each having a non-finitely generated ring of invariants (). Because these actions are of small degree, they induce linear actions of unipotent groups on for , and these invariant rings are also non-finitely generated. The smallest such action presented here is for the group acting linearly on .

     

Sets of Commuting Derivations and Simplicity

A ring R is simple under a set D of derivations if no nontrivial ideal of R is preserved by all derivations in D. Continuing previous joint work with C. J. Maxson, the author provides a computational test for the simplicity of k[x ₁,…,x _n ]/〈 x ₁ ^p ,…, x _n ^p 〉 (k a field of characteristic p > 0) under a set of commuting k-derivations. Specific rings are then examined for sets of commuting derivations, especially those under which the ring is simple. The possible sizes and minimality of such sets are also determined in particular cases.     

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r ^k ), r ^k ] _n  = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c ? α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.