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1.
Zhengxin Chen 《代数通讯》2013,41(2):738-769
Let L be a finite-dimensional complex simple Lie algebra, L ? be the ?-span of a Chevalley basis of L, and L R  = R ?? L ? be a Chevalley algebra of type L over a commutative ring R with identity. Let ?(R) be the solvable subalgebra of L R spanned by the basis elements of the maximal toral subalgebra and the root vectors associated with positive roots. In this article, we prove that under some conditions for R, any automorphism of ?(R) is uniquely decomposed as a product of a graph automorphism, a diagonal automorphism and an inner automorphism, and any derivation of ?(R) is uniquely decomposed as a sum of an inner derivation induced by root vectors and a diagonal derivation. Correspondingly, the automorphism group and the derivation algebra of ?(R) are determined.  相似文献   

2.
Zhengxin Chen  Bing Wang 《代数通讯》2013,41(5):2044-2061
Let L be a finite-dimensional complex simple Lie algebra, L ? be the ?-span of a Chevalley basis of L, and L R  = R ?? L ? be a Chevalley algebra of type L over a commutative ring R. Let 𝒩(R) be the nilpotent subalgebra of L R spanned by the root vectors associated with positive roots. A map ? of 𝒩(R) is called commuting if [?(x), x] = 0 for all x ∈ 𝒩(R). In this article, we prove that under some conditions for R, if Φ is not of type A 2, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a central derivation (resp., automorphism), and if Φ is of type A 2, then a derivation (resp., an automorphism) of 𝒩(R) is commuting if and only if it is a sum (resp., a product) of a graded diagonal derivation (resp., automorphism) and a central derivation (resp., automorphism).  相似文献   

3.
For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L 𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M d (L 𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ?, and consider its Lie subalgebra [S ?, S ?]. In our main result, we show that [S ?, S ?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L 𝕂(n)?, L 𝕂(n)?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.  相似文献   

4.
Mohammad Ashraf 《代数通讯》2017,45(10):4380-4395
Let ? be a commutative ring with identity and let 𝔄 = Tri(𝒜,?,?) be a triangular algebra consisting of unital algebras 𝒜,? over ? and an (𝒜,?)-bimodule ? which is faithful as a left 𝒜-module as well as a right ?-module. In this paper, we prove that under certain assumptions every nonlinear generalized Lie triple derivation GL:𝔄𝔄 is of the form GL = δ+τ, where δ:𝔄𝔄 is an additive generalized derivation on 𝔄 and τ is a mapping from 𝔄 into its center which annihilates all Lie triple products [[x,y],z].  相似文献   

5.
《代数通讯》2013,41(7):3271-3285
Abstract

Let k be a field with char k = p > 0 and G an abelian group with a bicharacter λ on G. For each p-(G,λ)-Lie color algebra L over k the p-universal enveloping algebra u(L) is a G-graded Hopf algebra,i.e.,a Hopf algebra in the category kG ? of kG-comodules. In this paper we describe a subcategory of kG ? which is equivalent to the category of the finite dimensional p-(G,λ)-Lie color algebras over k.  相似文献   

6.
Let L be a relatively free nilpotent Lie algebra over ? of rank n and class c, with n ≥ 2; freely generated by a set 𝒵. Give L the structure of a group, denoted by R, by means of the Baker–Campbell–Hausdorff formula. Let G be the subgroup of R generated by the set 𝒵 and N Aut(L)(G) the normalizer in Aut(L) of the set G. We prove that the automorphism group of L is generated by GL n (?) and N Aut(L)(G). Let H be a subgroup of finite index in Aut(G) generated by the tame automorphisms and a finite subset X of IA-automorphisms with cardinal s. We construct a set Y consisting of s + 1 IA-automorphisms of L such that Aut(L) is generated by GL n (?) and Y. We apply this particular method to construct generating sets for the automorphism groups of certain relatively free nilpotent Lie algebras.  相似文献   

7.
8.
Let L be a finite dimensional Lie algebra over a field F. It is well known that the solvable radical S(L) of the algebra L is a characteristic ideal of L if char F = 0, and there are counterexamples to this statement in case char F = p > 0. We prove that the sum S(L) of all solvable ideals of a Lie algebra L (not necessarily finite dimensional) is a characteristic ideal of L in the following cases: 1) char F = 0; 2) S(L) is solvable and its derived length is less than log2 p.  相似文献   

9.
ABSTRACT

Let n≥1 be a fixed integer, R a prime ring with its right Martindale quotient ring Q, C the extended centroid, and L a non-central Lie ideal of R. If F is a generalized skew derivation of R such that (F(x)F(y)?yx)n = 0 for all x,yL, then char(R) = 2 and R?M2(C), the ring of 2×2 matrices over C.  相似文献   

10.
Fei Li  Xianlong Bai 《代数通讯》2013,41(6):2109-2113
Let R ? ? be a GCD-domain. In this article, Weinberg's conjecture on the n × n matrix algebra M n (R) (n ≥ 2) is proved. Moreover, all the lattice orders (up to isomorphisms) on a full 2 × 2 matrix algebra over R are obtained.  相似文献   

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