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Let n ≥ 3. The complex Lie algebra, which is attached to a unit form q(x 1, x 2,..., x n) = $${\sum\nolimits_{i = 1}^n {x_i^2 + \sum\nolimits_{1 \leqslant i \leqslant j \leqslant n} {\left( { - 1} \right)} } ^{j - i}}{x_i}{x_j}$$ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type A n , and realized by the Ringel-Hall Lie algebra of a Nakayama algebra of radical square zero. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.  相似文献
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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.  相似文献
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Let n ≥ 4. The complex Lie algebra, which is attached to the unit form q(x1, x2,..., xn)■ and defined by generators and generalized Serre relations, is proved to be a finite-dimensional simple Lie algebra of type Dn, and realized by the Ringel-Hall Lie algebra of a Nakayama algebra. As its application of the realization, we give the roots and a Chevalley basis of the simple Lie algebra.  相似文献
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Let A be a domestic canonical algebra over a finite field. In this article, we prove that the composition algebra C(A) of A has a triangular decomposition 𝒫·𝒯· corresponding to the division of the indecomposable modules into the preprojectives, the regulars, and the preinjectives.  相似文献
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Let 𝔤 be a finite-dimensional simple Lie algebra over an algebraically closed field of characteristic zero. It is proved in this article that a bijective map ϕ on 𝔤 preserves Lie products if and only if it is a composition of a Lie algebra automorphism and a bijective map extended by an automorphism of the base field.  相似文献
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Zhengxin Chen 《代数通讯》2013,41(2):738-769
Let L be a finite-dimensional complex simple Lie algebra, L be the ℤ-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 ℬ(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 ℬ(R) is uniquely decomposed as a product of a graph automorphism, a diagonal automorphism and an inner automorphism, and any derivation of ℬ(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 ℬ(R) are determined.  相似文献