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Register a new userOn the characterization of the space of derivations in evolution algebras
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We study the space of derivations for some finite-dimensional evolution algebras, depending on the twin partition of an associated directed graph. For evolution algebras with a twin-free associated graph we prove that the space of derivations is zero. For the remaining families of evolution algebras we obtain sufficient conditions under which the study of such a space can be simplified. We accomplish this task by identifying the null entries of the respective derivation matrix. Our results suggest how strongly the associated graphs structure impacts in the characterization of derivations for a given evolution algebra. Therefore our approach constitutes an alternative to the recent developments in the research of this subject. As an illustration of the applicability of our results we provide some examples and we exhibit the classification of the derivations for non-degenerate irreducible $3$-dimensional evolution algebras.
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We characterize derivations and 2-local derivations from $M_{n}(mathcal{A})$ into $M_{n}(mathcal{M})$, $n ge 2$, where $mathcal{A}$ is a unital algebra over $mathbb{C}$ and $mathcal{M}$ is a unital $mathcal{A}$-bimodule. We show that every derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2,$ is the sum of an inner derivation and a derivation induced by a derivation from $mathcal{A}$ to $mathcal{M}$. We say that $mathcal{A}$ commutes with $mathcal{M}$ if $am=ma$ for every $ainmathcal{A}$ and $minmathcal{M}$. If $mathcal{A}$ commutes with $mathcal{M}$ we prove that every inner 2-local derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2$, is an inner derivation. In addition, if $mathcal{A}$ is commutative and commutes with $mathcal{M}$, then every 2-local derivation $D: M_{n}(mathcal{A}) to M_{n}(mathcal{M})$, $n ge 2$, is a derivation.
In the paper, a method of describing the outer derivations of the group algebra of a finitely presentable group is given. The description of derivations is given in terms of characters of the groupoid of the adjoint action of the group.
We initiate a study on a range of new generalized derivations of finite-dimensional Lie algebras over an algebraically closed field of characteristic zero. This new generalization of derivations has an analogue in the theory of associative prime rings and unites many well-known generalized derivations that have already appeared extensively in the study of Lie algebras and other nonassociative algebras. After exploiting fundamental properties, we introduce and analyze their interiors, especially focusing on the rationality of the corresponding Hilbert series. Applying techniques in computational ideal theory we develop an approach to explicitly compute these new generalized derivations for the three-dimensional special linear Lie algebra over the complex field.
We show that in the class of solvable Lie algebras there exist algebras which admit local derivations which are not ordinary derivation and also algebras for which every local derivation is a derivation. We found necessary and sufficient conditions under which any local derivation of solvable Lie algebras with abelian nilradical and one-dimensional complementary space is a derivation. Moreover, we prove that every local derivation on a finite-dimensional solvable Lie algebra with model nilradical and maximal dimension of complementary space is a derivation.
The space of derivations of finite dimensional evolution algebras associated to graphs over a field with characteristic zero has been completely characterized in the literature. In this work we generalize that characterization by describing the derivations of this class of algebras for fields of any characteristic.
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