Let g be a complex simple Lie algebra with a Borel subalgebra b. Consider the semidirect product Ib=b⋉b⁎, where the dual b⁎ of b is equipped with the coadjoint action of b and is considered as an ...abelian ideal of Ib. We describe the automorphism group Aut(Ib) of the Lie algebra Ib. In particular we prove that it contains the automorphism group of the extended Dynkin diagram of g. In type An, the dihedral subgroup was recently proved to be contained in Aut(Ib) by Dror Bar-Natan and Roland van der Veen in 1 (where Ib is denoted by Iun). Their construction is ad hoc and they asked for an explanation which is provided by this note. Let n denote the nilpotent radical of b. We obtain similar results for Ib‾=b⋉n⁎ that is both an Inönü-Wigner contraction of g and the quotient of Ib by its center.
We introduce the notion of anti-pre-Lie algebras as the underlying algebraic structures of nondegenerate commutative 2-cocycles which are the “symmetric” version of symplectic forms on Lie algebras. ...They can be characterized as a class of Lie-admissible algebras whose negative left multiplication operators make representations of the commutator Lie algebras. We observe that there is a clear analogy between anti-pre-Lie algebras and pre-Lie algebras by comparing them in terms of several aspects. Furthermore, it is unexpected that a subclass of anti-pre-Lie algebras, namely admissible Novikov algebras, correspond to Novikov algebras in terms of q-algebras. Consequently, there is a construction of admissible Novikov algebras from commutative associative algebras with derivations or more generally, admissible pairs. The correspondence extends to the level of Poisson type structures, leading to the introduction of the notions of anti-pre-Lie Poisson algebras and admissible Novikov-Poisson algebras, whereas the latter correspond to Novikov-Poisson algebras.
We describe transposed Poisson algebra structures on Block Lie algebras B(q) and Block Lie superalgebras S(q), where q is an arbitrary complex number. Specifically, we show that the transposed ...Poisson structures on B(q) are trivial whenever q∉Z, and for each q∈Z there is only one (up to an isomorphism) non-trivial transposed Poisson structure on B(q). The superalgebra S(q) admits only trivial transposed Poisson superalgebra structures for q≠0 and two non-isomorphic non-trivial transposed Poisson superalgebra structures for q=0. As a consequence, new Lie algebras and superalgebras that admit non-trivial Hom-Lie algebra structures are found.
We study sympathetic (i.e., perfect and complete) Lie algebras. Among other topics they arise in the study of adjoint Lie algebra cohomology. Here a motivation comes from a conjecture of Pirashvili, ...which says that a finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several general results for sympathetic Lie algebras and for the adjoint Lie algebra cohomology of arbitrary finite-dimensional Lie algebras in characteristic zero using a result of Hochschild and Serre. Moreover, for certain semidirect products we obtain explicit results for the adjoint cohomology.
We prove that every local derivation on a finite-dimensional semisimple Lie algebra over an algebraically closed field of characteristic zero is a derivation. We also give examples of ...finite-dimensional nilpotent Lie algebras L with dimL≥3 which admit local derivations which are not derivations.
Let
be a finite-dimensional complex Lie algebra and
be the affine variety of all multiplicative Hom-Lie algebras on
. We use a method of computational ideal theory to describe
, showing that
consists ...of two 1-dimensional and one 3-dimensional irreducible components and
for
. We construct a new family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra
and characterize the affine varieties
and
. We also study the derivation algebra
of a multiplicative Hom-Lie algebra D on
and, under some hypotheses on D, we prove that the Hilbert series
is a rational function.
In this research, we studied the affine Lie algebra aff(2,R). The aim of this research is to determine the 1-form in affine Lie algebra aff(2,R) which is associated with its symplectic structure so ...that affine Lie algebra aff(2,R) is a Frobenius Lie algebra. Realized the elements of the affine Lie algebra aff(2,R) in matrix form, then calculated the Lie brackets and formed the structure matrix of the affine Lie algebra aff(2,R). 1-form of the affine Lie algebra aff(2,R) is obtained from the determinant of the structure matrix of the affine Lie algebra aff(2,R). Furthermore, proved that the 2-form is symplectic and related to the 1-form. The result obtained is that the affine Lie algebra aff(2,R) has 1-form α=ε_12^*+ε_23^* on aff(2,R)^* which is related to its symplectic structure, β=ε_11^*∧ε_12^*+ε_12^*∧ε_22^*+ε_21^*∧ε_13^*+ε_22^*∧ε_23^* such that the affine Lie algebra aff(2,R) is a Frobenius Lie algebra. For further research, it can be developed into an affine Lie algebra with dimensions n(n+1).
In this research, we studied quasi-Frobenius Lie algebras and filiform Lie algebras of dimensions ≤ 5 over real field. The primary objective of this research is to classify the classification of ...filiform Lie algebras of dimensions ≤ 5 into quasi-Frobenius Lie algebras. The method employed in this research involves constructing a skew-symmetric 2-form in real Lie algebra, which also a nondegenerate 2-cocycle. The outcomes of this research reveal that there exists a class of filiform Lie algebras of dimensions $\le 5$ that can be classified as a quasi-Frobenius real Lie algebra. Furthermore, this research can be developed to classify higher dimensional filiform Lie algebras as quasi-Frobenius real Lie algebras.
В статье доказывается аналог теоремы Ф. Кубо 1 для почти локально разрешимых алгебр Ли с нулевым радикалом Джекобсона. Первый раздел направлен на выяснение некоторых аспектов гомологического описания ...радикала Джекобсона. Доказана теорема, обобщающая теорему Е. Маршалла на случай почти локально разрешимых алгебр Ли, следствием которой и является аналог теоремы Кубо. Во втором разделе исследуются некоторые свойства локально нильпотентного радикала алгебры Ли. Рассматриваются примитивные алгебры Ли. Приведены примеры, показывающие, что бесконечномерные коммутативные алгебры Ли являются примитивными над любыми полями; конечномерная абелева алгебра, размерности больше 1, над алгебраически замкнутым полем не является примитивной; пример неартиновой некоммутативной алгебры Ли являющейся примитивной. Показано, что для специальных алгебр Ли над полем характеристики нуль
PI
-неприводимо представленный радикал совпадает с локально нильпотентным. Приведен пример алгебры Ли, локально нильпотентный радикал которой не является ни локально нильпотентным, ни локально разрешимым. Даются достаточные условия примитивности алгебры Ли, приводятся примеры примитивных алгебр Ли и алгебры Ли не являющейся примитивной.
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