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Register a new userLie algebras and around: selected questions
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Authors
Pasha Zusmanovich
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Several open questions are discussed. The topics include cohomology of current and related Lie algebras, algebras represented as the sum of subalgebras, structures and phenomena peculiar to characteristic $2$, and variations on themes of Ado, Whitehead, and Banach.
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After recalling the notion of Lie algebroid, we construct these structures associated with contact forms or systems. We are then interested in particular classes of Lie Rinehart algebras.
In this paper, we introduce the notion Lie-derivation. This concept generalizes derivations for non-Lie Leibniz algebras. We study these Lie-derivations in the case where their image is contained in the Lie-center, call them Lie-central derivations. We provide a characterization of Lie-stem Leibniz algebras by their Lie-central derivations, and prove several properties of the Lie algebra of Lie-central derivations for Lie-nilpotent Leibniz algebras of class 2. We also introduce ${sf ID}_*-Lie$-derivations. A ${sf ID}_*-Lie$-derivation of a Leibniz algebra G is a Lie-derivation of G in which the image is contained in the second term of the lower Lie-central series of G, and that vanishes on Lie-central elements. We provide an upperbound for the dimension of the Lie algebra $ID_*^{Lie}(G)$ of $ID_*Lie$-derivation of G, and prove that the sets $ID_*^{Lie}(G)$ and $ID_*^{Lie}(G)$ are isomorphic for any two Lie-isoclinic Leibniz algebras G and Q.
We prove that free pre-Lie algebras, when considered as Lie algebras, are free. Working in the category of S-modules, we define a natural filtration on the space of generators. We also relate the symmetric group action on generators with the structure of the anticyclic PreLie operad.
The notion of rigidity of Lie algebra is linked to the following problem: when does a Lie brackets $mu$ on a vector space g satisfy that every Lie bracket $mu_1$ sufficiently close to $mu$ is of the form $mu_1 = P.mu $ for some P in GL(g) close to the identity? A Lie algebra which satisfies the above condition will be called rigid. The most famous example is the Lie algebra sl(2,C) of square matrices of order $2$ with vanishing trace. This Lie algebra is rigid, that is any close deformation is isomorphic to it. Let us note that, for this Lie algebra, there exists a quantification of its universal algebra. This led to the definition of the famous quantum group SL(2). Another interest of studying the rigid Lie algebras is the fact that there exists, for a given dimension, only a finite number of isomorphic classes of rigid Lie algebras. So we are tempted to establish a classification. This problem has been solved up to the dimension 8. To continue in this direction, properties must be established on the structure of these algebras. One of the first results establishes an algebraicity criterion cite{Carles}. However, the notion of algebraicity which is used is not the classical notion and it includes non-algebraic Lie algebras in the usual sense. The aim of this work is to show that a the Lie algebra is rigid, then its algebra of inner derivations is algebraic.
In this paper, we introduce the notion of a derivation of a Hom-Lie algebra and construct the corresponding strict Hom-Lie 2-algebra, which is called the derivation Hom-Lie 2-algebra. As applications, we study non-abelian extensions of Hom-Lie algebras. We show that iso- morphism classes of diagonal non-abelian extensions of a Hom-Lie algebra g by a Hom-Lie algebra h are in one-to-one correspondence with homotopy classes of morphisms from g to the derivation Hom-Lie 2-algebra DER(h).
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