Isotopisms of nilpotent Leibniz algebras
(2022) JCTS: Junior Category Theory Seminar — Location: Louvain-la-Neuve, Belgium (2.December.2022)
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- Abstract
- (en) Leibniz algebras were first introduced by J.-L. Loday in 1993 as a non-antisymmetric generalization of Lie algebras. Since then, many fundamental results from the theory of Lie algebras have been extended to the Leibniz setting. One of these is the Levi decomposition, which states that every finite-dimensional Leibniz algebra is the semidirect product of a solvable ideal and a semisimple subalgebra. This decomposition provides a first step toward the classification of Leibniz algebras. In general, deciding whether two Leibniz algebras are isomorphic is a difficult problem. A weaker and often more tractable notion is that of isotopism, introduced by A. A. Albert in 1942 in the study of non-associative algebras. An isotopism between two Leibniz algebras is a triple of linear isomorphisms (f, g, h) such that [f(x), g(y)] = h([x, y]) for all elements x and y. In this talk, we review the main results on the classification of two-step nilpotent Leibniz algebras. In particular, we recall that there are exactly three isomorphism classes of indecomposable nilpotent Leibniz algebras with one-dimensional commutator ideal. Just as Lie algebras arise as the infinitesimal counterparts of Lie groups, Leibniz algebras can be integrated into algebraic structures called Lie racks. We show that every real nilpotent Leibniz algebra admits a global integration into a Lie rack. Finally, although the classification of two-step nilpotent Leibniz algebras up to isomorphism remains open, we prove that there are only three isotopism classes of indecomposable nilpotent Leibniz algebras with one-dimensional commutator ideal. Moreover, we show how an isotopism between two such Leibniz algebras induces an isotopism between the corresponding global Lie racks. We conclude by discussing some open problems. This is joint work with Gianmarco La Rosa (University of Palermo) and Gábor P. Nagy (Budapest University of Technology).
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Mancini, M. (2022, December 2). Isotopisms of nilpotent Leibniz algebras. JCTS: Junior Category Theory Seminar, Louvain-la-Neuve, Belgium. https://hdl.handle.net/2078.5/279081