Malcev-like binary Lie algebras

Abstract

Abstract. The notion of a loop is generalization of the definition of a group such that all group axioms hold without the associative law. The first attempt to deal with analytic loops was to follow the ideas of Sophus Lie, to associate with an analytic loop an algebraic object, its tangent space at the identity, and to endow it with a reasonable algebraic structure derived from the manifold data and the loop operation. Once an algebraic object is associated with an analytic loop, then the algebraist is challenged to classify these algebras. This gives a first step towards a classification of local analytic loops. The next step is the passage from the local to the global theory: whether any local analytic loop can be embedded into a global one. This reseach was done successfully to differentiable Moufang loops by Kuzmin, Kerdman and Nagy (cf. [4], [6], [9]). By their results the theory of differentiable Moufang loops and their tangent Malcev algebras has developed significantly almost to the level of the theory of Lie groups and algebras. Since the Campbell-Hausdorff formula works also for binary Lie algebras (any two elements generate a Lie subalgebra), the theory of diassociative local analytic loops (any two elements generate a subgroup) can be treated fruifully using binary Lie algebras. The correspondence between local analytic Moufang and diassociative loops and their tangent algebras was the main motivation of Malcev for introducing the concepts of Malcev algebras (called them Moufang-Lie algebras) and binary Lie algebras [8]. Malcev algebras and the corresponding Moufang loops of dimension at most 5 were determined by Kuzmin in [5]. The minimum of the dimension of non-Lie binary Lie algebras is 4 and these algebras were classified by Gainov and Kuzmin in [3], [7]. One of the most natural generalizations of binary Lie and Malcev algebras is the anti-commutative algebra defined by an anti-symmetric bilinear operation on a vector space over a field. To the 4-dimensional binary Lie algebras one can associate a family of flags of subalgebras defined by algebraic properties. Figula and Nagy classified in [1] the 4-dimensional anti-commutative algebras having an analogous family of flags of subalgebras as the 4-dimensional non-Lie binary Lie algebras. The solvable 5-dimensional Malcev algebras have very similar decomposition properties as the 4-dimensional binary Lie algebras, they are extensions of a 1-dimensional algebra by a nilpotent Lie algebra and simultaneously extensions of a two-dimensional non-abelian Lie algebra by an abelian algebra. We call the 5-dimensional anti-commutative algebras that have the same ideal structures as solvable Malcev algebras Malcev-like algebras. These algebras can be regarded as close relatives of solvable Malcev algebras. The binary Lie algebras in the class of Malcevlike algebras, their normal forms and isomorphism classes were found in [2]. In the talk I would like to present the method what we used for the determination of the normal forms and isomorphism classes of Malcev-like binary Lie algebras and the results of our classification. These are joint results with Prof. Péter T. Nagy.