This paper deals with the computation of abelian subalgebras of the solvable Lie algebra hn, of n × n upper-triangular matrices. Firstly, we construct an algorithm to find abelian Lie subalgebras in a given Lie algebra hn. This algorithm allows us to compute an abelian subalgebra up to a certain dimension. Such a dimension is proved to be equal to the maximum for abelian subalgebras of hn.

If a Lie algebra g can be generated by M of its elements E1, . . . , EM , and if any other Lie algebra generated by M other elements F1, . . . , FM is a homomorphic image of g under the map Ei → Fi, we say that it is the free Lie algebra on M generators. The free nilpotent Lie algebra gM,r on M generators of rank r is the quotient of the free Lie algebra by the ideal gr+1 generated as follows: ...

The category of group-graded modules over an abelian group G is a monoidal category. For any bicharacter of G this category becomes a braided monoidal category. We define the notion of a Lie algebra in this category generalizing the concepts of Lie super and Lie color algebras. Our Lie algebras have n-ary multiplications between various graded components. They possess universal enveloping algeb...

We give a example of non nilpotent faithful representation of a filiform Lie algebra. This gives one counter-example of the conjecture saying that every affine connection on a filiform Lie group is complete. 1. Affine connection on a nilpotent Lie algebra 1.1. Affine connection on nilpotent Lie algebras. Definition 1. Let g be a n-dimensional Lie algebra over R. It is called affine if there is ...

A superconformal algebra is a simple complex Lie superalgebra g spanned by the coefficients of a finite family of pairwise local fields a(z) = ∑ n∈Z a(n)z , one of which is the Virasoro field L(z), [3, 8, 11]. Superconformal algebras play an important role in the string theory and conformal field theory. The Lie superalgebras K(N) of contact vector fields with Laurent polynomials as coefficient...

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Theorem 1. Let G ⊂ GL(n,R) be a real reductive group with Cartan decomposition g = k⊕ p, g being the Lie algebra of G. Let GC be the subgroup of GL(n,C) with Lie algebra g ⊕ ig and K̃ the subgroup of GC whose Lie algebra is k⊕ ip. Let ΩC be a complex homogeneous space for GC and φ a K̃-invariant strictly plurisubharmonic function on ΩC. If Ω is a G-orbit in ΩC and f = φ|Ω has a critical point, th...

‎In this paper‎, ‎we introduce the structure of a groupoid associated to a vector field‎ ‎on a smooth manifold‎. ‎We show that in the case of the $1$-dimensional manifolds‎, ‎our‎ ‎groupoid has a‎ ‎smooth structure such that makes it into a Lie groupoid‎. ‎Using this approach‎, ‎we associated to‎ ‎every vector field an equivalence‎ ‎relation on the Lie algebra of all vector fields on the smooth...