Some Properties on Isoclinism of Lie Algebras and Covers

Abstract

In this paper, we give some equivalent conditions for Lie algebras to be isoclinic. In particular, it is shown that if two Lie algebras L and K are isoclinic then L can be constructed from K and vice versa using the operations of forming direct sums, taking subalgebras, and factoring Lie algebras. We also study connection between isoclinic and the Schur multiplier of Lie algebras. In addition, we deal with some properties of covers of Lie algebras whose Schur multipliers are finite dimensional and prove that all covers of any abelian Lie algebra have Hopfian property. Finally, we indicate that if a Lie algebra L belongs to some certain classes of Lie algebras then so does its cover.

Isoclinic extensions of Lie algebras

“…In the above theorem, if we restrict ourselves to isoclinic Lie algebras, the following consequence is obtained, which was already proved in [12] using another technique.…”

“…Note that some results on the isoclinism and the Schur multiplier of Lie algebras also hold for the group case (see [5,12,14]). However, the results in Section 3 are not generally true in the case of groups (see Example…”

“…The result yields that the concepts of isoclinism and isomorphism between Lie algebras of the same finite dimension are identical. The structure of all Lie algebras occurring in an isoclinism family has been extensively studied in [12] and also some applications have been given in [2,11,12,14]. Furthermore, Salemkar et.…”

The Polynilpotent Multiplier of Lie Algebras

“…, c n )-covers of a Lie algebra whose polynilpotent multiplier is finite-dimensional to be isomorphic. We omit the proof which is quite similar to the proof of [18,Theorem 3.3].…”

Isoclinism of crossed modules

“…This work was detailed in Hall and Senior (1964). After then, some new results were given in many papers, such as Jones and Wiegold (1974); Modabbernia (2012); Mohammadzadeh et al (2013); Parvaneh et al (2011); Salemkar et al (2008); Tappe (1976). Also the relation between groups and their stem extensions with Schur multiplicators were given in Beyl and Tappe (1982) where they consider the isoclinism of central group extensions.…”