Knit Products of Graded Lie Algebras and Groups

If a graded Lie algebra is the direct sum of two graded sub Lie algebras, its bracket can be written in a form that mimics a ”double sided semidirect product”. It is called the knit product of the two subalgebras then. The integrated version of this is called a knit product of groups — it coincides with the ZappaSzép product. The behavior of homomorphisms with respect to knit products is investigated. Introduction If a Lie algebra is the direct sum of two sub Lie algebras one can write the bracket in a way that mimics semidirect products on both sides. The two representations do not take values in the respective spaces of derivations; they satisfy equations (see 1.1) which look ”derivatively knitted” — so we call them a derivatively knitted pair of representations. These equations are familiar for the Frölicher-Nijenhuis bracket of differential geometry, see [1] or [2, 1.10]. This paper is the outcome of my investigation of what formulas 1.1 mean algebraically. It was a surprise for me that they describe the general situation (Theorem 1.3). Also the behavior of homomorphisms with respect to knit products is investigated (Theorem 1.4). The integrated version of a knit product of Lie algebras will be called a knit product of groups — but it is well known to algebraists under the name ZappaSzép product, see [3] and the references therein. I present it here with different 1991 Mathematics Subject Classification. 17B65, 17B80, 20.