Knit products of graded Lie algebras and groups

Proceedings of the Winter School "Geometry and Physics", GDML_Books, (1990), p. [171]-175 / Harvested from

Résumé

Let $A = ⨁_{k} A_{k}$ and $B = ⨁_{k} B_{k}$ be graded Lie algebras whose grading is in $𝒵$ or $𝒵_{2}$ , but only one of them. Suppose that $(α, β)$ is a derivatively knitted pair of representations for $(A, B)$ , i.e. $α$ and $β$ satisfy equations which look “derivatively knitted"; then $A \oplus B : = ⨁_{k, l} (A_{k} \oplus B_{l})$ , endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra $A \oplus_{(α, β)} B$ . This graded Lie algebra is called the knit product of $A$ and $B$ . The author investigates the general situation for any graded Lie subalgebras $A$ and $B$ of a graded Lie algebra $C$ such that $A + B = C$ and $A \cap B = 0$ . He proves that $C$ as a graded Lie algebra is isomorphic to a knit product of $A$ and $B$ . Also he investigates the behaviour of homomorphisms with respect to knit products. The integrated version of a knit product of Lie algebras is called a knit product of group!

MR MR1061798 | Zbl 0954.17508

EUDML-ID : urn:eudml:doc:221177
Mots clés:

@article{701470,
     title = {Knit products of graded Lie algebras and groups},
     booktitle = {Proceedings of the Winter School "Geometry and Physics"},
     series = {GDML\_Books},
     publisher = {Circolo Matematico di Palermo},
     address = {Palermo},
     year = {1990},
     pages = {[171]-175},
     mrnumber = {MR1061798},
     zbl = {0954.17508},
     url = {http://dml.mathdoc.fr/item/701470}
}

Michor, Peter W. Knit products of graded Lie algebras and groups, dans Proceedings of the Winter School "Geometry and Physics", GDML_Books,  (1990), pp. [171]-175. http://gdmltest.u-ga.fr/item/701470/