Knit products of graded Lie algebras and groups

Michor, Peter W.. "Knit products of graded Lie algebras and groups." Proceedings of the Winter School "Geometry and Physics". Palermo: Circolo Matematico di Palermo, 1990. [171]-175. <http://eudml.org/doc/221177>.

@inProceedings{Michor1990,
abstract = {Let $A=\bigoplus _kA_k$ and $B=\bigoplus _kB_k$ be graded Lie algebras whose grading is in $\mathcal \{Z\}$ or $\mathcal \{Z\}_2$, but only one of them. Suppose that $(\alpha ,\beta )$ is a derivatively knitted pair of representations for $(A,B)$, i.e. $\alpha $ and $\beta $ satisfy equations which look “derivatively knitted"; then $A\oplus B:=\bigoplus _\{k,l\}(A_k\oplus B_l)$, endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra $A\oplus _\{(\alpha ,\beta )\}B$. 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!},
author = {Michor, Peter W.},
booktitle = {Proceedings of the Winter School "Geometry and Physics"},
keywords = {Geometry; Physics; Proceedings; Winter school; Srní (Czechoslovakia)},
location = {Palermo},
pages = {[171]-175},
publisher = {Circolo Matematico di Palermo},
title = {Knit products of graded Lie algebras and groups},
url = {http://eudml.org/doc/221177},
year = {1990},
}

TY - CLSWK
AU - Michor, Peter W.
TI - Knit products of graded Lie algebras and groups
T2 - Proceedings of the Winter School "Geometry and Physics"
PY - 1990
CY - Palermo
PB - Circolo Matematico di Palermo
SP - [171]
EP - 175
AB - Let $A=\bigoplus _kA_k$ and $B=\bigoplus _kB_k$ be graded Lie algebras whose grading is in $\mathcal {Z}$ or $\mathcal {Z}_2$, but only one of them. Suppose that $(\alpha ,\beta )$ is a derivatively knitted pair of representations for $(A,B)$, i.e. $\alpha $ and $\beta $ satisfy equations which look “derivatively knitted"; then $A\oplus B:=\bigoplus _{k,l}(A_k\oplus B_l)$, endowed with a suitable bracket, which mimics semidirect products on both sides, becomes a graded Lie algebra $A\oplus _{(\alpha ,\beta )}B$. 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!
KW - Geometry; Physics; Proceedings; Winter school; Srní (Czechoslovakia)
UR - http://eudml.org/doc/221177
ER -