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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! (English) |
