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Note this is a part of physics paper arXiv-9909150. And please assume that i am not familiar with advanced algebraic knowledge(Group theory, abstract algebra, etc..).

Excerpt from above paper,


The proposed U-duality group is then generated by

\begin{align} G(Z)=SL(2,Z) \bowtie SO(d,d;Z), \label{orig} \end{align}

where $\bowtie$ refers to the non-commuting action of both groups.


I want to understand $\bowtie$ symbol, explicitly. They said $\bowtie$ refers to the non-commuting action of both groups, but i am not pretty sure about its meaning.

First, is $G(Z)$ is kind of direct product of $SL(2,Z)$ and $SO(d,d;Z)$? I mean G(Z) satisfies the both properties of $SL(2,Z)$ and $SO(d,d;Z)$?

Second, non-commuting action means, $SL(2,Z) \bowtie SO(d,d;Z)$ and $SO(d,d;Z) \bowtie SL(2,Z)$ are different?

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  • $\begingroup$ @arctictern, thanks. Now i can understand above notation. !!! $\endgroup$ – phy_math Aug 4 '16 at 6:19
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I once wrote out a Math.SE answer about these in general here. This product goes by many names: Zappa-Szep product, the general product, the knit product, etc. They are the "next obvious thing" to define once you have defined direct products ($H,K\lhd G$, with $H\cap K=1$, $HK=G$) and semidirect products ($H\lhd G$, $K<G$, with $H\cap K=1$, $HK=G$). For Zappa-Szep we require that $H, K<G$, with $H\cap K=1$, $HK=G$. There are no conditions on the normality of the subgroups.

As an intriguing motivation, they can be studied for many, many algebraic structures: see this fascinating paper of Matt Brin for details!

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For more details see the paper Knit products of graded Lie algebras and groups by Peter W. Michor, see section $2$. The knit product of $A$ and $B$ is $A\bowtie B$, and it coincides with the Zappa-Sz'ep product.

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