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What does it mean when we say a topological group $\Gamma$ has linear type?

Is it an algebraic property or a topology property?

I wonder if anyone could give some references.

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    You should probably tell us what reference you're looking at; different authors may mean different things. I would _guess_ it means that the group admits a continuous injective homomorphism into $\text{GL}_n(\mathbb{C})$ for some $n$, but who knows?2012-05-22

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Another answer: I found in Annals of Mathematics, vol. 40, no. 3, July, 1939 that:

An "Abelian, convex, connected, and sequentially complete Hausdorff group such that it possesses no elements of finite order may be called a linear topological group.

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    Thanks, that's a definition I could understand.2012-05-30
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This might not be the right answer, but maybe it is helpful.

An algebraic group is an algebraic variety $G$ together two maps $\begin{align} &\mu: G \times G \to G & &(x,y) \mapsto xy\\ &i: G \to G & &x \mapsto x^{-1} \end{align} $ which are both morphisms of varieties.

Now if the varieties are affine, then we say that $G$ is a linear algebraic group.

Now this might not be what you are studying. Even thougth there is an underlying topology (the Zariski topology) on $G$, it doesn't mean that the group is a topological group.