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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? –  Qiaochu Yuan May 22 '12 at 20:22

2 Answers 2

up vote 1 down vote accepted

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. –  Qijun Tan May 30 '12 at 5:38

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.

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