Embedding linear algebraic groups of a given dimension into a fixed $\mathrm{GL}_N$ Given $n$, can $n$-dimensional linear algebraic groups over $\mathbb{C}$ be embedded into $\mathrm{GL}(N,\mathbb{C})$ for a uniformly bounded $N$?
Thanks so much for your reply!
 A: As Dave points out, it's very risky to include finite (dimension 0) affine algebraic groups (in the traditional sense) in your question.   So I'm assuming $G$ is connected, in which case the answer to the question is YES.    
Even more than in the case of Cayley's theorem for embedding finite groups into symmetric groups, the old theorem of Chevalley (in arbitrary characteristic) embeds an arbitrary affine algebraic group into a general linear group without any control over the size of that linear group.    But if you appeal to the classication of semisimple groups along with the Levi decomposition in characteristic 0, you can control the size better.   To start with, there are only finitely many (connected) semisimple groups of a given dimension $n$, so that already bounds $N$ for them.   On the other hand, all $n$-dimensional algebraic tori are isomorphic to the direct product of $n$ copies of the multiplicative group, so they are also well-behaved with respect to embedding.    At least in characteristic 0, all $n$-dimensional unipotent groups are isomorphic to vector groups and have bounded embeddings.  Combined with the Levi decomposition over $\mathbb{C}$ (in the algebraic group setting), these individual results and the general Borel-Tits structure theory for reductive groups will provide $N$ indirectly.  
ADDED: My initial thinking about the question was not precise enough to handle unipotent radicals or semidirect product decompositions here.  So I would stick to reductive (connected) groups, where the structure and classification theorems easy to apply.   Aside from that it's worth asking why people don't work more often with affine algebraic groups in linear embeddings: probably it's because such embeddings (though possible by Chevalley's theorem) are too loosely structured and seldom of much help in applications.   Intrinsic properties such as Jordan-Chevalley decomposition and Bruhat decomposition are typically more useful in practice. 
