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I have seen at many places the notions that Lie Algebras are infinitesimal objects and they look really close at a point. But I never understood this. They are abstract algebraic objects different from rings in that they are equipped with a weird sort of product and a weird Jacobi identity. Any hints on how to make this connection to infinitesimals?

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Interesting question! I don't know the answer myself but Sophus Lie heavily uses infinitesimals in his paper on Lie groups (which I am currently reading). –  anon Aug 9 '10 at 20:08
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In fact Abraham Robinson showed how to rigorize the classical intuitive arguments via nonstandard analysis - see my recent post for references math.stackexchange.com/questions/1828 –  Bill Dubuque Aug 9 '10 at 20:20
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Are there classical intuitive arguments that were not easily expressible using standard approaches (e.g., calling an "infinitesimal" a sequence or function tending to zero, or calculating with nilpotents in Taylor series)? –  T.. Aug 9 '10 at 21:31
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3 Answers 3

The direct infinitesimal analogue of a given Lie group is not the Lie algebra with its bracket operation and Jacobi identity, but the Lie algebra (thought of as just the space of tangent vectors at the identity, without the added structure of a bracket operation) with addition of elements being the group operation. Addition is commutative but the group generally is not. To capture the noncommutativity you need to squeeze additional information from the group down to the infinitesimal level of the Lie algebra (by taking second-order information in the series expansion of group elements near the identity; the tangent space is first-order). The Lie multiplication $[x,y]$ is the second-order infinitesimal analogue of the commutator $x y x^{-1} y^{-1}$, and the Jacobi identity is an analogue of an identity for the commutator. Historically, the Jacobi identity for algebras (that is, for Lie algebras whose bracket is $XY - YX$ in an associative algebra) must have come first, and is used mainly in algebras, but you can think of it as coming from the group.

Locally, second-order information is enough: the Lie algebra determines the structure of the group, up to some questions of a different, "global" nature (topology) about connectivity and covering spaces.

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This is all carefully explained in Chapter 8 of Fulton and Harris. The key fact is that if $G$ is a connected Lie group, then $G$ is generated by the elements in any neighborhood of the identity. This implies that a morphism out of $G$ is determined by what it does to elements arbitrarily close to the identity (and it is a general principle in category theory that an object is determined by the morphisms out of it). Since we have a tangent space at the identity, we can say even more: it turns out that a morphism $f : G \to H$ is determined by its differential $df : T_e G \to T_e H$, where $T_e$ is the tangent space at the identity, otherwise known as the Lie algebra.

This differential is just a linear map between finite-dimensional vector spaces, so it's much easier to handle than the original map $f$. The problem is then to characterize which linear maps can occur. As a necessary condition, $f$ must preserve the Lie bracket on $T_e G$, and if $G$ is simply connected this is both necessary and sufficient. So essentially the whole point of the wacky definition of a Lie algebra is to make this theorem true.

This means, roughly speaking, that Lie algebras capture the local, or infinitesimal, structure of a Lie group. The Lie algebra of a Lie group can't capture the global topological structure, but being able to separate out the easy part and the hard part of understanding a Lie group is very valuable.

(The connection to Charles' answer is that a tangent vector at the identity determines, by translation, a left-invariant vector field on $G$. You can think of such a vector field as specifying, at each element of $G$, a direction in which something can flow.)

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Of course there is much more to this story, and if you want to understand it you really should pick up a book like Fulton and Harris. If the level is wrong for you, I also recommend John Stillwell's Naive Lie Theory. –  Qiaochu Yuan Aug 9 '10 at 21:54
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Lie algebras are infinitesimal in the sense that they naturally arise (over $\mathbb{R}$) as tangent space at the identity of a Lie group, which is also the same as the left-invariant vector fields (infinitesimal flows) on the Lie group.

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Why are vector fields the same as infinitesimal flows? –  Tenny Aug 9 '10 at 20:31
    
They tell you, at each point, what "direction" to move in, and often they can be integrated to actual flows (details available in most differential geometry books) –  Charles Siegel Aug 10 '10 at 1:17
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